Math 241 Homework 5 Solutionskmanguba/math241-2/HW5.pdf · 2017-12-10 · Problem 37. Verify that the given point is on the curve and nd the lines that are (a) tangent and (b) normal

Math 241 Homework 5 Solutions

Section 3.6

Problem 1. Use implicit differentiation to find dy/dx: x2y + xy2 = 6

Solution

x2y + xy2 = 6⇒ 2xy + x2dy

dx+ y2 + x(2y

dy

dx) = 0

⇒ x2dy

dx+ 2xy

dy

dx= −2xy − y2

⇒dy

dx(x2 + 2xy) = −2xy − y2

⇒dy

dx=−2xy − y2

x2 + 2xy

Problem 3. Use implicit differentiation to find dy/dx: 2xy + y2 = x + y

Solution

2xy + y2 = x + y⇒ 2y + 2xdy

dx+ 2y

dy

dx= 1 +

dy

dx

⇒ 2xdy

dx+ 2y

dy

dx−dy

dx= 1 − 2y

⇒dy

dx(2x + 2y − 1) = 1 − 2y

⇒dy

dx=

1 − 2y

2x + 2y − 1

1

Problem 7. Use implicit differentiation to find dy/dx: y2 =x − 1

x + 1

Solution

y2 =x − 1

x + 1⇒ 2y

dy

dx=(x + 1)(1) − (x − 1)(1)

(x + 1)2

⇒dy

dx=�x + 1 −�x + 1

(x + 1)2⋅

1

2y

⇒dy

dx=

�2

�2y(x + 1)2

⇒dy

dx=

1

y(x + 1)2

Problem 9. Use implicit differentiation to find dy/dx: x = tan y

Solution

x = tan y⇒ 1 = sec2 ydy

dx

⇒dy

dx=

1

sec2 y

2

Problem 13. Use implicit differentiation to find dy/dx: y sin(1

y) = 1 − xy

Solution

y sin(1

y) = 1 − xy⇒

dy

dxsin(

1

y) + y cos(

1

y)d

dx(

1

y) = −y − x

dy

dx

⇒dy

dxsin(

1

y) + y cos(

1

y)(−

1

y2)dy

dx= −y − x

dy

dx

⇒dy

dxsin(

1

y) − cos(

1

y)

1

y

dy

dx= −y − x

dy

dx

⇒dy

dxsin(

1

y) − cos(

1

y)

1

y

dy

dx+ x

dy

dx= −y

⇒dy

dx

⎛

⎝sin(

1

y) −

1

ycos(

1

y) + x

⎞

⎠= −y

⇒dy

dx= −

y

sin(1/y) − (1/y) cos(1/y) + x

⇒dy

dx= −

y2

y sin(1/y) − cos(1/y) + xy

Problem 15. Find dr/dθ: θ1/2 + r1/2 = 1

Solution

θ1/2 + r1/2 = 1⇒1

2θ−1/2 +

1

2r−1/2

dr

dθ= 0

⇒1

2r−1/2

dr

dθ= −

1

2θ−1/2

⇒1

2√r

dr

dθ= −

1

2√θ

⇒dr

dθ= −

�2√r

�2√θ

⇒dr

dθ= −

√r√θ

3

Problem 25. If x3 + y3 = 16, find the value of d2y/dx2 at the point (2,2).

Solution

x3 + y3 = 16⇒ 3x2 + 3y2dy

dx= 0

⇒ 3y2dy

dx= −3x2

⇒dy

dx=−�3x

2

�3y2

⇒dy

dx= −

x2

y2

⇒d2y

dx2= −

y2(2x) − x2(2y(dy/dx))

y4

⇒d2y

dx2= −

2xy2 − 2x2y(−x2/y2)

y4

⇒d2y

dx2= −

2xy4 + 2x4y

y6

Plugging in x = 2 and y = 2 we havedy

dx= −

4

4= −1

andd2y

dx2= −

2(2)(24) + 2(2)4(2)

26=

26 + 26

26= 2

4

Problem 37. Verify that the given point is on the curve and find the lines that are (a) tangentand (b) normal to the curve at the given point: y = 2 sin(πx − y), (1,0)

Solution

Verify:

Plugging in the point we have 0 = 2 sin(π) which is true. Thus the point is on the curve.

y = 2 sin(πx − y)⇒dy

dx= 2 cos(πx − y)(π −

dy

dx)

Plugging in x = 1 and y = 0 we have

dy

dx= 2 cos(π)(π −

dy

dx)⇔

dy

dx= −2π + 2

dy

dx

⇔dy

dx= 2π

(a)

y = 2π(x − 1) = 2πx − 2π

(b)

y = −1

2π(x − 1) = −

1

2πx +

1

2π

5

Problem 41. Find the slopes of the curve y4 = y2 − x2 at the two points shown here.

In Exercises 29 and 30, find the slope of the curve at the given points.

29. y2 + x2 = y4 - 2x at (-2, 1) and (-2, -1)

30. (x2 + y2)2 = (x - y)2 at (1, 0) and (1, -1)

Slopes, Tangents, and NormalsIn Exercises 31–40, verify that the given point is on the curve and find the lines that are (a) tangent and (b) normal to the curve at the given point.

31. x2 + xy - y2 = 1, (2, 3)

32. x2 + y2 = 25, (3, -4)

33. x2y2 = 9, (-1, 3)

34. y2 - 2x - 4y - 1 = 0, (-2, 1)

35. 6x2 + 3xy + 2y2 + 17y - 6 = 0, (-1, 0)

36. x2 - 23xy + 2y2 = 5, 123, 22 37. 2xy + p sin y = 2p, (1, p>2)

38. x sin 2y = y cos 2x, (p>4, p>2)

39. y = 2 sin (px - y), (1, 0)

40. x2 cos2 y - sin y = 0, (0, p)

41. Parallel tangents Find the two points where the curve x2 + xy + y2 = 7 crosses the x-axis, and show that the tangents to the curve at these points are parallel. What is the common slope of these tangents?

42. Normals parallel to a line Find the normals to the curve xy + 2x - y = 0 that are parallel to the line 2x + y = 0.

43. The eight curve Find the slopes of the curve y4 = y2 - x2 at the two points shown here.

x

y

0

1

−1

y4 = y2 − x2

"

34

"

32

,

"

34

12

,a b

a b

44. The cissoid of Diocles (from about 200 b.c.) Find equations for the tangent and normal to the cissoid of Diocles y2(2 - x) = x3 at (1, 1).

x

y

1

1

(1, 1)

0

y2(2 − x) = x3

45. The devil’s curve (Gabriel Cramer, 1750) Find the slopes of the devil’s curve y4 - 4y2 = x4 - 9x2 at the four indicated points.

x

y

3−3

2

−2

(3, 2)

(3, −2)

(−3, 2)

(−3, −2)

y4 − 4y2 = x4 − 9x2

46. The folium of Descartes (See Figure 3.28.)

a. Find the slope of the folium of Descartes x3 + y3 - 9xy = 0 at the points (4, 2) and (2, 4).

b. At what point other than the origin does the folium have a horizontal tangent?

c. Find the coordinates of the point A in Figure 3.28 where the folium has a vertical tangent.

Theory and Examples 47. Intersecting normal The line that is normal to the curve

x2 + 2xy - 3y2 = 0 at (1, 1) intersects the curve at what other point?

48. Power rule for rational exponents Let p and q be integers with q 7 0. If y = x p>q, differentiate the equivalent equation yq = xp implicitly and show that, for y ≠ 0,

ddx

x p>q =pq x(p>q) - 1.

49. Normals to a parabola Show that if it is possible to draw three normals from the point (a, 0) to the parabola x = y2 shown in the accompanying diagram, then a must be greater than 1>2. One of the normals is the x-axis. For what value of a are the other two normals perpendicular?

x

y

0 (a, 0)

x = y2

3.7 Implicit Differentiation 165

Solution

y4 = y2 − x2 ⇒ 4y3dy

dx= 2y

dy

dx− 2x

⇔dy

dx(4y3 − 2y) = −2x

⇔dy

dx= −

2x

4y3 − 2y

At⎛

⎝

√3

4,

√3

2

⎞

⎠:

dy

dx= −

2(√

3/4)

4(√

3/2)3 − 2(√

3/2)

= −

√3

3√

3 − 2√

3

= −1

At⎛

⎝

√3

4,

√3

2

⎞

⎠:

dy

dx= −

2(√

3/4)

4(1/2)3 − 2(1/2)

= −

√3/2

1/2 − 1

=√

3

6

Problem 42. Find equations for the tangent and normal to the cissoid of Diocles y2(2 − x) = x3

at (1,1)


29. y2 + x2 = y4 - 2x at (-2, 1) and (-2, -1)

30. (x2 + y2)2 = (x - y)2 at (1, 0) and (1, -1)


31. x2 + xy - y2 = 1, (2, 3)

32. x2 + y2 = 25, (3, -4)

33. x2y2 = 9, (-1, 3)

34. y2 - 2x - 4y - 1 = 0, (-2, 1)

35. 6x2 + 3xy + 2y2 + 17y - 6 = 0, (-1, 0)

36. x2 - 23xy + 2y2 = 5, 123, 22 37. 2xy + p sin y = 2p, (1, p>2)

38. x sin 2y = y cos 2x, (p>4, p>2)

39. y = 2 sin (px - y), (1, 0)

40. x2 cos2 y - sin y = 0, (0, p)




x

y

0

1

−1

y4 = y2 − x2

"

34

"

32

,

"

34

12

,a b

a b


x

y

1

1

(1, 1)

0

y2(2 − x) = x3


x

y

3−3

2

−2

(3, 2)

(3, −2)

(−3, 2)

(−3, −2)

y4 − 4y2 = x4 − 9x2








ddx

x p>q =pq x(p>q) - 1.


x

y

0 (a, 0)

x = y2


Solution

y2(2 − x) = x3 ⇒ 2ydy

dx(2 − x) + y2(−1) = 3x2

Plugging in x = 1 and y = 1 we have

2dy

dx− 1 = 3⇔

dy

dx= 2

Thus the tangent line isy − 1 = 2(x − 1)⇔ y = 2x − 1

And the normal line is

y − 1 = −1

2(x − 1)⇔ y = −

1

2x +

3

2

7

Problem 43. Find the slopes of the devil’s curve y4 − 4y2 = x4 − 9x2 at the four indicated points.


29. y2 + x2 = y4 - 2x at (-2, 1) and (-2, -1)

30. (x2 + y2)2 = (x - y)2 at (1, 0) and (1, -1)


31. x2 + xy - y2 = 1, (2, 3)

32. x2 + y2 = 25, (3, -4)

33. x2y2 = 9, (-1, 3)

34. y2 - 2x - 4y - 1 = 0, (-2, 1)

35. 6x2 + 3xy + 2y2 + 17y - 6 = 0, (-1, 0)

36. x2 - 23xy + 2y2 = 5, 123, 22 37. 2xy + p sin y = 2p, (1, p>2)

38. x sin 2y = y cos 2x, (p>4, p>2)

39. y = 2 sin (px - y), (1, 0)

40. x2 cos2 y - sin y = 0, (0, p)




x

y

0

1

−1

y4 = y2 − x2

"

34

"

32

,

"

34

12

,a b

a b


x

y

1

1

(1, 1)

0

y2(2 − x) = x3


x

y

3−3

2

−2

(3, 2)

(3, −2)

(−3, 2)

(−3, −2)

y4 − 4y2 = x4 − 9x2








ddx

x p>q =pq x(p>q) - 1.


x

y

0 (a, 0)

x = y2


Solution

y4 − 4y2 = x4 − 9x2⇔ 4y3dy

dx− 8y

dy

dx= 4x3 − 18x

⇔dy

dx(4y3 − 8y) = 4x3 − 18x

⇔dy

dx=

4x3 − 18x

4y3 − 8y

At (-3,2):

dy

dx=

4(−3)3 − 18(−3)

4(2)3 − 8(2)

=4(−27) + 54

32 − 16

=−54

16

= −27

8

8

At (-3,-2):

dy

dx=

4(−3)3 − 18(−3)

4(−2)3 − 8(−2)

=−108 + 54

−32 + 16

=54

16

=27

8

At (3,2):

dy

dx=

4(3)3 − 18(3)

4(2)3 − 8(2)

=108 − 54

32 − 16

=54

16

=27

8

At (3,-2):

dy

dx=

4(3)3 − 18(3)

4(−2)3 − 8(−2)

=108 − 54

−32 + 16

=54

−16

= −27

8

9

Section 3.7

Problem 1. Suppose that the radius r and area A = πr2 of a circle are differentiable functions oft. Write an equation that relates dA/dt to dr/dt.

Solution Differentiating with respect to t we have:

A = πr2 ⇒dA

dt= 2πr

dr

dt

10

Problem 5. The voltage V (volts), current I (amperes), and resistance R (ohms) of an electriccircuit like the one shown here are related by the equation V = IR. Suppose that V is increasing ata rate of 1 volt/sec while I is decreasing at the rate of 1/3 amp/sec. Let t denote time in seconds.

so that

(25 + h)2 = 841, or 25 + h = 29.

Equation (2) now gives

dhdt = 21

29# 4 = 84

29 ≈ 2.9 ft>sec

as the rate at which the weight is being raised when x = 21 ft.

1. Area Suppose that the radius r and area A = pr2 of a circle are differentiable functions of t. Write an equation that relates dA>dt to dr>dt.

2. Surface area Suppose that the radius r and surface area S = 4pr2 of a sphere are differentiable functions of t. Write an equation that relates dS>dt to dr>dt.

3. Assume that y = 5x and dx>dt = 2. Find dy>dt.

4. Assume that 2x + 3y = 12 and dy>dt = -2. Find dx>dt.

5. If y = x2 and dx>dt = 3, then what is dy>dt when x = -1?

6. If x = y3 - y and dy>dt = 5, then what is dx>dt when y = 2?

7. If x2 + y2 = 25 and dx>dt = -2, then what is dy>dt when x = 3 and y = -4?

8. If x2y3 = 4>27 and dy>dt = 1>2, then what is dx>dt when x = 2?

9. If L = 2x2 + y2, dx>dt = -1, and dy>dt = 3, find dL>dt when x = 5 and y = 12.

10. If r + s2 + y3 = 12, dr>dt = 4, and ds>dt = -3, find dy>dt when r = 3 and s = 1.

11. If the original 24 m edge length x of a cube decreases at the rate of 5 m>min, when x = 3 m at what rate does the cube’s

a. surface area change?

b. volume change?

12. A cube’s surface area increases at the rate of 72 in2>sec. At what rate is the cube’s volume changing when the edge length is x = 3 in?

13. Volume The radius r and height h of a right circular cylinder are related to the cylinder’s volume V by the formula V = pr2h.

a. How is dV>dt related to dh>dt if r is constant?

b. How is dV>dt related to dr>dt if h is constant?

c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant?

14. Volume The radius r and height h of a right circular cone are related to the cone’s volume V by the equation V = (1>3)pr2h.

a. How is dV>dt related to dh>dt if r is constant?

b. How is dV>dt related to dr>dt if h is constant?

c. How is dV>dt related to dr>dt and dh>dt if neither r nor h is constant?

15. Changing voltage The voltage V (volts), current I (amperes), and resistance R (ohms) of an electric circuit like the one shown here are related by the equation V = IR. Suppose that V is

increasing at the rate of 1 volt>sec while I is decreasing at the rate of 1>3 amp>sec. Let t denote time in seconds.

V

R

I

+ −

a. What is the value of dV>dt?

b. What is the value of dI>dt?

c. What equation relates dR>dt to dV>dt and dI>dt?

d. Find the rate at which R is changing when V = 12 volts and I = 2 amps. Is R increasing, or decreasing?

16. Electrical power The power P (watts) of an electric circuit is related to the circuit’s resistance R (ohms) and current I (amperes) by the equation P = RI2.

a. How are dP>dt, dR>dt, and dI>dt related if none of P, R, and I are constant?

b. How is dR>dt related to dI>dt if P is constant?

17. Distance Let x and y be differentiable functions of t and let s = 2x2 + y2 be the distance between the points (x, 0) and (0, y) in the xy-plane.

a. How is ds>dt related to dx>dt if y is constant?

b. How is ds>dt related to dx>dt and dy>dt if neither x nor y is constant?

c. How is dx>dt related to dy>dt if s is constant?

18. Diagonals If x, y, and z are lengths of the edges of a rectangular box, the common length of the box’s diagonals is s = 2x2 + y2 + z2.

a. Assuming that x, y, and z are differentiable functions of t, how is ds>dt related to dx>dt, dy>dt, and dz>dt?

b. How is ds>dt related to dy>dt and dz>dt if x is constant?

c. How are dx>dt, dy>dt, and dz>dt related if s is constant?

19. Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure u is

A = 12 ab sin u.

Exercises 3.10

3.10 Related Rates 187

(a) What is the value of dV /dt?

(b) What is the value of dI/dt?

(c) What equation relates dR/dt to dV /dt and dI/dt?

(d) Find the rate at which R is changing when V = 12 voltes and I = 2 amp. Is R increasing, ordecreasing?

Solution

(a) We are given that V is increasing at a rate of 1 volt/sec so

dV

dt= 1 volt/sec

(b) We are given that I is decreasing at the rate of 1/3 amp/sec so

dI

dt= −

1

3amp/sec

(c)

V = IR⇒dV

dt=dI

dtR + I

dR

dt

(d) Since V = IR we have

R =V

I=

12

2= 6

Plugging these into the equation from (c) we have

1 = −1

3(6) + 2

dR

dt⇔

dr

dt=

3

2ohms/sec

Since it is positive, it is increasing.

11

Problem 11. The length l of a rectangle is decreasing at a rate of 2 cm/sec while the width wis increasing at a rate of 2 cm/sec. When l = 12cm and w = 5cm, find the rates of change of (a)the area, (b) the perimeter, and (c) the lengths of the diagonals of the rectangle. Which of thesequantities are decreasing, and which are increasing?

Solution

(a)

A = lw⇒dA

dt=dl

dtw + l

dw

dt

Plugging in l = 12, w = 5,dl

dt= −2, and

dw

dt= 2 we have

dA

dt= −2(5) + (12)(2) = 14 cm2/sec, which is increasing.

(b)

P = 2l + 2w⇒dP

dt= 2

dl

dt+ 2

dw

dt

Plugging indl

dt= −2 and

dw

dt= 2 we have

dP

dt= 2(−2) + 2(2) = 0 cm/sec, which is neither decreasing nor increasing

(c) The diagonal is related to the sides by the pythagorean theorem:

D2= l2 +w2

⇒ 2ddD

dt= 2l

dl

dt+ 2w

dw

dt

When I = 12 and w = 5 we have

D2= 144 + 25 = 169⇒D = 13

Plugging in l = 12, w = 5, D = 13,dl

dt= −2, and

dw

dt= 2 we have

26dD

dt= 24(−2) + 10(2)⇔

dD

dt= −

14

13cm/sec, which is decreasing

12

Problem 12. Suppose that the edge lengths x, y, and z of a closed rectangular box are changingat the following rates:

dx

dt= 1 m/sec,

dy

dt= −2 m/sec,

dz

dt= 1 m/sec.

Find the rates at which the box’s (a) volume, (b) surface area, and (c) diagonal length s =√x2 + y2 + z2

are changing at the instant when x = 4, y = 3, and z = 2.

Solution

(a)

V = xyz ⇒dV

dt=dx

dtyz + x

dy

dtz + xy

dz

dt

Plugging indx

dt= 1,

dy

dt= −2,

dz

dt= 1, x = 4, y = 3, and z = 2 we have

dV

dt= 1(3)(2) + (4)(−2)(2) + (4)(3)(1) = 2m3/sec

(b)

S = 2xy + 2xz + 2yz ⇒dS

dt= 2x

dy

dt+ 2

dx

dty + 2x

dz

dt+ 2

dx

dtz + 2y

dz

dt+ 2

dy

dtz

Plugging indx

dt= 1,

dy

dt= −2,

dz

dt= 1, x = 4, y = 3, and z = 2 we have

dS

dt= 2(4)(−2) + 2(1)(3) + 2(4)(2) + 2(1)(2) + 2(3)(1) + 2(−2)(2)

= 8 m/sec

(c)

s =√x2 + y2 + z2 ⇒

ds

dt=

1

2(x2 + y2 + z2)−1/2 (2x

dx

dt+ 2y

dy

dt+ 2z

dz

dt)

Plugging indx

dt= 1,

dy

dt= −2,

dz

dt= 1, x = 4, y = 3, and z = 2 we have

ds

dt=

1

2(42 + 32 + 22)−1/2 (2(4)(1) + 2(3)(2) + 2(2)(1))

=1

2(29)−1/2(24)

=12√

29m/s

13

Problem 15. A girl flies a kite at a height of 300 ft, the wind carrying the kite horizontally awayfrom her at a rate of 25 ft/sec. How fast must she let out the string when the kite is 500 ft awayfrom her?

Solution The picture is the following:

Using the pythagorean theorem we have

x2 + 3002 = z2 ⇒ 2xdx

dt= 2z

dz

dt

When z = 500 we have x2 + 3002 = 5002 ⇒ x = 400. We are also givendx

dt= 25 and we want to find

dz

dt. Plugging in our known information we have

2(400)(25) = 2(500)dz

dt⇔

dz

dt= 20 ft/sec

14

Problem 17. Sand falls from a conveyor belt at the rate of 10 m3/min onto the top of a conicalpile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) heightand (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.

Solution We have the formula

V =1

3πr2h

We are given that h = (3/8)(2r) = 3r/4⇔ r = (4/3)h. Thus we have

V =1

3π (

4

3h)

2

⇔ V =16πh3

27

Differentiating with respect to t we have

dV

dt=

16π(3)h2

27

dh

dt=

16πh2

9

dh

dt

(a) We are given that dV /dt = 10. Plugging this and h = 4 in we have

10 =16(4)2π

9

dh

dt⇔

dh

dt= 0.1119 m/min = 11.19 cm/min

(b)

r =4

3h⇒

dr

dt=

4

3

dh

dt

From (a) we havedh

dt= 11.19⇔

dr

dt=

4

3(11.19) = 14.92 cm/min

15

Problem 19. Water is flowing at the rate of 6 m3/min from a reservoir shaped like a hemisphericalbowl of radius 13 m, shown here in profile. Answer the following questions, given that the volumeof water in a hemispherical bowl of radius R is V = (π/3)y2(3R − y) when the water is y metersdeep.

a. How is dA>dt related to du>dt if a and b are constant?

b. How is dA>dt related to du>dt and da>dt if only b is constant?

c. How is dA>dt related to du>dt, da>dt, and db>dt if none of a, b, and u are constant?

20. Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm >min. At what rate is the plate’s area increasing when the radius is 50 cm?

21. Changing dimensions in a rectangle The length l of a rectan-gle is decreasing at the rate of 2 cm>sec while the width w is increasing at the rate of 2 cm>sec. When l = 12 cm and w = 5 cm, find the rates of change of (a) the area, (b) the perim-eter, and (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing, and which are increasing?

22. Changing dimensions in a rectangular box Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates:

dxdt

= 1 m>sec, dydt

= -2 m>sec, dzdt

= 1 m>sec.

Find the rates at which the box’s (a) volume, (b) surface area, and (c) diagonal length s = 2x2 + y2 + z2 are changing at the instant when x = 4, y = 3, and z = 2.

23. A sliding ladder A 13-ft ladder is leaning against a house when its base starts to slide away (see accompanying figure). By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft>sec.

a. How fast is the top of the ladder sliding down the wall then?

b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?

c. At what rate is the angle u between the ladder and the ground changing then?

x0

y

13-ft ladder

y(t)

x(t)

u

24. Commercial air traffic Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane B is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when A is 5 nautical miles from the intersection point and B is 12 nautical miles from the intersection point?

25. Flying a kite A girl flies a kite at a height of 300 ft, the wind car-rying the kite horizontally away from her at a rate of 25 ft>sec. How fast must she let out the string when the kite is 500 ft away from her?

26. Boring a cylinder The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-thousandth of an inch every 3 min. How rapidly is the cylinder volume increas-ing when the bore (diameter) is 3.800 in.?

27. A growing sand pile Sand falls from a conveyor belt at the rate of 10 m3>min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.

28. A draining conical reservoir Water is flowing at the rate of 50 m3>min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m.

a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep?

b. How fast is the radius of the water’s surface changing then? Answer in centimeters per minute.

29. A draining hemispherical reservoir Water is flowing at the rate of 6 m3>min from a reservoir shaped like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following ques-tions, given that the volume of water in a hemispherical bowl of radius R is V = (p>3)y2(3R - y) when the water is y meters deep.

r

y

13

Center of sphere

Water level

a. At what rate is the water level changing when the water is 8 m deep?

b. What is the radius r of the water’s surface when the water is y m deep?

c. At what rate is the radius r changing when the water is 8 m deep?

30. A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop’s radius increases at a constant rate.

31. The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 100p ft3>min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing?

32. Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft>sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?

b. At what rate is the angle u changing at this instant (see the figure)?

Ring at edgeof dock

6'

u

188 Chapter 3: Derivatives

(a) At what rate is the water level changing when the water is 8 m deep?

(b) What is the radius r of the water’s surface when the water is y m deep?

(c) At what rate is the radius r hanging when the water is 8 m deep?

Solution

V =π

3y2(39 − y)⇒

dV

dt=

2π

3ydy

dt(39 − y) +

π

3y2 (−

dy

dt)

(a) We are given that dV /dt = −6, r = 13, y = 8. Plugging these in we have

−6 =2π

3(8)

dy

dt(39 − 8) −

π

3(8)2

dy

dt⇔ −6 =

496π

3

dy

dt−

64π

3

dy

dt

⇔dy

dt= −

6

(432π/3)

⇔dy

dt= −

1

24πm/min

(b) From the photo we have

(13 − y)2 + r2 = 132⇔ r2 = 169 − (13 − y)2 ⇒ r =√

169 − (13 − y)2 =√

26y − y2 m

(c) From c we havedr

dt=

1

2(26y − y2)−1/2 (26

dy

dt− 2y

dy

dt)

Plugging in y = 8 and our answer from (a) we have

dr

dt=

1

2√

26(8) − 82(−

26

24π+

16

24π) = −

5

288πm/min

16

Problem 22. A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock6 ft above the bow. The rope is hauled in at the rate of 2 ft/sec.

(a) How fast is the boat approaching the dock when 10 ft of rope are out?

(b) At what rate is the angle θ changing at this instant (see the figure)?

a. How is dA>dt related to du>dt if a and b are constant?

b. How is dA>dt related to du>dt and da>dt if only b is constant?

c. How is dA>dt related to du>dt, da>dt, and db>dt if none of a, b, and u are constant?

20. Heating a plate When a circular plate of metal is heated in an oven, its radius increases at the rate of 0.01 cm >min. At what rate is the plate’s area increasing when the radius is 50 cm?

21. Changing dimensions in a rectangle The length l of a rectan-gle is decreasing at the rate of 2 cm>sec while the width w is increasing at the rate of 2 cm>sec. When l = 12 cm and w = 5 cm, find the rates of change of (a) the area, (b) the perim-eter, and (c) the lengths of the diagonals of the rectangle. Which of these quantities are decreasing, and which are increasing?

22. Changing dimensions in a rectangular box Suppose that the edge lengths x, y, and z of a closed rectangular box are changing at the following rates:

dxdt

= 1 m>sec, dydt

= -2 m>sec, dzdt

= 1 m>sec.

Find the rates at which the box’s (a) volume, (b) surface area, and (c) diagonal length s = 2x2 + y2 + z2 are changing at the instant when x = 4, y = 3, and z = 2.

23. A sliding ladder A 13-ft ladder is leaning against a house when its base starts to slide away (see accompanying figure). By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft>sec.

a. How fast is the top of the ladder sliding down the wall then?

b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?

c. At what rate is the angle u between the ladder and the ground changing then?

x0

y

13-ft ladder

y(t)

x(t)

u

24. Commercial air traffic Two commercial airplanes are flying at an altitude of 40,000 ft along straight-line courses that intersect at right angles. Plane A is approaching the intersection point at a speed of 442 knots (nautical miles per hour; a nautical mile is 2000 yd). Plane B is approaching the intersection at 481 knots. At what rate is the distance between the planes changing when A is 5 nautical miles from the intersection point and B is 12 nautical miles from the intersection point?

25. Flying a kite A girl flies a kite at a height of 300 ft, the wind car-rying the kite horizontally away from her at a rate of 25 ft>sec. How fast must she let out the string when the kite is 500 ft away from her?

26. Boring a cylinder The mechanics at Lincoln Automotive are reboring a 6-in.-deep cylinder to fit a new piston. The machine they are using increases the cylinder’s radius one-thousandth of an inch every 3 min. How rapidly is the cylinder volume increas-ing when the bore (diameter) is 3.800 in.?

27. A growing sand pile Sand falls from a conveyor belt at the rate of 10 m3>min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in centimeters per minute.

28. A draining conical reservoir Water is flowing at the rate of 50 m3>min from a shallow concrete conical reservoir (vertex down) of base radius 45 m and height 6 m.

a. How fast (centimeters per minute) is the water level falling when the water is 5 m deep?

b. How fast is the radius of the water’s surface changing then? Answer in centimeters per minute.

29. A draining hemispherical reservoir Water is flowing at the rate of 6 m3>min from a reservoir shaped like a hemispherical bowl of radius 13 m, shown here in profile. Answer the following ques-tions, given that the volume of water in a hemispherical bowl of radius R is V = (p>3)y2(3R - y) when the water is y meters deep.

r

y

13

Center of sphere

Water level

a. At what rate is the water level changing when the water is 8 m deep?

b. What is the radius r of the water’s surface when the water is y m deep?

c. At what rate is the radius r changing when the water is 8 m deep?

30. A growing raindrop Suppose that a drop of mist is a perfect sphere and that, through condensation, the drop picks up moisture at a rate proportional to its surface area. Show that under these circumstances the drop’s radius increases at a constant rate.

31. The radius of an inflating balloon A spherical balloon is inflated with helium at the rate of 100p ft3>min. How fast is the balloon’s radius increasing at the instant the radius is 5 ft? How fast is the surface area increasing?

32. Hauling in a dinghy A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow. The rope is hauled in at the rate of 2 ft>sec.

a. How fast is the boat approaching the dock when 10 ft of rope are out?

b. At what rate is the angle u changing at this instant (see the figure)?

Ring at edgeof dock

6'

u


Solution

(a) If we call the length of the rope z and the distance from the dock x we have

x2 + 62 = z2 ⇒ x =√z2 − 36⇒

dx

dt=

1

2(z2 − 36)−1/2 (2z

dz

dt)

We are given that z = 10 anddz

dt= −2, thus

dx

dt=

1

2(100 − 36)−1/2(2(10)(−2) = −

5

2

(b) We have

tan θ =x

6⇒ sec2 θ

dθ

dt=

1

6

dx

dt⇒

dθ

dt=

1

6

dx

dtcos2 θ

When the rope is 10ft out we have

cos θ =6

10=

3

5⇒ cos2 θ =

9

25

Plugging this along with the answer from (a) we have

dθ

dt= −

1

6⋅5

2⋅

9

25= −

3

20radian/sec

17

Problem 23. A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec.Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/secpasses under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?

33. A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft>sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft>sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?

y

x0

y(t)

s(t)

x(t)

34. Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3>min.

a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep?

b. How fast is the level in the cone falling then?

6″

6″

6″

How fastis thislevel rising?

How fastis thislevel falling?

35. Cardiac output In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 L>min. At rest it is likely to be a bit under 6 L>min. If you are a trained marathon runner running a mar-athon, your cardiac output can be as high as 30 L>min.

Your cardiac output can be calculated with the formula

y =QD ,

where Q is the number of milliliters of CO2 you exhale in a minute and D is the difference between the CO2 concentration (ml>L) in the blood pumped to the lungs and the CO2 concentration in the blood returning from the lungs. With Q = 233 ml>min and D = 97 - 56 = 41 ml>L,

y =233 ml>min

41 ml>L ≈ 5.68 L>min,

fairly close to the 6 L>min that most people have at basal (rest-ing) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.)

Suppose that when Q = 233 and D = 41, we also know that D is decreasing at the rate of 2 units a minute but that Q remains unchanged. What is happening to the cardiac output?

36. Moving along a parabola A particle moves along the parabola y = x2 in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m>sec. How fast is the angle of inclination u of the line joining the particle to the origin changing when x = 3 m?

37. Motion in the plane The coordinates of a particle in the metric xy-plane are differentiable functions of time t with dx>dt =-1 m>sec and dy>dt = -5 m>sec. How fast is the particle’s dis-tance from the origin changing as it passes through the point (5, 12)?

38. Videotaping a moving car You are videotaping a race from a stand 132 ft from the track, following a car that is moving at 180 mi>h (264 ft>sec), as shown in the accompanying figure. How fast will your camera angle u be changing when the car is right in front of you? A half second later?

u

Car

Camera

132′

39. A moving shadow A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. (See accompanying figure.) How fast is the shadow of the ball moving along the ground 1>2 sec later? (Assume the ball falls a distance s = 16t2 ft in t sec.)

x

Light

30

Shadow

0

50-ftpole

Ball at time t = 0

1/2 sec later

x(t)


Solution Using the pythagorean theorem we have

x2 + y2 = s2 ⇒ 2xdx

dt+ 2y

dy

dt= 2s

ds

dt

Since the bicycle is going a constant speed, 3 seconds later the distance x = 17 ⋅3 = 51 and since theballoon is rising at a constant rate from 65ft above we have that 3 seconds later it is at a height ofx = 64 + 1 ⋅ 3 = 68. Using the pythagorean theorem we have s =

√512 + 682 = 85. We are also given

thatdx

dt= 17 and

dy

dt= 1. Plugging these in we have

2(51)(17) + 2(68)(1) = 2(85)ds

dt⇒

ds

dt= 11 ft/sec

18

Problem 27. A particle moves along the parabola y = x2 in the first quadrant in such a way thatits x-coordinate (measured in meters) increases at a steady 10 m/sec. How fast is the angle ofinclination θ of the line joining the particle to the origin changing when x = 3 m?

Solution The figure looks like the following:

Thus we have

tan θ =y

x⇒ sec2 θ

dθ

dt=x(dy/dt) − (dx/dt)y

x2

We are also given that

y = x2 ⇒dy

dt= 2x

dx

dt

We are given that x = 3 and dx/dt = 10 Thus

dy

dt= 60, y = 32 = 9, and θ = tan−1 (

9

3) = tan−1(3)

Plugging this into the differential equation we have:

sec2(tan−1(3))dθ

dt=

3(60) − (10)(9)

9⇔

dθ

dt= 1 radian/sec

19

Problem 31. A light shines from the top of a pole 50 ft high. A ball is dropped from the sameheight from a point 30 ft away from the light. (See accompanying figure.) How fast is the shadowof the ball moving along the ground 1/2 sec later? (Assume the ball falls a distance s = 16t2 ft in tsec.)

33. A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft>sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft>sec passes under it. How fast is the distance s(t) between the bicycle and balloon increasing 3 sec later?

y

x0

y(t)

s(t)

x(t)

34. Making coffee Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in3>min.

a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep?

b. How fast is the level in the cone falling then?

6″

6″

6″

How fastis thislevel rising?

How fastis thislevel falling?

35. Cardiac output In the late 1860s, Adolf Fick, a professor of physiology in the Faculty of Medicine in Würzberg, Germany, developed one of the methods we use today for measuring how much blood your heart pumps in a minute. Your cardiac output as you read this sentence is probably about 7 L>min. At rest it is likely to be a bit under 6 L>min. If you are a trained marathon runner running a mar-athon, your cardiac output can be as high as 30 L>min.

Your cardiac output can be calculated with the formula

y =QD ,

where Q is the number of milliliters of CO2 you exhale in a minute and D is the difference between the CO2 concentration (ml>L) in the blood pumped to the lungs and the CO2 concentration in the blood returning from the lungs. With Q = 233 ml>min and D = 97 - 56 = 41 ml>L,

y =233 ml>min

41 ml>L ≈ 5.68 L>min,

fairly close to the 6 L>min that most people have at basal (rest-ing) conditions. (Data courtesy of J. Kenneth Herd, M.D., Quillan College of Medicine, East Tennessee State University.)

Suppose that when Q = 233 and D = 41, we also know that D is decreasing at the rate of 2 units a minute but that Q remains unchanged. What is happening to the cardiac output?

36. Moving along a parabola A particle moves along the parabola y = x2 in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 10 m>sec. How fast is the angle of inclination u of the line joining the particle to the origin changing when x = 3 m?

37. Motion in the plane The coordinates of a particle in the metric xy-plane are differentiable functions of time t with dx>dt =-1 m>sec and dy>dt = -5 m>sec. How fast is the particle’s dis-tance from the origin changing as it passes through the point (5, 12)?

38. Videotaping a moving car You are videotaping a race from a stand 132 ft from the track, following a car that is moving at 180 mi>h (264 ft>sec), as shown in the accompanying figure. How fast will your camera angle u be changing when the car is right in front of you? A half second later?

u

Car

Camera

132′

39. A moving shadow A light shines from the top of a pole 50 ft high. A ball is dropped from the same height from a point 30 ft away from the light. (See accompanying figure.) How fast is the shadow of the ball moving along the ground 1>2 sec later? (Assume the ball falls a distance s = 16t2 ft in t sec.)

x

Light

30

Shadow

0

50-ftpole

Ball at time t = 0

1/2 sec later

x(t)


Solution Let s be the distance the ball has fallen. From the picture we have

x − 30

x=

50 − s

50⇔ 50x − 1500 = 50x − sx⇔ sx = 1500⇔ x =

1500

s

Differentiating with respect to t we have

dx

dt= −

1500

s2ds

dt

We are given s = 16(t2)⇒ ds/dt = 32t. At t = 1/2, s = 16(1/4) = 4 and ds/dt = 32/2 = 16. Pluggingthese in we have

dx

dt= −

1500

16(16) = −1500 ft/sec

20

Problem 35. On a morning of a day when the sun will pass directly overhead, the shadow ofan 80-ft building on level ground is 60 ft long. At the moment in question, the angle θ the sunmakes with the ground is increasing at a rate of 0.27○/min. At what rate is the shadow decreasing?(Remember to use radians. Express your answer in inches per minute, to the nearest tenth.)

40. A building’s shadow On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 ft long. At the moment in question, the angle u the sun makes with the ground is increasing at the rate of 0.27°>min. At what rate is the shadow decreasing? (Remember to use radians. Express your answer in inches per minute, to the nearest tenth.)

80′

u

41. A melting ice layer A spherical iron ball 8 in. in diameter is coated with a layer of ice of uniform thickness. If the ice melts at the rate of 10 in3>min, how fast is the thickness of the ice decreasing when it is 2 in. thick? How fast is the outer surface area of ice decreasing?

42. Highway patrol A highway patrol plane flies 3 mi above a level, straight road at a steady 120 mi>h. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi, the line-of-sight distance is decreasing at the rate of 160 mi>h. Find the car’s speed along the highway.

43. Baseball players A baseball diamond is a square 90 ft on a side. A player runs from first base to second at a rate of 16 ft>sec.

a. At what rate is the player’s distance from third base changing when the player is 30 ft from first base?

b. At what rates are angles u1 and u2 (see the figure) changing at that time?

c. The player slides into second base at the rate of 15 ft>sec. At what rates are angles u1 and u2 changing as the player touches base?

90′

Second base

Player

Home

30′ Firstbase

Thirdbase

u1

u2

44. Ships Two ships are steaming straight away from a point O along routes that make a 120° angle. Ship A moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yd). Ship B moves at 21 knots. How fast are the ships moving apart when OA = 5 and OB = 3 nautical miles?

45. Clock’s moving hands At what rate is the angle between a clock’s minute and hour hands changing at 4 o’clock in the after-noon?

46. Oil spill An explosion at an oil rig located in gulf waters causes an elliptical oil slick to spread on the surface from the rig. The slick is a constant 9 in. thick. After several days, when the major axis of the slick is 2 mi long and the minor axis is 3/4 mi wide, it is deter-mined that its length is increasing at the rate of 30 ft/hr, and its width is increasing at the rate of 10 ft/hr. At what rate (in cubic feet per hour) is oil flowing from the site of the rig at that time?

3.11 Linearization and Differentials

Sometimes we can approximate complicated functions with simpler ones that give the accuracy we want for specific applications and are easier to work with. The approximating functions discussed in this section are called linearizations, and they are based on tangent lines. Other approximating functions, such as polynomials, are discussed in Chapter 9.

We introduce new variables dx and dy, called differentials, and define them in a way that makes Leibniz’s notation for the derivative dy>dx a true ratio. We use dy to estimate error in measurement, which then provides for a precise proof of the Chain Rule (Section 3.6).

Linearization

As you can see in Figure 3.50, the tangent to the curve y = x2 lies close to the curve near the point of tangency. For a brief interval to either side, the y-values along the tangent line give good approximations to the y-values on the curve. We observe this phenomenon by zooming in on the two graphs at the point of tangency or by looking at tables of values for the difference between ƒ(x) and its tangent line near the x-coordinate of the point of tan-gency. The phenomenon is true not just for parabolas; every differentiable curve behaves locally like its tangent line.


Solution From the picture we have

tan θ =80

x⇒ sec2 θ

dθ

dt= −

80

x2dx

dt

At the instant we are looking at we have θ = tan−180

60and

dθ

dt= 0.27○ ⋅

π

180Plugging these in we

have

sec2 (tan−18

6)(

0.27π

180) = −

80

602dx

dt⇔

dx

dt= −0.589 ft/min = −7.1in/min

Since the problem want the rate it’s decreasing we have that the shadow is decreasing at a rate of

7.1 in/min

21

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Math 241 Homework 5 Solutionskmanguba/math241-2/HW5.pdf · 2017-12-10 · Problem 37. Verify that the given point is on the curve and nd the lines that are (a) tangent and (b) normal