Homework Homework Assignment #28 Read Section 4.7 Page 265, Exercises: 31, 39, 43, 47, 53, 59 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

Homework

Homework Assignment #28 Read Section 4.7 Page 265, Exercises: 31, 39, 43, 47, 53, 59

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Example, Page 265


190 /10 190 900 /10 100

190 910 /10 99,....The formula is correct.

n r n

n

31. Let r be the monthly rent in an apartment building with 100 units. All units can be rent when r = $900 and one unit becomes vacant for every $10 increase in rent. The average monthly maintenance cost per occupied unit is $100.(a) Show that the number of units rented is n = 190 – r/10 for 900 ≤ r ≤ 1900.

(b) Find a formula for the net cash intake (revenue minus maintenance)and determine the rent that maximizes intake.

2

100 190 /10 100 19000 20010

200 2 /10 0 200 2 /10 2 2000

$1,000

rR nr n r r r

R r r r

r

Example, Page 265


A five-sided box is to be constructed from a rectangle of cardboard of sides A and B by cutting squares of side h from each corner and foldingup the resulting tabs.39. Find the value of h that maximizes the volume of the box if A = 15 and B = 24. What are the dimensions of the resulting box?

2 3 2

2 2 2

3

15 2 24 2 360 102 4 4 102 360

12 204 360 12 17 30 0 17 30 0

15 2 0 2, 20, 11, 440

V h h h h h h h h h

V h h h h V h h

h h h l w V units

Example, Page 265


43. Use calculus to show that among all right triangles with hypotenuse of length 1, the isosceles triangle has maximum area.

2 2

1 1sin Let side have length 1 and = 90 .

2 21

Then side cos and cos sin2

1 1cos cos sin sin cos sin

2 21

cos 2 0 cos 2 0 2 90 , 452

A bh ab C a A

b C A C C

A C C C C C C

A C C C C

Example, Page 265


43. Can you see more directly why this must be true from Figure 23?

If the diameter of the semicircle is 1,and the vertex is any point on the circumference, the angle at the vertexis 90º. When x = y, h has the largest value and so does area.

Example, Page 265


47. Find the area of the largest isosceles triangle that can be inscribed in a circle of radius r.

2

12 cos30 3

23

sin 302

1 3 33 3

2 2 4

A bh b r r

h r r r

A r r r

3030

30

30 30

30

r

rr

r/2

Example, Page 265


53. A small blood vessel of radius r branches off at an angle of θfrom a larger vessel of radius R to supply blood along a path from Ato B. According to Poiseuille’s Law, the total resistance to blood flowis proportional to

where a and b are as in Figure 26. Show that total resistance is minimized when cos θ =(r/R)4.

4 4

cot csca b bT

R r

Example, Page 265


53. Continued.

24 4 4 4

42

4 4 4 2

44 4 4

4 4 4

cot csccsc csc cot

csc cot0 csc csc cot

csccos

cot sin cos1csc

sin

a b b b bT T

R r R r

b b rT

R r R

r r r r

R R R R

Example, Page 265


59. Find the maximum length of a pole that can be carried around a corner joining corridors of widths 8ft and 4 ft. See Figure 30.

1 2

1 2

2 2

3 3

3 1 3

8 4sin ,cos

8 4

sin cos4 sin8cos

sin cos

0 4sin 8cos

8tan tan 2 51.561

48 4

16.648sin51.561 cos51.561

l l

l l l l

l

l

l


Jon Rogawski

Calculus, ETFirst Edition

Chapter 4: Applications of the DerivativeSection 4.7: L'Hopital's Rule


In the case of the limit of a rational function at a point where therational function has an indeterminate value, 0/0 or ∞/∞, we may use L’Hôpital’s Rule to find the derivative.


0

1 1Evaluate: lim

sinx x x


0Evaluate: lim x

xx

Example, Page 277Show that L’Hôpital’s Rule is applicable and use it to evaluate the limit.


2

52

2 102. lim

2 40 16x

x x

x x

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Homework Homework Assignment #28 Read Section 4.7 Page 265, Exercises: 31, 39, 43, 47, 53, 59 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company