Topic: U2L5 Quadratic Word Problems

EQ: Can I solve and interpret the solutions of a quadratic function in the context of a problem?

There are basically two types of quadratic word problems:

Those that ask you to find the vertex

Those that ask you to find the roots

By giving you the equation

There are basically two types of quadratic word problems:

Those that ask you to find the vertex

Those that ask you to find the roots

By giving you the information to find the equation

We have seen that the vertex of a quadratic in general form is given by We have seen that the vertex of a quadratic in general form is given by

a

bf

a

b

2,

2

The y-value of the vertex is either the maximum value the function can have or it’s the minimum value the function can have.

The y-value of the vertex is either the maximum value the function can have or it’s the minimum value the function can have.

y-value’s a min when a >0

y-value’s a max when a<0

a. What price would maximize the revenue? a. What price would maximize the revenue?

The word maximize screams “FIND THE VERTEX!!”The word maximize screams “FIND THE VERTEX!!”

808.0

64

)4.0(2

642

p

p

p

a

bp

Thus, a price of $80 would maximize the revenue.

Thus, a price of $80 would maximize the revenue.

Example 1

A small business’ profits over the last year have been related to the price of the only product. The relationship is R(p) = -0.4p2 +64p-2400, where R is the revenue measured in thousands of dollars and p is the price of the product measured in dollars.

Example 1

A small business’ profits over the last year have been related to the price of the only product. The relationship is R(p) = -0.4p2 +64p-2400, where R is the revenue measured in thousands of dollars and p is the price of the product measured in dollars.

b. What is the maximum revenue possible? b. What is the maximum revenue possible?

The answer to this question is the y-value of the vertex.The answer to this question is the y-value of the vertex.

160)80(

240051202560)80(

24005120)6400(4.0)80(

2400)80(64)80(4.0)80(

802

R

R

R

R

p

The maximum revenue is $160 000.The maximum revenue is $160 000.

Example 1

A small business’ profits over the last year have been related to the price of the only product. The relationship is R(p) = -0.4p2 +64p-2400, where R is the revenue measured in thousands of dollars and p is the price of the product measured in dollars.

c. How much money would they lose if they gave the product away? c. How much money would they lose if they gave the product away?

This question is talking about a price of 0 or p = 0This question is talking about a price of 0 or p = 0

2400)0(

2400)0(64)0(4.0)0(

02

R

R

p

The business would lose $2 400 000.The business would lose $2 400 000.

A lifeguard has 75m of rope to section off the supervised area of the beach. What is the largest rectangular swimming area possible?

A lifeguard has 75m of rope to section off the supervised area of the beach. What is the largest rectangular swimming area possible?

2w + L = 75 A = wL

L= 75-2w

A=w(75-2w)

A = 75w-2w2A = 75w-2w2

75.18

)2(2

752

w

w

a

bw

A = 75(18.75)-2(18.75)2 A =1406.25-703.125A= 703.125

A = 75(18.75)-2(18.75)2 A =1406.25-703.125A= 703.125

Example 2Ex

We know that we can find the roots of a quadratic function by setting one side equal to zero and

We know that we can find the roots of a quadratic function by setting one side equal to zero and

Factoring (sometimes)Factoring (sometimes)

Using the quadratic root formulaUsing the quadratic root formula

This ALWAYS works for a quadratic in general form and is easy to do.

This ALWAYS works for a quadratic in general form and is easy to do.

a

acbbx

2

42

Example 1

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in meters and x represents the time in seconds.

a. How long was the duck underwater?a. How long was the duck underwater?

The duck is no longer underwater when the depth is 0. We can plug in y= 0 and solve for x.

)4(20

420 2

xx

xx

x20 40 x

So x = 0 or 4

The duck was underwater for 4 seconds

The duck was underwater for 4 seconds

Example 1

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres from the water and x represents the time in seconds.

b. When was the duck at a depth of 5m?b. When was the duck at a depth of 5m?We can plug in y= -5 and solve for x.

4

244

4

40164

)2(2

)5)(2(4)4()4(

2

4

2

2

x

x

x

a

acbbx

We cannot solve this because there’s a negative number under the square root.

We conclude that the duck is never 5m below the water.

5420

4252

2

xx

xx

Example 1

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres from the water and x represents the time in seconds.

b. When was the duck at a depth of 5m?b. When was the duck at a depth of 5m?We can check this by finding the minimum value of y.

2

)1(4)1(2

14

4

)2(2

)4(2

2

y

y

x

x

x

a

bx

We conclude that the duck is never 5m below the water.

Example 1

A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the position of the duck in metres and x represents the time in seconds.

c. How long was the duck at least 0.5m below the water’s surface?c. How long was the duck at least 0.5m below the water’s surface?

We can plug in y= -0.5 and solve for x.

The duck was 0.5m below at t = 0.14s and at t = 1.87s

This will give us the times when the duck is at 0.5 m below.

This will give us the times when the duck is at 0.5 m below.

5.0420

425.02

2

xx

xx

sorx

x

x

x

x

a

acbbx

87.114.0

4

46.344

124

4

4164

)2(2

)5.0)(2(4)4()4(

2

4

2

2

Therefore it was below 0.5m for 1.73s

Roots Word Problems

Example 2: no equation

Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?

Let’s define our variables: S = one of the numbers G= the other number

The question indicates two equations relating these two variables.

S – G = 18S – G = 18

S2+ G2 = 194S2+ G2 = 194

Again, we have a substitution situation. Solve the simpler equation for a variable and plug it in to the other equation.

S=18+GS=18+G

(18+G)2 + G2 = 194

Roots Word Problems

Example 3: no equation

Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?

(18+G)2 + G2 = 194

We need to solve this equation for G. Then, use the quadratic formula.

Let’s FOIL and make one side equal to 0.

324+36G+ G2 + G2 = 194

2G2 +36G+324-194 =0

2G2 +36G+130=0 135

4

1636

)2(2

)130)(2(4)36()36(

2

4

2

2

orG

G

G

a

acbbG

S = 18 + (-5) = 13 or S = 18+(-13)=5

Roots Word Problems

Example 3: no equation

Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?Two numbers have a difference of 18. The sum of their squares is 194. What are the numbers?

(18+G)2 + G2 = 194

The numbers are either -5 and 13

or -13 and 5.

Roots Word Problems: Try one

A rectangle is 8 feet long and 6 feet wide. If the same number of feet increases each dimension, the area of the new rectangle formed is 32 square feet more than the area of the original rectangle. How many feet increased each dimension?

A rectangle is 8 feet long and 6 feet wide. If the same number of feet increases each dimension, the area of the new rectangle formed is 32 square feet more than the area of the original rectangle. How many feet increased each dimension?

The new area is 32+6∙8=32+48=80

The dimensions of the new rectangle are

6+x and 8+x6+x and 8+x

So 80 = (6+x)(8+x)

So 80 = (6+x)(8+x)

80=48+14x+x2

0=x2 +14x-32

x =2 or x = -16

6

8