Question

Claire and Richard are both artists who use square canvases. Claire

uses the polynomial 50%? + 250 to decide how much to charge for her paintings

and Richard uses the polynomial 40x² + 350 to decide how much to charge for

his paintings. In each polynomial, x is the height of the painting in feet.

a. How much does Claire charge for a 20-foot-tall painting?

b. How much does Richard charge for a 15-foot-tall painting?

c. To the nearest tenth, for what height will both Claire and Richard charge

the same amount for a painting? Explain how to find the answer.

d. When both Claire and Richard charge the same amount for a painting,

how much does each charge?

Answer 1

A.
`Claire = 20250`

B.
`Richard = 9350`

C. Height = 3.2 feet

D. Charges = $760

Explanation:

Given

`Claire = 50x^2 + 250`

`Richard = 40x^2 + 350`

Solving (a): Claire's 20ft charges

In this case, x = 20

Substitute 20 for x in
`Claire = 50x^2 + 250`

`Claire = 50(20)^2 + 250`

`Claire = 50(400) + 250`

`Claire = 20000 + 250`

`Claire = 20250`

Solving (b): Richard's 15ft charges

In this case, x = 15

Substitute 20 for x in
`Richard = 40x^2 + 350`

`Richard = 40(15)^2 + 350`

`Richard = 40(225) + 350`

`Richard = 9000 + 350`

`Richard = 9350`

Solving (c): Height which they both charge the same;

This implies that

`50x^2 + 250 = 40x^2 + 350`

Solving for x [Collect Like Terms\

`50x^2 - 40x^2 = 350 - 250`

`10x^2 = 100`

Divide both sides by 10

`x^2 = 10`

Take square roots of both sides

`x = √(10)`

`x = 3.2ft` (Approximated)

Hence; Height = 3.2 feet

Solving (d): How much they charge when the charge the same amount.

Substitute 3.2 for x in any of the given equation

`Claire = 50x^2 + 250`

`Claire = 50(3.2)^2 + 250`

`Claire = 50*10.24 + 250`

`Claire = 512 + 250`

`Claire = 762`

`Richard = 40x^2 + 350`

`Richard = 40(3.2)^2 + 350`

`Richard = 40 * 10.24 + 350`

`Richard = 409.6 + 350`

`Richard = 759.6`

The reason for the difference is due to approximation

Hence, they both charge approximately 760