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

Answer:

A.  $Claire = 20250$

B.   $Richard = 9350$

C.    Height = 3.2 feet

D. Charges = $760

Step-by-step 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

$x = \sqrt{10}$

$x = 3.2ft$  (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