Question

A rectangular box has a square base. The combined length of a side of the square base, and the height is 20 in. Let x be the length of a side of the base of the box.
a. Write a polynomial function in factored form modeling the volume V of the box.
b. What is the maximum possible volume of the box?

Answer 1

Answer:

a. V = (20-x) $x^{2}$ $in^{3}$  

b . 1185.185 $in^{3}$

Step-by-step explanation:

Given that:

• The height:  20  - x (in )
• Let x be the length of a side of the base of the box (x>0)

a. Write a polynomial function in factored form modeling the volume V of the box.

As we know that, this is a rectangular box has a square base so the Volume of it is:

V = h *$x^{2}$ $in^{3}$

<=> V = (20-x) $x^{2}$  $in^{3}$

b. What is the maximum possible volume of the box?

To  maximum the volume of it, we need to use first derivative of the volume.

<=> dV / Dx = -3$x^{2}$ + 40x

Let dV / Dx = 0, we have:

-3$x^{2}$ + 40x  = 0

<=> x = 40/3

=>the height h = 20/3

So  the maximum possible volume of the box is:

V = 20/3 * 40/3 *40/3

= 1185.185 $in^{3}$