Answer:
a. V = (20-x) [tex]x^{2}[/tex] [tex]in^{3}[/tex]
b . 1185.185 [tex]in^{3}[/tex]
Step-by-step explanation:
Given that:
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 *[tex]x^{2}[/tex] [tex]in^{3}[/tex]
<=> V = (20-x) [tex]x^{2}[/tex] [tex]in^{3}[/tex]
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[tex]x^{2}[/tex] + 40x
Let dV / Dx = 0, we have:
-3[tex]x^{2}[/tex] + 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 [tex]in^{3}[/tex]
a) The polynomial function in factored form that models the volume[tex]V=x^2(20-x)[/tex]
b) The maximum possible volume is [tex]\approx 1185.19 \text{ in.}^3[/tex]
If x is the length of the side on the base of the box, and h is the height, then the whole volume, V, is given by the formula
[tex]V=x^2h[/tex]
a) From the question,
[tex]x+h=20\implies h=20-x[/tex]
Substituting back into the formula for the volume, we have
[tex]V=x^2(20-x)[/tex]
Which gives the formula of the volume in factored polynomial form.
b) Removing brackets from the formula for the volume, and differentiating;
[tex]V=20x^2-x^3\\\\\frac{dV}{dx}=40-3x^2[/tex]
The function has maxima and minima when [tex]\frac{dV}{dx}=0[/tex]
Solving
[tex]\frac{dV}{dx}=0\\\implies 40x-3x^2=0\\\implies x(40-3x)=0\\\implies \text{ either } x=0 \text{ or } x=\frac{40}{3}[/tex]
For maxima, [tex]\frac{d^2V}{dx^2}<0[/tex]
So, if we substitute each value of x gotten from the previous line in turn, We will get that the maxima occurs when [tex]x=\frac{40}{3}[/tex]
That is, the maximum possible volume occurs when the height is [tex]\frac{40}{3}[/tex]
So the maximum possible volume will be
[tex]V=x^2(20-x)\\V=(\frac{40}{3})^2(20-\frac{40}{3})\approx 1185.19 \text{ in.}^3[/tex]
Learn more about maxima and minima here: https://brainly.com/question/2541083?
