Question

The following equation represents the volume of a rectangular prism with a width of winches:V= 2w^3 - 7w^2 +3wa. What is the volume if the width is 5 inches?b. Factor this polynomial completely and describe what each factor means in terms of thedimensions of the rectangular prism.c. If the width is 5 inches, what are the other dimensions? How does this relate to youranswer to part a?d. Graph the polynomial on a graphing calculator or an online graphing application. Whatare the x-intercepts? What do these mean in terms of the situation?e. What are the domain and range in terms of the situation? Justify your answers.

Answer 1

Solution:

Given:

$V=2w^3-7w^2+3w$

a) When the width is 5 inches,

Substitute w = 5 into the equation.

$\begin{gathered} V=2(5^3)-7(5^2)+3(5) \\ V=2(125)-7(25)+15 \\ V=250-175+15 \\ V=90\text{ }in^3 \end{gathered}$

Therefore, the volume of the prism is 90 cubic inches.

b) Factor the polynomial

$\begin{gathered} 2w^3-7w^2+3w=w(2w^2-7w+3) \\ w(2w^2-w-6w+3)=w(w(2w-1)-3(2w-2)) \\ =w(w-3)(2w-1) \end{gathered}$

Therefore, the completely factored polynomial is w(w-3)(2w-1)

$\begin{gathered} w\text{ is the width} \\ 2w-1\text{ can be the length} \\ w-3\text{ can be the height} \end{gathered}$

c) If w = 5 inches;

$\begin{gathered} l=2w-1 \\ l=2(5)-1 \\ l=10-1 \\ l=9inches \\ \\ \\ \\ h=w-3 \\ h=5-3 \\ h=2inches \end{gathered}$

It relates using the formula of volume of a rectangular prism;

$\begin{gathered} V=lbh \\ V=9\times5\times2 \\ V=90in^3 \\ \\ The\text{ volume in part \lparen a\rparen is also }90in^3 \end{gathered}$

d) The graph of the function is shown below;

The x-intercepts are;

$w=0,w=0.5,w=3$

In terms of the situation, the x-intercepts means when