Solution:
Given:
[tex]V=2w^3-7w^2+3w[/tex]a) When the width is 5 inches,
Substitute w = 5 into the equation.
[tex]\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}[/tex]Therefore, the volume of the prism is 90 cubic inches.
b) Factor the polynomial
[tex]\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}[/tex]Therefore, the completely factored polynomial is w(w-3)(2w-1)
[tex]\begin{gathered} w\text{ is the width} \\ 2w-1\text{ can be the length} \\ w-3\text{ can be the height} \end{gathered}[/tex]c) If w = 5 inches;
[tex]\begin{gathered} l=2w-1 \\ l=2(5)-1 \\ l=10-1 \\ l=9inches \\ \\ \\ \\ h=w-3 \\ h=5-3 \\ h=2inches \end{gathered}[/tex]It relates using the formula of volume of a rectangular prism;
[tex]\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}[/tex]d) The graph of the function is shown below;
The x-intercepts are;
[tex]w=0,w=0.5,w=3[/tex]In terms of the situation, the x-intercepts means when
