Chapter 8.1, Problem 80SR

Solution

a)

Find the polynomial function that represents the cargo space.

The polynomial function that represents the cargo space is $V\left(h\right)=2h^3-8h^2-64h$

Given information: length of the cargo space is $h+4,$ the width of the cargo space is $2h-16$

Calculation:

Let $h$ be the height of the cargo space.

The length of the cargo space is $h+4$

The width of the cargo space is $2h-16$

Volume of the cargo space is $lbh$

  $\begin{array}{l}V\left(h\right)=h\left(h+4\right)\left(2h-16\right)\\ =2h\left(h+4\right)\left(h-8\right)\\ =2h\left(h^2-4h-32\right)\\ =2h^3-8h^2-64h\end{array}$

Thus, the polynomial function that represents the cargo space is $V\left(h\right)=2h^3-8h^2-64h$

b)

Find the package long, wide and tall will not fit the cargo space.

Find the polynomial function that represents the cargo space is $V\left(h\right)=2h^3-8h^2-64h$

The package with 34 in long, 44 in wide and 34 in tall will not fit the cargo space since the dimensions of the cargo space should be at least 34 inches long.

Given information: Given the polynomial function that represents the cargo space is $V\left(h\right)=55296$

Calculation:

The polynomial function that represents the cargo space is $V\left(h\right)=2h^3-8h^2-64h$

Given the polynomial function that represents the cargo space is $V\left(h\right)=55296$

  $\begin{array}{l}55296=2h^3-8h^2-64h\\ h=32\end{array}$

Hence $h=32in$

The length of the cargo space is $h+4=32+4=36$

The width of the cargo space is $2h-16=2\left(32\right)-16=48$

Thus, the package with 34 in long, 44 in wide and 34 in tall will not fit the cargo space since the dimensions of the cargo space should be at least 34 inches long.