Express the length and the width of the cargo space in terms of its height.
B
b
Use the function found in Part A to determine the dimensions of the cargo space.
A
a
V=2h^3-8h^2-64h
B
b
No, see solution.
Practice makes perfect
a
We can represent the cargo space as a prism. Recall the formula for the volume of a prism.
V=Bh
Here B is the area of the base of the prism and h is the height perpendicular to that base. In our case the base of the prism is a rectangle. Recall the formula for the area of a rectangle.
A=l wHere l is the length of the rectangle and w is its width. We will use this information to rewrite the formula for the volume of the cargo space.
V=l wh
We want to write a polynomial function that represents the volume of the cargo space. The length and the width of the cargo space are described in terms of the height h. Thus, our function will depend on one variable, h. Let's express the length and the width in terms of h. The length is 4 inches greater than the height.
l=h + 4
The width is 16 inches less than twice the height.
w= 2h-16
Finally we will substitute the two above expressions into the formula.
The polynomial function that represents the volume of the cargo space in terms of its height h is V=2h^3-8h^2-64h. Please note that h is expressed in inches.
b
First we have to find the dimensions of the cargo space. Then we can compare these dimensions to the dimensions of the package. Let's do it!
Finding the Dimensions of the Cargo Space
To find the dimensions of the cargo space, we will use the function found in Part A that represents the volume V of the cargo space in terms of its height h.
V=2h^3-8h^2-64h
We know that the volume of the cargo space is 55 296 cubic inches.
55 296=2h^3-8h^2-64h
Since the height h can only be positive, we are looking for a positive realzero. Let's start by rewriting our polynomial function in standard form with the leading coefficient equal to 1.
We will use a graphing calculator to find the positive real zero of the function.
We can see that the only positive real zero is x=32, so the height h of the cargo space is 32 inches. Knowing the height we can find the length l and the width w of the cargo space. In Part A we expressed l and w in terms of h.
ll=h+4 w=2h-16 ⇒ ll= 32+4 w=2( 32)-16
We have that the length of the cargo space is 36 inches and its width is 48 inches.
Comparing the Dimensions
Let's compare the dimensions of the cargo space and the package.
Dimension
The Cargo Space
The Package
Length
36 inches
34 inches
Width
48 inches
44 inches
Height
32 inches
34 inches
We can see that the length and the width of the package are less than the length and the width of the cargo space, respectively. However, the height of the package is greater than the height of the cargo space. Therefore, the package is too tall to fit in the cargo space.
