Question

You are designing an aquarium for your new office. The dimensions of the aquarium are restricted as shown in the diagram below.

Dimensions:
Height: (w+9)in.
Width: w in.
Length: (49-w)in.

Write a polynomial expression that represents the volume of the aquarium according to the specified dimensions. (I got w(w+9)(49-w))

You need the aquarium to hold 17,640 cubic inches of water. Find the possible dimensions of the aquarium.

Answer 1

Part 1)

`V=w(w+9)(49-w)`

Part 2)

We can have the aquarium have a width of 40 inches, a height of 49 inches, and a length of 9 inches.

Or we can have the aquarium have a width of 21 inches, a height of 30 inches, and a length of 28 inches.

Explanation:

The dimensions of the aquarium is as follows:

`\text{Height}= (w+9)\text{ inches}\\\text{Width} = w\text{ inches} \text{ and } \\\text{Length} = (49-w)\text{ inches}`

Part 1)

We want to find a polynomial expression that represents the volume of the aquarium according to the specified dimensions.

Since the aquarium is a rectangular prism, the volume will simply be the product of all the dimensions. Hence:

`V=w(w+9)(49-w)`

Part 2)

We want to aquarium to hold 17, 640 cubic inches of water. In other words, the volume V should be 17640. Thus:

`17640=w(w+9)(49-w)`

First, we can distribute the right-hand side:

`17640=(w^2+9w)(49-w)`

Distribute:

`(w^2+9w)(49)+(w^2+9w)(-w)=17640`

Distribute:

`(49w^2+441w)+(-w^3-9w^2)=17640`

Simplify:

`-w^3+40w^2+441w-17640=0`

Now, we can factor. From the first two terms, factor out a -w² and from the second two terms, we can factor out 441:

`-w^2(w-40)+441(w-40)=0`

Grouping:

`(441-w^2)(w-40)=0`

Zero Product Property:

`441-w^2=0\text{ or } w-40=0`

Solve for each case:

`w=21\text{ or } w= 40`

So, we have two possible sets of dimensions.

We can have the aquarium have a width of 40 inches, a height of 49 inches, and a length of 9 inches.

Or we can have the aquarium have a width of 21 inches, a height of 30 inches, and a length of 28 inches.