Question

You have been asked to construct an aquarium that holds 1000 gallons. It will be a standard rectangular prism with restrictions that the length must be 1 foot longer than the width and the height must be 4.6 feet less than the length. Find the height of the aquarium to the nearest hundredth of an inch.

Answer 1

The height of the aquarium is
`33.75\ in`

Explanation:

Remember that

`1\ gal=231\ in^3`

`1\ ft=12\ in`

we know that

The volume of a rectangular prism is equal to

`V=LWH`

In this problem we have

`V=1,000\ gal=1,000(231)=231,000\ in^3`

so

`231,000=LWH` ----> equation A

`L=W+1(12)` --->
`W=L-12` --->equation B (convert ft to in)

`H=L-4.6(12)`--->
`H=L-55.2` ---> equation C (convert ft to in)

substitute equation B and equation C in equation A

`231,000=L(L-12)(L-55.2)`

solve for L

Apply distributive property

`231,000=(L^2-12L)(L-55.2)`

`231,000=L^3-55.2L^2-12L^2+662.4L`

`L^3-67.2L^2+662.4L-231,000=0`

Solve the cubic equation by graphing

using a graphing tool

The solution is L=88.95 in

see the attached figure

Find the height of the aquarium H

`H=L-55.2`

substitute the value of L

`H=88.95-55.2=33.75\ in`

Find the value of W

`W=88.95-12=76.95\ in`