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.

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calculista calculista · Answer 1 · 2020-01-10T20:54:25+01:00

Answer:

The height of the aquarium is [tex]33.75\ in[/tex]

Step-by-step explanation:

Remember that

[tex]1\ gal=231\ in^3[/tex]

[tex]1\ ft=12\ in[/tex]

we know that

The volume of a rectangular prism is equal to

[tex]V=LWH[/tex]

In this problem we have

[tex]V=1,000\ gal=1,000(231)=231,000\ in^3[/tex]

so

[tex]231,000=LWH[/tex] ----> equation A

[tex]L=W+1(12)[/tex] ---> [tex]W=L-12[/tex] --->equation B (convert ft to in)

[tex]H=L-4.6(12)[/tex]--->[tex]H=L-55.2[/tex] ---> equation C (convert ft to in)

substitute equation B and equation C in equation A

[tex]231,000=L(L-12)(L-55.2)[/tex]

solve for L

Apply distributive property

[tex]231,000=(L^2-12L)(L-55.2)[/tex]

[tex]231,000=L^3-55.2L^2-12L^2+662.4L[/tex]

[tex]L^3-67.2L^2+662.4L-231,000=0[/tex]

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

[tex]H=L-55.2[/tex]

substitute the value of L

[tex]H=88.95-55.2=33.75\ in[/tex]

Find the value of W

[tex]W=88.95-12=76.95\ in[/tex]