Homework Help: calculus

Posted by Megan on Friday, July 8, 2011 at 10:02am.

A rectangular aquarium is to be built with an open top (no material needed) and a square base. You have 60 square feet of material to build it with. (We are assuming that the sides and bottom are perfectly flat). What is the maximum volume that this aquarium might have? Note-draw a picture, and note that the area of the four sides and bottom can be calculated: this is the amount of material that you will use.

--any help would be very appreciated--

calculus - Reiny, Friday, July 8, 2011 at 10:39am
let the base by x by x and the height y
(make a sketch)

surface area = x^2 + 4xy
x^2 + 4xy = 60
y = (60- x^2)/(4x)

volume= x^2(y) = x^2(60-x^2)/(4x)
= 15x - (1/4)x^3
d(volume)/dx = 15 - (3/4)x^2
= 0 for a max of volume

(3/4)x^2 = 15
x^2 = 20
x = √20

so max volume = 15√20 - (1/4)20√20 = 10√20 or 20√5
calculus - Reiny, Friday, July 8, 2011 at 11:00am
let the base by x by x and the height y
(make a sketch)

surface area = x^2 + 4xy
x^2 + 4xy = 60
y = (60- x^2)/(4x)

volume= x^2(y) = x^2(60-x^2)/(4x)
= 15x - (1/4)x^3
d(volume)/dx = 15 - (3/4)x^2
= 0 for a max of volume

(3/4)x^2 = 15
x^2 = 20
x = √20

so max volume = 15√20 - (1/4)20√20 = 10√20 or 20√5
calculus - Megan, Friday, July 8, 2011 at 11:33am
thank you very much

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