SOLUTION: A rectangular box has a square base with edge at least one inch long. It has no top, and the total area of its five sides is 300 in2. What is the maximum possible volume of such a

Algebra ->  Volume -> SOLUTION: A rectangular box has a square base with edge at least one inch long. It has no top, and the total area of its five sides is 300 in2. What is the maximum possible volume of such a       Log On


   

Question 843003: A rectangular box has a square base with edge at least one inch long. It has no top, and the total area of its five sides is 300 in2. What is the maximum possible volume of such a box?
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
x = side of base which is square shape
y = height

Total area for five sides without top: x%5E2%2B4xy=300.
We will use this solved for y soon: y=%28300-x%5E2%29%2F4x.

V = Volume.
V=x%5E2%2Ay
V=x%5E2%2A%28300-x%5E2%29%2F4x
highlight%28V=75x-%281%2F4%29x%5E3%29
This might also be expressed as
highlight%28V=x%2875-%281%2F4%29x%5E2%29%29 although the previous form allows use of calculus derivative.

Derivative.
dV%2Fdx=75-%283%2F4%29x%5E2.
The maximum occurs for 75-%283%2F4%29x%5E2=0
%283%2F4%29x%5E2=75
x%5E2=75%284%2F3%29
x%5E2=100
highlight%28x=10%29
-
Knowing the x value, find the y value from again, y=%28300-x%5E2%29%2F4x;
y=%28300-100%29%2F%284%2A10%29
y=200%2F40
highlight%28y=5%29
-
Maximum Volume: x%5E2%2Ay=100%2A5=highlight%28highlight%28500%29%29