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--
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
thank you very much
To find the maximum volume of the aquarium, we need to determine the dimensions of the square base and the height of the aquarium.
Let's assume that the side length of the square base is x, and the height of the aquarium is h.
The area of the four sides and the bottom can be calculated by finding the sum of the areas of the sides and the bottom of the aquarium.
The bottom has an area equal to x^2 (since it is a square), and there are four sides, each with an area equal to xh (length times height).
So the total area is the sum of the bottom and the four sides:
Area = x^2 + 4xh
We are given that the total material we have to build the aquarium is 60 square feet. Therefore, we can set up the following equation:
60 = x^2 + 4xh
Now, we want to find the maximum volume of the aquarium, which is given by the formula:
Volume = x^2h
To find the maximum volume, we need to express the equation for the area (60 = x^2 + 4xh) in terms of h.
Let's solve the area equation for h:
60 - x^2 = 4xh
Divide both sides of the equation by 4x:
(60 - x^2) / (4x) = h
Now, substitute this value of h into the equation for volume:
Volume = x^2h
Volume = x^2 * [(60 - x^2) / (4x)]
Simplify the equation:
Volume = (x^2 * (60 - x^2)) / (4x)
Volume = (60x - x^3) / 4
To find the maximum volume, we need to find the critical points of the volume equation. We can do this by finding the derivative of the volume equation, setting it equal to zero, and solving for x.
Take the derivative of the volume equation:
dV/dx = (60 - 3x^2) / 4
Set the derivative equal to zero and solve for x:
(60 - 3x^2) / 4 = 0
60 - 3x^2 = 0
3x^2 = 60
x^2 = 20
x = √20 or x = -√20 (we discard the negative value)
Therefore, the critical point is x = √20.
Now, substitute this value of x back into the volume equation to find the maximum volume:
Volume = (60x - x^3) / 4
Volume = (60√20 - (√20)^3) / 4
Simplify the equation:
Volume = (60√20 - 20√20) / 4
Volume = (40√20) / 4
Volume = 10√20
Rationalizing the square root:
Volume = 10√(4 * 5)
Volume = 10 * 2√5
Volume = 20√5
Therefore, the maximum volume that this aquarium might have is 20√5 cubic feet.
I hope this explanation helps you understand how to find the maximum volume of the rectangular aquarium!