amriaf
Dec 16, 2007, 06:57 PM
A cylindrical container costs $3.00 per square foot for the sides and $4.00 a square foot for the top and bottom. The container must hold 100 cubic feet of material. What are the dimensions of the most economical container.
Nosnosna
Dec 16, 2007, 07:24 PM
To solve this problem, you must find the minimum value for one of the variables in a specialized from of the formula the surface area of a cylinder, which is known to be
A = 2 * pi * r ^2 + 2 * pi * r * h
The specialized form for the price, based on the values in the given problem is:
P = (2 * pi * r^2) * 4 + (2 * pi * r * h) * 3
r and h are related in the formula for the volume, which is:
V = h * pi * r^2
Since you know that V = 100, you can solve for either of the variables in the volume function:
100 = h * pi * r^2
h = 100 / (pi * r^2)
Substitute that into the formula with values above, and you get
P = (2 * pi * r^2) * 4 + (2 * pi * r * (100 / (pi * r^2))) * 3
Now you just have to find the value of r that gives the smallest value for A