Making a cylinder is costing your company $2/in^2 for top and bottom and $6/in^2 for the body(side). If the cylinder is to have 166.25oz volume, what are the most economical cylinder that you can make and what will be the cost of making it.
let radius be r inches
let the height be h inches
vol = πr^2h
166.25 = πr^2 h
h = 166.25/(πr^2)
"economical cylinder" is a function of the surface area, SA
SA = 2 circles + rectangles of the side
cost = 2(2πr^2) + 6(2πrh)
= 4πr^2 + 12πrh
= 4πr^2 + 12πr(166.25/(πr^2))
= 4πr^2 + 1995/r
d(cost)/dr = 8πr - 1995/r^2 = 0 for a max/min of SA
8πr = 1995/r^2
r^3 = 249.375/π
r = appr 4.298 inches
h = appr 2.865 inches
check my arithmetic
To find the most economical cylinder and the cost of making it, we need to calculate the surface area of different cylinders and compare their costs.
Let's denote:
- r as the radius of the cylinder's top and bottom (in inches),
- h as the height of the cylinder (in inches),
- A_top_bottom as the surface area of the top and bottom of the cylinder, and
- A_body as the surface area of the body (side) of the cylinder.
For a cylinder, the volume V is given by V = π * r^2 * h.
We are given that the volume should be 166.25 oz, but we need to convert it to cubic inches since our costs are given in terms of inches. There are 1 fluid ounce (oz) equals 1.80469 cubic inches. Therefore, the volume is V = 166.25 * 1.80469 = 299.78156 cubic inches.
Using the formula for the volume of a cylinder, we can solve for h:
299.78156 = π * r^2 * h
Now, let's calculate the surface areas:
A_top_bottom = 2 * π * r^2 (since there are two top/bottom surfaces)
A_body = 2 * π * r * h (since the body is a rectangular shape when unrolled)
Now, let's find the most economical cylinder by considering the cost.
The cost C is given by:
C = Cost_top_bottom * A_top_bottom + Cost_body * A_body
We are given that the cost for the top and bottom is $2/in^2 and the cost for the body is $6/in^2.
To find the most economical cylinder, we need to minimize the cost function while maintaining the volume constraint.
1. Determine the range of values for r and h that make sense for a cylinder, considering practical constraints.
2. Using the volume equation, solve for h in terms of r and substitute the result into the cost equation.
3. Differentiate the cost function with respect to r and set it equal to zero to find the critical points.
4. Analyze those critical points to find which one corresponds to the minimum cost.
By following these steps, we can find the dimensions (radius and height) and the cost of the most economical cylinder.