a right circular cylinder has a volume 600cubic cm. if it costs $2 for the material needed to make the top and bottom and $3 for the material needed to make the sides, what is the radius and height of the cylinder that would cost the least?
To find the dimensions of the cylinder that would cost the least, we need to consider the cost of both the top and bottom (which are in the shape of circles) and the sides (which form a rectangle when unrolled).
Let's denote the radius of the cylinder as r and the height as h.
The volume of a right circular cylinder is given by the formula:
V = πr^2h
Given that the volume is 600 cubic cm, we have:
600 = πr^2h
To minimize the cost, we can express it as a function of r and h. The cost consists of the material required for the top and bottom (2 circles) and the material required for the sides (rectangle).
Cost = (2πr^2)($2) + (2πrh)($3)
Simplifying this expression, we get:
Cost = 4πr^2 + 6πrh
Now, we can substitute the volume equation (600 = πr^2h) into the cost equation:
Cost = 4πr^2 + (6πr(600/πr^2))
= 4πr^2 + (3600/r)
To find the minimum cost, we need to take the derivative of the cost equation with respect to r, set it equal to zero, and solve for r:
dCost/dr = 8πr - (3600/r^2) = 0
Simplifying and solving for r, we get:
8πr = 3600/r^2
r^3 = 450/π
r ≈ 6.768 cm
Now that we have the value of r, we can substitute it back into the volume equation to find the height (h):
600 = π(6.768)^2h
Simplifying and solving for h, we get:
h ≈ 4.221 cm
Therefore, the dimensions of the cylinder that would cost the least are approximately r = 6.768 cm and h = 4.221 cm.