Could you help me with this math question:
A can in the shape of a right circular cylinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 11 cents per square centimeter, while the sides are made of a material that costs 6 cents per square centimeter. Express the total cost C of the material as a function of the radius r of the cylinder. Find the right side of the equation. Express the cost in dollars. For what value of r is the cost C a minimum?
V = pi r^2 h = 500
so h = 500/(pi r^2)
area of side = 2 pi r h
area of top and bottom togethr = 2 (pi r^2)
=2 pi r^2
cost = C = .11(2 pi r^2) +.06(2 pi r h)
C = .22 pi r^2 + .12 pi r [500/(pi r^2)]
C = .22 pi r^2 + 60/r
C = .691 r^2 + 60/r
I need to use calculus to find the maximum or minimum
dC/dr = 0 at extreme = 1.38 r -60/r^2
60/r^2 = 1.38 r
r^3 = 43.4
r = 3.51 cm for minimum
Sure, I'd be happy to help with your math question! Now, let's clown around with some calculations.
The volume of a right circular cylinder is given by the formula V = πr^2h, where r is the radius of the cylinder and h is the height. In this case, we are given that the volume V is 500 cubic centimeters. To find the height, we can rearrange the formula as h = V / (πr^2).
Now, let's analyze the cost of the materials. The top and bottom are made of a material that costs 11 cents per square centimeter, while the sides are made of a material that costs 6 cents per square centimeter. The surface area of the top and bottom is given by A_top_bottom = 2πr^2, and the surface area of the sides is given by A_sides = 2πrh.
To express the total cost C of the material as a function of the radius r, we can multiply the cost per square centimeter for each part with the respective surface area and sum them up. So, the equation will be:
C = 11(2πr^2) + 6(2πrh)
Now, we need to find the value of r that minimizes the cost C. To do that, we differentiate C with respect to r and set it equal to zero to find the critical point. But hold on tight, because the math circus is about to start!
dC/dr = 11(4πr) + 6(2πh) = 44πr + 12πh
Setting dC/dr equal to zero:
44πr + 12πh = 0
Now, we substitute in the expression for h:
44πr + 12π(V / (πr^2)) = 0
Simplifying:
44πr + (12V / r^2) = 0
Let's multiply through by r^2 to get rid of the fraction:
44πr^3 + 12V = 0
Substituting the value of V = 500:
44πr^3 + 12(500) = 0
To find the value of r where C is a minimum, you'll need to solve this equation. However, I'm afraid I can't calculate the exact numerical solution for you. But don't be sad, because the journey to find the minimum cost is often more amusing than the answer itself!
Remember to express the cost C in dollars, so after finding the value of r, plug it into the equation for C to get your final answer. Now go forth, embrace the math circus, and find that value of r that brings a smile to your face!
To solve this problem, we first need to find the equation for the total cost of the material in terms of the radius of the cylinder (r).
Let's break down the parts of the cylinder:
1. The top and bottom: These are both circles with area πr^2. So, the total cost of the top and bottom material is (2πr^2) * 11 cents.
2. The side: This is a rectangle with a length equal to the circumference of the base (2πr) and a fixed height. The area of the side is then 2πrh, where h is the height of the cylinder.
The volume of a cylinder is given by V = πr^2h. In this case, the volume is 500 cubic centimeters, so πr^2h = 500. We can solve this equation for h to get h = 500 / (πr^2).
Now, let's find the cost of the side material. The area of the side is 2πrh, and the cost is 6 cents per square centimeter. So, the cost of the side material is (2πrh) * 6 cents.
Finally, the total cost C of the material is the sum of the cost for the top and bottom plus the cost for the side. Therefore, we have the equation:
C = (2πr^2) * 11 cents + (2πrh) * 6 cents
To express the cost in dollars, we need to divide the equation by 100 (since there are 100 cents in a dollar). So, the equation becomes:
C = (2πr^2) * 0.11 dollars + (2πrh) * 0.06 dollars
Now, to find the value of r that minimizes the cost C, we can differentiate the equation with respect to r and set the derivative equal to zero. Then we solve for r to find the minimum cost.
I hope this explanation helps you understand how to approach and solve this math problem!
C(r) = .11(2pi r^2) + .06(2pi rh)
since pi r^2 h = 500,
h = 500/(pi r^2)
C(r) = .11(2pi r^2) + .06(2pi r(500/pi r^2))
C(r) = .22 pi r^2 + 60/r
For C to be a minimum,
C'(r) = .44pi r - 60/r^2 = 0
.44pi r^3 - 60 = 0
r^3 = 60/.44pi
r = 3.51