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This is a good calculus question. Have you bothered trying this at all? Or are you just giving up completely?
You know that the volume of a cylinder is pi r^2 h. If r and h are in centimeters, then the volume will be in cubic centimeters, which is the same as mL. So we at least have to pick r and h such that pi r^2 h = 355. That means h = 355 / (pi r^2).
The surface area of a cylinder is the top (pi r^2) plus the bottom (also pi r^2) times the lateral side (2 pi r h). So the total surface area is 2 pi r^2 + 2 pi r^2 + 2pi r h, or 4 pi r^2 + 2 pi r h. Substitute h = 355 / (pi r^2) into this to get the surface area in terms of just r.
Now that you have the surface area in terms of just r, you can find the value of r that minimizes this by taking the derivative, stetting it equal to 0, and solving for r. Use that to find h. You'll find that the dimensions are different from an actual soda can, but I'm sure you can think of why this is the case.
THE MATH PROBLEM:
The surface area of a cylindrical aluminum can is measure of how much aluminum the can requires. If the can has a radius r and a height h, its surface area A and its volume V are given by the equations:
A=2(pi)r^2 + 2(pi)rh and V= (pi)r^2h
A) The volume, V, of a 12 oz cola can is 355cm^3. A cola can is approximately cylindrical. Express the cola can's surface area A as a function of its radius r, where r is measured in centimeters. Simlify your answer. (Hint: Your function should only contain two variables, A and r).
B) The manufacturers wish to use the least amount of aluminum (in centimeters squared) necessary to make hte 12 oz cola can. Use your answer in part A to find the minimum amount of aluminum needed. State the values of r and h that minimize the amount of aluminum used.
PART A
First find an expression to represent h, since you do not want h in the function. Plug 355 in as the volume and solve for h:
v = pi (r^2) (h)
355 = pi (r^2) (h)
355 / (pi r^2) = h
Use
You know that the volume of a cylinder is pi r^2 h. If r and h are in centimeters, then the volume will be in cubic centimeters, which is the same as mL. So we at least have to pick r and h such that pi r^2 h = 355. That means h = 355 / (pi r^2).
The surface area of a cylinder is the top (pi r^2) plus the bottom (also pi r^2) times the lateral side (2 pi r h). So the total surface area is 2 pi r^2 + 2 pi r^2 + 2pi r h, or 4 pi r^2 + 2 pi r h. Substitute h = 355 / (pi r^2) into this to get the surface area in terms of just r.
Now that you have the surface area in terms of just r, you can find the value of r that minimizes this by taking the derivative, stetting it equal to 0, and solving for r. Use that to find h. You'll find that the dimensions are different from an actual soda can, but I'm sure you can think of why this is the case.
THE MATH PROBLEM:
The surface area of a cylindrical aluminum can is measure of how much aluminum the can requires. If the can has a radius r and a height h, its surface area A and its volume V are given by the equations:
A=2(pi)r^2 + 2(pi)rh and V= (pi)r^2h
A) The volume, V, of a 12 oz cola can is 355cm^3. A cola can is approximately cylindrical. Express the cola can's surface area A as a function of its radius r, where r is measured in centimeters. Simlify your answer. (Hint: Your function should only contain two variables, A and r).
B) The manufacturers wish to use the least amount of aluminum (in centimeters squared) necessary to make hte 12 oz cola can. Use your answer in part A to find the minimum amount of aluminum needed. State the values of r and h that minimize the amount of aluminum used.
PART A
First find an expression to represent h, since you do not want h in the function. Plug 355 in as the volume and solve for h:
v = pi (r^2) (h)
355 = pi (r^2) (h)
355 / (pi r^2) = h
Use