A company wants to build a cylindrical container with a semi-sphere lid.
For a fixed volume V , the company wants to use a minimal amount of material for container
and lid combined. Which radius r and height h of the container minimize the surface area for
container and lid combined?
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Sorry about that, computer must've lagged and double posted or something!
To find the dimensions that minimize surface area for the container and lid combined, we need to consider the volume and surface area formulas for a cylinder and a hemisphere.
Let's analyze the problem step by step:
1. Define the variables:
- r: The radius of the cylinder and the hemisphere (since they share the same radius)
- h: The height of the cylinder
2. Formulate the given constraints:
- The volume of the cylindrical container is fixed to V.
- The volume of the cylinder is given by V1 = πr^2h.
- The volume of the hemisphere is given by V2 = (2/3)πr^3.
3. Express height h in terms of r:
- From the cylindrical volume formula, we have h = V/(πr^2).
4. Express the surface area (S) in terms of r:
- For the cylinder, the surface area of the curved part is A1 = 2πrh.
- The surface area of the top and bottom is A2 = 2πr^2.
- For the hemisphere, the surface area is A3 = 2πr^2 (since it is only a half-sphere).
So, the total surface area is S = A1 + A2 + A3 = 2πrh + 2πr^2 + 2πr^2 = 2πrh + 4πr^2.
5. Substitute the expression for h in terms of r into the surface area equation:
- S = 2πr(V/(πr^2)) + 4πr^2 = 2V/r + 4πr^2.
6. We need to minimize S, so we take the derivative with respect to r and set it to zero to find the critical points:
dS/dr = -2V/r^2 + 8πr = 0.
Simplifying this equation:
8πr = 2V/r^2,
r^3 = V/(4π),
r = (V/(4π))^(1/3).
7. Once we have the value of r, we can substitute it back into the equation for h in terms of r:
h = V/(πr^2).
h = (4πV^2/4π^2)^(1/3) = (V/π)^(1/3).
Therefore, the radius that minimizes the surface area is r = (V/(4π))^(1/3), and the corresponding height is h = (V/π)^(1/3).
This means that for a fixed volume V, the company should build a cylindrical container with a semi-sphere lid using a radius of (V/(4π))^(1/3) and a height of (V/π)^(1/3) to minimize the surface area for the container and lid combined.