A silo​ (base not​ included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is 9 times as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.
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Part 1
If r is the radius of the​ hemisphere, h the height of the​ cylinder, and V the​ volume, then requals

enter your response here and hequals

enter your response here.
​(Type exact​ answers, using pi as​ needed.)

Let's solve this problem step by step.

### Part 1: Expressing \( r \) and \( h \) in terms of \( V \).

#### Volume Equation:
The volume \( V \) of the silo is the sum of the volume of the cylindrical part and the hemispherical part.

- Volume of the cylinder = \( \pi r^2 h \)
- Volume of the hemisphere = \( \frac{2}{3} \pi r^3 \)

Thus, the total volume \( V \) is:

\[
V = \pi r^2 h + \frac{2}{3} \pi r^3
\]

#### Express \( h \) in terms of \( V \) and \( r \):
We rearrange the equation for \( h \):

\[
h = \frac{V - \frac{2}{3} \pi r^3}{\pi r^2}
\]

Thus, we have:

\[
h = \frac{V}{\pi r^2} - \frac{2}{3} r
\]

Now, we have both \( r \) and \( h \) expressed in terms of the volume \( V \) and the radius \( r \).

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A silo​ (base not​ included) is to be constructed in the form of a cylinder surmounted by a hemisphere. The cost of construction per square unit of surface area is 9 times as great for the hemisphere as it is for the cylindrical sidewall. Determine the dimensions to be used if the volume is fixed and the cost of construction is to be kept to a minimum. Neglect the thickness of the silo and waste in construction.

Learn how to solve an optimization problem involving a silo constructed as a cylinder surmounted by a hemisphere. This problem involves minimizing construction costs while maintaining a fixed volume. Key mathematical concepts include volume calculation, surface area, and cost minimization using calculus. Suitable for advanced high school or undergraduate students.

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