Questions: A grain silo consists of a cylindrical concrete tower surmounted by a metal hemispherical dome. The metal in the dome costs 1.8 times as much as the concrete (per unit of surface area). If the volume of the silo is 800 m^3, what are the dimensions of the silo (radius and height of the cylindrical tower) that minimize the cost of the materials? Assume the silo has no floor and no flat ceiling under the dome. What is the function of the cost of the silo, C, in terms of the radius, n c ≡ (Type an expression. Type an exact answer, using z as needed.)

Solution Steps

To solve this problem, we need to:

1. Express the volume of the silo in terms of the radius and height of the cylindrical part.
2. Express the surface area of the cylindrical part and the hemispherical dome.
3. Formulate the cost function in terms of the radius and height.
4. Use calculus to minimize the cost function.

1. The volume of the silo is the sum of the volume of the cylindrical part and the hemispherical dome.
2. The surface area of the cylindrical part and the hemispherical dome needs to be calculated.
3. The cost function is formed by multiplying the surface areas by their respective costs.
4. Use calculus to find the minimum cost by taking the derivative of the cost function and setting it to zero.

The total volume \( V \) of the silo is given by the sum of the volume of the cylindrical part and the volume of the hemispherical dome:

\[ V = \pi r^2 h + \frac{2}{3} \pi r^3 = 800 \]

Rearranging the volume equation to solve for the height \( h \) in terms of the radius \( r \):

\[ h = \frac{800 - \frac{2}{3} \pi r^3}{\pi r^2} = -\frac{2}{3} r + \frac{800}{\pi r^2} \]

The surface area \( A \) of the cylindrical part and the hemispherical dome is given by:

\[ A_{\text{cylinder}} = 2 \pi r h \] \[ A_{\text{hemisphere}} = 2 \pi r^2 \]

Substituting \( h \) into the surface area of the cylinder:

\[ A_{\text{cylinder}} = 2 \pi r \left(-\frac{2}{3} r + \frac{800}{\pi r^2}\right) = -\frac{4}{3} \pi r^2 + \frac{1600}{r} \]

The cost function \( C \) is defined as:

\[ C = A_{\text{cylinder}} + 1.8 A_{\text{hemisphere}} \]

Substituting the surface area expressions:

\[ C = \left(-\frac{4}{3} \pi r^2 + \frac{1600}{r}\right) + 1.8 \left(2 \pi r^2\right) \]

This simplifies to:

\[ C = \pi \left(2.2667 r^3 + 509.2958\right) \cdot \frac{1}{r} \]

Final Answer