Questions: 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 4 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. If r is the radius of the hemisphere, h the height of the cylinder, and V the volume, then r= and h= . (Type exact answers, using π as needed.)

Solution Steps

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 part. This can be expressed as: \[ V = \pi r^2 h + \frac{2}{3} \pi r^3 \]

From the volume constraint, we can solve for the height \( h \) in terms of the radius \( r \): \[ h = \frac{V}{\pi r^2} - \frac{2}{3} r \]

The total cost \( C \) of construction is based on the surface area of the cylindrical sidewall and the hemisphere. The cost function can be expressed as: \[ C = 2 \pi r h + 4 \cdot 2 \pi r^2 \] Substituting \( h \) from Step 2 into the cost function gives: \[ C = 2 \pi r \left( \frac{V}{\pi r^2} - \frac{2}{3} r \right) + 8 \pi r^2 \]

To find the optimal dimensions that minimize the cost, we differentiate the cost function \( C \) with respect to \( r \) and set the derivative equal to zero: \[ \frac{dC}{dr} = 0 \]

Solving the derivative equation yields the critical points for \( r \): \[ r = \frac{V^{1/3}}{2.754} \quad \text{(approximately)} \] This can be expressed as: \[ r = 0.36278316785978 V^{1/3} \]

Substituting the optimal radius back into the expression for height \( h \) gives: \[ h = 2.17669900715869 V^{1/3} \] This can be expressed as: \[ h = \frac{V^{1/3}}{0.459} \quad \text{(approximately)} \]

The optimal dimensions for the silo, in terms of the fixed volume \( V \), are: \[ r = 0.36278316785978 V^{1/3} \quad \text{and} \quad h = 2.17669900715869 V^{1/3} \]

Final Answer