Q:

# A silo is in the shape of a right circular cylinder roofed with a hemisphere. If the silo is to have a capacity of 200 cubic feet, determine the value of H for the silo that requires the least material of construction.

Accepted Solution

A:
Let's start by finding the formula for the volume of the silo. The volume consists of the cylinder and the hemisphere on top of it. Let r be the radius of the cylinder and the hemisphere and let h be the height of the cylinder. Then: Volume of cylinder = πr²h Volume of hemisphere = (2/3)πr³/2 = (1/3)πr³ Total volume of silo = πr²h + (1/3)πr³ We want to find the value of h that requires the least material of construction, which means we want to minimize the surface area of the silo. The surface area consists of the curved surface of the cylinder and the curved surface of the hemisphere. Let S be the surface area, then: Surface area of cylinder = 2πrh Surface area of hemisphere = 2πr² Total surface area of silo = 2πrh + 2πr² Now we can use the volume formula to solve for h in terms of r: 200 = πr²h + (1/3)πr³ h = (200 - (1/3)πr³) / πr² Substituting this expression for h into the surface area formula, we get: S = 2πr((200 - (1/3)πr³) / πr²) + 2πr² S = 400/r + (2/3)πr² To find the value of r that minimizes S, we take the derivative of S with respect to r and set it equal to zero: dS/dr = -400/r² + (4/3)πr = 0 400/r² = (4/3)πr r³ = 300/(π/3) r = 4.44 feet (rounded to two decimal places) Now we can use the formula for h to find the corresponding value of h: h = (200 - (1/3)π(4.44)³) / π(4.44)² h = 7.78 feet (rounded to two decimal places) Therefore, the value of H for the silo that requires the least material of construction is approximately 7.78 feet.