A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder must have a volume of 4000 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize cost.
I got as far as finding the radius to equal around 13.3067 but after plugging that value into my h function, i'm getting a negative number, I think I'm doing something wrong?
Found the answer, it was a calculator error but r is actually~6.2035 leading h~24.8141
To solve this problem, we need to find the dimensions (radius and height) of the solid that will minimize the cost. Let's consider the following steps:
Step 1: Express the volume of the solid.
The given volume is 4000 cubic feet. Since the solid comprises a cylinder and two hemispheres, we can express the volume as follows:
Volume = Volume of Cylinder + 2 * Volume of Hemisphere
The volume of a cylinder is given by the formula:
Volume of Cylinder = π * radius^2 * height
The volume of a hemisphere is given by the formula:
Volume of Hemisphere = (2/3) * π * radius^3
Plugging these equations into the given volume equation, we get:
4000 = π * radius^2 * height + 2 * (2/3) * π * radius^3
Step 2: Express the cost function.
The cost of the sides of the solid is C1, and the cost of the hemispheres is twice that of the sides, so the cost of the hemispheres is 2C1.
The cost of the sides can be expressed as:
Cost of Sides = C1 * Surface Area of Sides
The cost of the hemispheres can be expressed as:
Cost of Hemispheres = 2C1 * Surface Area of Hemispheres
The surface area of the sides of the solid is given by the formula:
Surface Area of Sides = 2 * π * radius * height
The surface area of a hemisphere is given by the formula:
Surface Area of Hemisphere = 2 * π * radius^2
Plugging these equations into the cost function, we get:
Cost = C1 * 2 * π * radius * height + 2C1 * 2 * π * radius^2
Step 3: Express either the height or the radius in terms of the other variable.
Since we already have an equation with the volume and the cost, we can express the height in terms of the radius. To do this, solve the volume equation for height:
height = (4000 - 2 * (2/3) * π * radius^3) / (π * radius^2)
Step 4: Substitute the expression for height into the cost function.
By substituting the expression for height into the cost function, we obtain a cost function in terms of the radius only.
Step 5: Differentiate the cost function with respect to the radius and solve for the critical points.
Differentiate the cost function with respect to the radius and set it equal to zero to find the critical points. This will give you the radius value that minimizes the cost.
Step 6: Substitute the optimal radius value back into the height expression to find the corresponding height value.
Step 7: Calculate the optimal cost by substituting the optimal radius and height values into the cost function.
By following these steps, you should be able to find the dimensions that will minimize the cost. Keep in mind that the negative number you mentioned may indicate a calculation error, so be sure to double-check your calculations.