Question

Help! So confused D:

You plan to build a cylindrical container with no top. The material used for the lateral sides costs $8 per square foot while the material for the bottom costs $7 per square foot.

(a) Express the total cost of building the cylinder, C in terms of the cylinder's radius, r, and height, h.

C =

(b) If the volume of the cylindrical container is 350 cubic feet, express the total cost of building the cylinder, C in terms of the cylinder's radius, r.

C(r) =

Answer 1

The total cost of building the cylinder is represented by C = 8(2πrh) + 7(πr²). When the volume is 350 cubic feet, the cost in terms of the radius r is C(r) = (16 × 350)/r + 7πr².

Step-by-step explanation:

To calculate the total cost of building the cylinder, we need to calculate the surface area of the lateral sides and the bottom separately, as they have different costs per square foot. The lateral surface area is the perimeter of the base circle (which is 2πr) times the height (h), and the area of the bottom is the area of the circle (πr²).

Therefore:

The cost of the lateral sides is (2πrh) × $8 and the cost of the bottom is (πr²) × $7. Adding these two costs together, the total cost of building the cylinder C is:

C = 8(2πrh) + 7(πr²)

To express the total cost in terms of the cylinder's radius r when the volume is 350 cubic feet, recall that V = πr²h. We solve for height h to get:

h = V/(πr²)

Substituting this value of h into the previous cost equation, we get:

C(r) = 8(2πr(V/(πr²))) + 7(πr²)

C(r) = 16V/r + 7πr²

Given V = 350 cubic feet, we then have:

C(r) = (16 × 350)/r + 7πr² which expresses the cost as a function of the radius r.

Answer 2

The area (A) of the cylinder is the sum of two areas of its base (A1) (because it has two bases) and lateral surface area (A2):
A = 2A1 + A2
However, our cylinder has no top, so its area is: A = A1 + A2
The area of the circular base is: A1 = r²π
The area of lateral surface is: A2 = 2rhπ

A) The answer is C = 7r²π + 16rπh

The cost C of building the cylinder is:
C = 7 * A1 + 8 * A2
C = 7 * r²π + 8 * 2rπh
C = 7r²π + 16rπh

B) The answer is C = 7r²π + 5600/r

The volume of the cylinder is: V = r²πh
We have: V = 350 ft³
Hence: 350 = r²πh
h = 350/r²π

From the previous task (a) we have the total cost:
C = 7r²π + 16rπh
Let's just substitute h:
C = 7r²π + 16rπ(350/r²π)
C = 7r²π + 5600/r