A company plans to manufacture a container having the shape of a right circular cylinder, open at the top, and having a capacity of 24pi cubic inches. If the cost of the material for the bottom is $.30 per square inch and that for the curved sides is $.10 per square inch, express the total cost C of the material as a function of the radius r of the base of the container.
Did they not ask for r and h for minimum cost?
No. This is exactly how the question was presented.
OK, wait til next year :)
To express the total cost of the material as a function of the radius of the base of the container, we first need to find the height of the cylinder.
The volume of a right circular cylinder is given by the formula:
V = πr²h
In this case, the capacity of the container is given as 24π cubic inches, so we have:
24π = πr²h
Dividing both sides of the equation by πr², we get:
h = 24/r²
Now, to calculate the cost of the materials, we need to find the area of the bottom and the curved sides of the container.
The area of the bottom (base) of the cylinder is given by the formula:
A_bottom = πr²
The cost of the material for the bottom is given as $0.30 per square inch, so the cost of the bottom is:
Cost_bottom = 0.30 * A_bottom = 0.30πr²
The area of the curved sides of the cylinder can be found by calculating the lateral surface area, which is given by the formula:
A_sides = 2πrh
Substituting the value of h we found earlier, we get:
A_sides = 2πr*(24/r²) = 48π/r
The cost of the material for the curved sides is given as $0.10 per square inch, so the cost of the sides is:
Cost_sides = 0.10 * A_sides = 0.10 * (48π/r) = 4.8π/r
Finally, the total cost of the material is the sum of the costs for the bottom and the sides:
C = Cost_bottom + Cost_sides = 0.30πr² + 4.8π/r
Therefore, the total cost C of the material is a function of the radius r of the base of the container and can be expressed as:
C(r) = 0.30πr² + 4.8π/r
C = .30*pi r^2 + .1* 2 pi r h
pi r^2 h = 24 pi
r^2 h = 24
h = 24/r^2
C = .30 pi r^2 + .2 pi r (24)/r^2
C = pi (.3 r^2 + 4.8/r)