Respuesta :
Answer: radius r = 2 inches
height h = 6 inches
Step-by-step explanation:
Given;
Volume V = 24πin3
Volume of a cylinder is given by
V = πr^2h
h = V/πr^2. ....1
Where, h = height and r = radius of cylinder
For the surface area of the cylinder with open top. we have,
S = 2πrh + πr^2
For the cost of materials used, let k represent the cost of materials used for the body of the cylinder.
Then, for the bottom will be 3k
Total cost will be represented by C, which gives
C = 2πrhk + 3πr^2k. .....2
Substituting eqn 1 to 2, we have;
C = 2πrVk/πr^2 + 3πr^2k
C = 2Vk/r + 3πr^2k
The material cost is minimum at dC/dr = 0
dC/dr = -2Vk/r^2 + 6πrk =0
6πrk = 2Vk/r^2
r^3 = 2V/6π
r = (2×24π/6π)^-3
r = (8)^-3
r = 2
Substituting r = 2 into eqn1
h = 24π/π(2^2)
h = 24/4 = 6
h = 6
In this exercise we have to use the knowledge of geometry to calculate the radius and height of the cylinder through the volume:
- Radius: [tex]r = 2 inches[/tex]
- Height: [tex]h = 6 inches\\[/tex]
Given Volume by:
[tex]V = 24\pi[/tex]
Volume of a cylinder is given by:
[tex]V = \pi r^2h[/tex]
[tex]h = V/\pi r^2[/tex]
For the surface area of the cylinder with open top. we have,
[tex]S = 2\pi rh + \pi r^2[/tex]
For the cost of materials used, let k represent the cost of materials used for the body of the cylinder. Total cost will be represented by C, which gives:
[tex]C = 2\pi rhk + 3\pi r^2k.[/tex]
Substituting the equation , we have;
[tex]C = 2\pi rVk/\pi r^2 + 3\pi r^2k[/tex]
[tex]C = 2Vk/r + 3\pi r^2k[/tex]
The material cost is minimum at:
[tex]dC/dr = -2Vk/r^2 + 6\pi rk =0[/tex]
[tex]6\pi rk = 2Vk/r^2[/tex]
[tex]r^3 = 2V/6 \pi[/tex]
[tex]r = (2*24\pi /6\pi)^{-3}[/tex]
[tex]r = (8)^{-3}[/tex]
[tex]r = 2[/tex]
Substituting r into:
[tex]h = 24\pi /\pi(2^2)[/tex]
[tex]h = 24/4 = 6[/tex]
[tex]h = 6[/tex]
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