Question

Find the dimensions of a cylinder of volume 17 m3 of minimal cost if the top and bottom are made of material that costs twice as much as the material for the side.

Answer 1

Answer:

The answer is below

Step-by-step explanation:

The volume of the cylinder with radius (r) and height (h) is given as:

Volume = πr²h

17 = πr²h

h = 17 / πr²

Let k represent the cost of the side and 2k the cost of the bottom.

The area of the side = 2πrh, hence the cost of the side = k(2πrh)

The area of the top and bottom = 2πr², hence the cost of the top and bottom = 2k(2πr²)

The total cost (T) = cost of side + cost of the top and bottom

T = k(2πrh) + 2k(2πr²)

T = 2kπ(rh + 2r²)

Put h = 17/πr²

T = 2kπ (r×17/πr² + 2r²)

T = 2kπ(17/πr + 2r²)

The minimal cost is at dT/dr = 0

$\frac{dT}{dr} =0=\frac{d}{dr} [2k\pi(\frac{17}{\pi r} +2r^2)]\\\\4r-\frac{17}{\pi r^2} =0\\\\4r=\frac{17}{\pi r^2}\\\\r^3=\frac{17}{4\pi }=1.35\\\\r=\sqrt[3]{1.35}\\\\r=1.1\ m\\\\h=\frac{17}{\pi r^2} =\frac{17}{\pi (1.1)^2} \\\\h=4.42\ m$