The Question is at the details. Please answer this question.?

Question

The Muhibbah Company is a manufacturer of cylindrical aluminium tins. The manager plans to reduce the cost of production. The production cost is proportional to the area of the aluminium sheet used. The volume that each tin can hold is 1000 cm cube (1 liter).
1. Determine the value of h,r and hence calculate the ratio of h/r when the total surface area of each tin is minimum. Here, h cm denotes the height and r cm the radius of the tin.
2. Investigate cases where the top and bottom surfaces are cut from
(i) Equliteral triangle.
(ii)Regular hexagon.
Find the ratio of h/r for each case.

Answer 1

Are we to assume that the tin is a cylinder? The minimum surface are for a given volume is a sphere.

Assuming a cylinder...

V = (pi) r^2 h

SA = 2 (pi) r^2 + 2 (pi) r h

The volume is 1000cm^3, so substitute that into the volume equation and solve for h. That gives us

h = 1000 / (pi) r^2

Next substitute that result into the SA equation for the h. That gives us

SA = 2 (pi) r^2 + 2 (pi) r (1000 / (pi) r^2)

Simplify that last ugly mess and we get

SA = 2 (pi) r^2 + (2000 / r)

Now, depending on what math you're taking, you can find the minimum in a variety of ways... inspecting the graph, successive approximation, or differentiating.

For part 2, use the area formulae for the triangle or hexagon, and perimeter instead of circumference in the SA formula, and solve in the same way.

Good luck.

Answer 2

Surely you know the volume as a function of h and r.

Setting that equal to 1000 cubic cm, you get h as a function of r.

Surely you also know the surface area as a function of h and r -- but therefore you also know it as a function of r.

Take the derivative, set it equal to 0, and you will be most of the way done!