A given quantity of metal is to be cast into a half cylinder with a rectangular base and semi - circular ends. Show that in order that the total surface area may be minimum the ratio of the length of the cylinder to the diameter of its semi - circular ends is π : (π + 2).

Question

Philoid · Accepted Answer

Let ‘h’ be the height ‘or’ length of half cylinder, ‘r’ be the radius of half cylinder and ‘d’ be the diameter.

We know that,

⇒ Volume of half cylinder (V) =

⇒ …… (1)

Now we find the Total surface area (TSA) of the half cylinder,

⇒ TSA = Lateral surface area of the half cylinder + Area of two semi - circular ends + Area of the rectangular base

⇒

From (1)

⇒

We need total surface area to be minimum and let us take the TSA as the function of r,

For maxima and minima,

⇒

Differentiating TSA again,

⇒

At

⇒

⇒ >0(Minima)

We have got Total surface area minimum for

We know that diameter is twice of radius

⇒

∴ Thus proved.