Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.ORA tank with rectangular base and rectangular sides, open at the top is to be constructed sothat its depth is ...

Question

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r.
OR

A tank with rectangular base and rectangular sides, open at the top is to be constructed sothat its depth is 2 m and volume is 8 m³. If building of tank costs Rs. 70 per sq. metre for thebase and Rs. 45 per sq. metre for sides, what is the cost of least expensive tank?

Philoid · Accepted Answer

Let a right circular cylinder of radius “R” and height “H” is inscribed in the sphere of given radius “r”.

Let V be the volume of the cylinder.

Then, V = πR²H

…. (1)

Differentiating both sides w.r.t H to get,

….. (2)

For maximum value put dV/dH = 0

Again, differentiating w.r.t H we get,

At ,

So, volume is maximum when height of cylinder is .

Substitute in (1) to get,

OR

Let the length and breadth of the tank are L and B.

∴ V = 8

2LB = 8

…. (1)

Total surface area S = Area of base + Area of 4 walls

= LB + 2(B+L).2

= LB+4B+4L

The cost of constructing the tank is:

C = 70(LB) + 45(4B + 4L)

….. (2)

Differentiating both sides w.r.t L we get,

…. (3)

For minimization dC/dL = 0,

⇒ L² = 4

⇒ L = 2

Differentiate (3) w.r.t L to get,

∴ Cost is minimum when L = 2.

From (1),

B = 2

From (2),

= 280 + 720

= Rs 1000