Q. 254.5( 2 Votes )

# A right circular

Answer :

Given,

• A right circular cylinder is inscribed inside a cone.

• The curved surface area is maximum.

Let us consider,

• ‘r_{1}’ be the radius of the cone.

• ‘h_{1}’ be the height of the cone.

• ‘r’ be the radius of the inscribed cylinder.

• ‘h’ be the height of the inscribed cylinder.

DF = r, and AD = AL – DL = h_{1} – h

Now, here ΔADF and ΔALC are similar,

Then

----- (1)

Now let us consider the curved surface area of the cylinder,

S = 2πrh

Substituting h in the formula,

---- (2)

For finding the maximum/ minimum of given function, we can find it by differentiating it with r and then equating it to zero. __This is because if the function S(r) has a maximum/minimum at a point c then S’(c) = 0.__

Differentiating the equation (2) with respect to r:

[Since ]

------- (3)

To find the critical point, we need to equate equation (3) to zero.

__Now to check if this critical point will determine the maximum volume of the inscribed cylinder, we need to check with second differential which needs to be negative.__

Consider differentiating the equation (3) with r:

----- (4)

[Since ]

Now let us find the value of

As , so the function S is maximum at

Substituting r in equation (1)

--- (5)

As S is maximum, from (5) we can clearly say that h_{1} = 2h and

r_{1} = 2r

this means the radius of the cone is twice the radius of the cylinder or equal to diameter of the cylinder.

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