A balloon, which always remains spherical, has a variable diameter 3/2 (2x + 1). Find the rate of change of its volume with respect to x

Solution:

Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the

change in the other quantity.

They are used in many situations like finding maxima or minima of a function, finding the slope of the curve, and even inflection points.

We know that Volume of sphere is given by 4/3π r³

It is given that diameter,

d = 3/2 (2x + 1)

Hence,

r = 3/4 (2x + 1)

Therefore,

On Substituting the value of r, we get

V = 4/3 π (3/4)³ (2x + 1)³

= 9/16 π (2x + 1)³

Thus,

dV/dx = 9/16 π d/dx (2x + 1)³

= (9/16) x (π) x (3) x (2) x (2x+1)²

= 27/8 π (2x + 1)²

Therefore,

the rate of change of its volume with respect to x is 27/8 π (2x + 1)²

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.1 Question 13

A balloon, which always remains spherical, has a variable diameter 3/2 (2x + 1). Find the rate of change of its volume with respect to x

Summary:

Given that a balloon, which always remains spherical, has a variable diameter 3/2 (2x + 1). Hence, the rate of change of its volume with respect to x is 27/8 π (2x + 1)²