Ex 6.1, 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.Let d be the diameter of the balloon
Given that
Diameter = d = 3/2 (2x + 1)
Let r be the radius of the balloon
r = 𝑑/2 = 𝟑/𝟒 (2x + 1)
The balloon is a spherical
Volume of the balloon = 4/3 𝜋𝑟^3
We need to find rate of change of volume with respect to x i.e. 𝑑𝑉/𝑑𝑥
Now,
𝑑𝑉/𝑑𝑥 = 𝑑/𝑑𝑥 (4/3 𝜋𝑟^3 )
= 4𝜋/3 × (𝑑𝑟^3)/𝑑𝑥
= 4𝜋/3 × 𝑑/𝑑𝑥 (27/64 (2𝑥+1)^3 )
= 9𝜋/16 × (𝑑(2𝑥 + 1)^3)/𝑑𝑥
= 9𝜋/16 × 3(2x + 1)2 × 2
= 𝟐𝟕𝝅/𝟖 (𝟐𝒙+𝟏)^𝟐

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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