Application of Derivatives

Rishav Sharma

Student

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A circular disc of radius 3 cm is being heated. Due to expansion, its radius increases at a rate of 0.05 cm per second. Find the rate at which its area is increasing if the radius is 3.2 cm.

  • 25 Dec, 2023
Replies (2)
Vidhyamath Online

Instructor

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Let us assume that “r” be the radius of the given disc and “A” be the area, then the area is given as:

A = πr2

By using the chain rule,

Then dA/dt = 2πr(dr/dt)

Thus, the approximate rate of increase of radius = dr = (dr/dt) ∆t = 0.05 cm per second

Hence, the approximate rate of increase in area is:

dA = (dA/dt)(∆t) = 2πr[(dr/dt) ∆t ]

= 2π (3.2) (0.05)

= 0.320π cm2 per second.

Therefore, when r= 3.2 cm, then the area is increasing at a rate of 0.320π cm2/second.

  • 25 Dec, 2023
Vidhyamath Online

Instructor

  • 45
  • 0

Practice Problems

Solve the practice problems given below:

  1. Water is dropping out at a constant rate of 1 cubic cm/sec through a small hole at the vertex of the conical container, whose axis is vertical. If the slant height of water in the vessel is 4 cm, find the rate of decrease of slant height, where the vertical angle of the conical vessel is π/6.
  2. Show that the function f(x) = log x/x has maximum at x=e.
  3. Determine the approximate variation in the surface area of a cube of side x metres caused by decreasing the side by 1%.
  4. The radius of a sphere is estimated as 9 cm with an error of 0.03 cm, then calculate the approximate error in measuring its volume.
  5. Calculate two positive numbers whose sum is 15 and the sum of whose squares is minimum.

  • 25 Dec, 2023

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