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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

(a) The volume of a growing spherical cell is V = 4 3 π r 3 , where the radius r is measured in micrometers (1 μm = 10–6 m). Find the average rate of change of V with respect to r when r changes from

(i) 5 to 8μm

(ii) 5 to 6μm

(iii) 5 to 5.1μm

(b) Find the instantaneous rate of change of V with respect to r when r = 5μm.

(c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise 13(c).

(a)

To determine

To find: The average rate of change of volume of sphere V with respect to when r, when r changes from

  1. i. 5to8μm
  2. ii. 5to6μm
  3. iii. 5to5.1μm
Explanation

Calculation:

Volume of a sphere is as below.

V(r)=43πr3

Calculate the average rate of change of the volume of a sphere, when the radius of sphere changes from 5μm to 8μm.

V(r)=V(8)V(5)85

V(r)=43π(8)343π(5)33V(r)=43π(512)43π(125)3V(r)=172πμm3/μm

Therefore, the average rate of change of the volume of a sphere is 172πμm3/μm_.

Calculate the average rate of change of the volume of a sphere, when the radius of sphere changes from 5μm to 6μm.

V(r)=V(6)V(5)65V(r)=43π(6)343π(5)31V(r)=43π(216)43π(125)1V(r)=121

(b)

To determine

To find: The instantaneous rate of change when radius r is 5μm.

(c)

To determine

To find: The rate of change of the volume of a sphere with respect to its radius is equal to the surface area of the volume and why it is geometrically true.

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