   Chapter 2.7, Problem 12E

Chapter
Section
Textbook Problem

(a) Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, calculate dV/dx when x = 3 mm and explain its meaning.(b) Show that the rate of change of the volume of a cube with respect to its edge length is equal to half the surface area of the cube. Explain geometrically why this result is true by arguing by analogy with Exercise 11(b).

To determine

Part (a):

To calculate: The rate of change of the volume V(x) with respect to side length x that means dVdx at x = 3 and explain its meaning.

Explanation

1) Concept

Use concept of derivative to find the rate of change of the volume V(x) of a cube with respect to its edge length x

2) Formula:

i) The volume of a cube with edge length x is, V(x)=x3

ii) Power Rule:ddxxn=nxn-1

3) Given:

V is the volume of a cube and side length x = 3mm

4) Calculation:

To find dVdx use power rule of derivative

dVdx=ddxx3=3x2

Thus, the rate of change of the volume of a cube with respect to its edge length x is,

dVdx=3x2

To find dV

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