(a) Sodium chlorate crystals are easy to grow in the shape...

(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 $x=3 \mathrm{mm}$ 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$(b)$

(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 $x=3 \mathrm{mm}$ 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$(b)$

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Key Concepts

Surface Area of a Cube

The surface area of a cube is calculated as 6x^2. Establishing the connection between the derivative of the volume (3x^2) and half the surface area (which is 3x^2) reveals an intrinsic geometrical relationship between the volume’s rate of change and the area over which the change is effectively distributed.

Geometric Interpretation of Derivatives

This concept provides insight into how a small increment in the cube’s edge length results in adding a thin outer layer. The volume of this thin layer is approximately the product of the cube’s surface area and the small change in edge length. In this case, the derivative being equal to half the total surface area reflects this geometric idea, as it accounts for the effective area contributing to the volume change.

Derivative of a Function

The derivative measures the instantaneous rate of change of one quantity with respect to another. In the context of this problem, it represents how fast the volume of a cube increases as its edge length increases.

Power Rule in Differentiation

This rule is used to differentiate functions of the form x^n. For a cube with volume V = x^3, applying the power rule gives dV/dx = 3x^2, which quantifies the rate of change of the volume with respect to changes in the edge length.

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