Question 4 of 82intermediate🔧 ApplyNumerical3 marks

An edge of a variable cube is increasing at the rate of 3 cm/s. How fast is the volume of the cube increasing when the edge is 10 cm long?

Correct Answer

The volume of the cube is increasing at the rate of 900 cm³/s when the edge is 10 cm long.

Exercise: EXERCISE 6.1 | Q: 4 | (Page 4)
For More Understanding

Explanation

This is a direct application of related rates using the chain rule, similar to Example 2 and Example 3 in the textbook context. The student needs to differentiate the volume formula with respect to time and substitute the given values.

Solution Steps

  1. Step 1: Let x be the length of the edge and V be the volume of the cube. Then, V = x³, where x is a function of time t.

  2. Step 2: Given: dx/dt = 3 cm/s (rate at which edge is increasing)

  3. Step 3: Differentiating V = x³ with respect to t using the Chain Rule: dV/dt = d/dt(x³) = d/dx(x³) · dx/dt = 3x² · dx/dt

  4. Step 4: Substituting the given values when x = 10 cm: dV/dt = 3(10)² × 3 = 3×1003 \times 100 × 3 = 900 cm³/s

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