Question 8 of 82intermediate🔧 ApplyNumerical4 marks

A balloon, which always remains spherical on inflation, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is 15 cm.

Correct Answer

1/π cm/s

Exercise: EXERCISE 6.1 | Q: 8 | (Page 5)
For More Understanding

Explanation

This is a related rates problem. The volume of a sphere is V = (4/3)πr³. Differentiating with respect to time gives dV/dt = 4πr² dr/dt. Substituting dV/dt = 900 cm³/s and r = 15 cm yields dr/dt = 1/π cm/s. The method follows the chain rule as illustrated in Example 31 of the provided context.

Solution Steps

  1. Step 1: Write the volume of a sphere: V = (4/3)πr³.

  2. Step 2: Differentiate with respect to time t: dV/dt = 4πr² (dr/dt).

  3. Step 3: Substitute the given values: dV/dt = 900 cm³/s and r = 15 cm.

  4. Step 4: Compute: 900 = 4π × (15)² × dr/dt = 4π × 225 × dr/dt = 900π × dr/dt.

  5. Step 5: Solve for dr/dt: dr/dt = 900/(900π) = 1/π cm/s.

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