Applications of Derivatives
Rate of Change of Quantities
Grade 12
Question:
<p>A spherical balloon is filled with \(4500\pi\) cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of \(72\pi\) cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is</p>
<p>\(\dfrac{9}{7}\)</p>
<p>\(\dfrac{7}{9}\)</p>
<p>\(\dfrac{2}{9}\)</p>
<p>\(\dfrac{9}{2}\)</p>
Step-by-Step Solution
Key Concept: Use the volume formula V = (4/3)πr³ to relate volume and radius, then differentiate with respect to time to connect dV/dt with dr/dt. Find the radius at t=49 using V(t) = 4500π - 72π(49), then substitute into the derivative equation.
<p><strong>Step 1:</strong> Set up the volume-radius relationship. For a sphere: V = (4/3)πr³</p><p><strong>Step 2:</strong> Differentiate both sides with respect to time: dV/dt = 4πr² · dr/dt</p><p><strong>Step 3:</strong> Find the volume at t = 49 minutes: V(49) = 4500π - 72π(49) = 4500π - 3528π = 972π cubic meters</p><p><strong>Step 4:</strong> Find the radius at t = 49 using V = (4/3)πr³: 972π = (4/3)πr³ → r³ = 729 → r = 9 meters</p><p><strong>Step 5:</strong> Substitute into the differentiated equation with dV/dt = -72π (negative because gas escapes): -72π = 4π(9)² · dr/dt → -72π = 324π · dr/dt → dr/dt = -72/324 = -2/9 meters per minute</p><p><strong>Step 6:</strong> The rate at which radius decreases is the absolute value: |dr/dt| = 2/9 meters per minute</p><p>∴ Answer: C</p>
Correct Answer: C