Applications of Derivatives
Rate of Change
Grade 12
Question:
<p>The height <em>h</em> of a right circular cone is 20 cm and is decreasing at the rate of 4 cm/s. At the same time, the radius <em>r</em> is 10 cm and is increasing at the rate of 2 cm/s. Find the rate of change of the volume in cm³/s ________ (up to four decimal places).</p>
Step-by-Step Solution
Key Concept: Volume of cone is V = (1/3)πr²h. Since both r and h are functions of time, use the product rule: dV/dt = (1/3)π[2r(dr/dt)·h + r²(dh/dt)] with given values at the specific instant.
<p><strong>Step 1:</strong> Write the volume formula for a cone:</p><p>V = (1/3)πr²h</p><p><strong>Step 2:</strong> Differentiate both sides with respect to time using the product rule on r²h:</p><p>dV/dt = (1/3)π[2r(dr/dt)·h + r²(dh/dt)]</p><p><strong>Step 3:</strong> Substitute the given values at the specific instant:</p><p>• r = 10 cm, h = 20 cm</p><p>• dr/dt = 2 cm/s (increasing)</p><p>• dh/dt = -4 cm/s (decreasing)</p><p><strong>Step 4:</strong> Calculate:</p><p>dV/dt = (1/3)π[2(10)(2)(20) + (10)²(-4)]</p><p>dV/dt = (1/3)π[800 - 400]</p><p>dV/dt = (1/3)π(400)</p><p>dV/dt = (400π)/3</p><p>dV/dt = 400 × 3.14159.../3</p><p>∴ Answer: <strong>418.0794</strong> cm³/s
Correct Answer: 418