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practice(1)

# practice(1) - Examples on Related Rates(Section 3.9 Example...

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Examples on Related Rates (Section 3.9) Example 1 Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm 3 / s. How fast is the radius of the balloon increasing when the diameter is 50 cm? Guidelines 1. Identify Given information : The rate of increase of the volume of air is 100 cm 3 / s. Unknown : The rate of increase of the radius when the diameter is 50 cm. Draw a diagram . 2. Express these quantities mathematically Let V be the volume of the balloon and let r be its radius. The rate of increase of the volume with respect to time is the derivative dV dt . The rate of increase of the radius is dr dt . 3. Restate the given and the unknown as follows Given dV dt = 100 cm 3 / s. Unknown dr dt when 2 · r = 50 cm. 4. Connect dV dt and dr dt The volume of a sphere: V = 4 3 πr 3 . To use the given information, we differentiate each side of the equation with respect to t (the Chain Rule): 5. Substitute the given information into the resulting equation and solve for the unknown dr dt 1

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Example 2 A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away
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