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Understanding Related Rates: Calculating the Volume Change of a Balloon Over Time

This guide explores related rates problems, focusing on determining the rate of change of the volume of a balloon as it is filled with air. By using the formula for the volume of a sphere and applying the Chain Rule, we analyze how the volume (V) and radius (r) of the balloon relate over time (t). A practical example illustrates how to find out how fast the volume of a balloon is increasing when the radius is 6 inches and the diameter is expanding at a specific rate. Clear units and mathematical processes are also discussed to enhance understanding.

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Understanding Related Rates: Calculating the Volume Change of a Balloon Over Time

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  1. Related Rates These problems find the rates of change of two or more related variables that are changing with respect to time, t. To begin, let’s examine the Volume of a balloon as it is filled with air.

  2. The Volume of a sphere is given by: As a balloon is being filled, both the volume, V, and radius, r, increase with time, t. To examine the “rates of change” we need the derivative with respect to time, t.

  3. For the derivative we want: Notice, the variables do not agree. The derivative is with respect to time so the Chain Rule must be followed.

  4. On the left we get: Chain Rule On the right:

  5. This gives the “Related rate”: Question: At a given time, what does the rate of change of the Volume depend upon? Answer: The radius, r, and its rate of change, dr/dt

  6. Find how fast the volume of a balloon is changing when the radius is 6 inches and the diameter of the balloon is increasing at a rate of 3 inches per second. From the problem: r = 6 in and dr/dt = 1.5 in/sec

  7. = 678.6 What are the units? Look at the units in the equation: (inches)2(inches/sec) = in3/sec Which gives Volume/time

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