Question
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A spherical balloon is inflated and its volume increases at a rate of . What is the rate of change of its radius when the radius is 9 in.?
The balloon’s radius is increasing at a rate of
(Round to three decimal places as needed.)

Ask by Chan Gibbs. in the United States
Mar 12,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The balloon’s radius is increasing at a rate of approximately 0.023 inches per minute.

Solution

We start with the formula for the volume of a sphere:
Differentiate both sides with respect to time :
We are given that in /min and the radius in, so substitute these values into the equation:
Simplify the equation:
Now solve for :
Calculating the numerical value:
Rounded to three decimal places, the rate of change of the balloon’s radius is:

Answered by UpStudy AI and reviewed by a Professional Tutor

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The Deep Dive

To find the rate of change of the radius when the radius is 9 inches, we start with the formula for the volume of a sphere:
We know that the volume is changing over time, so we can differentiate both sides with respect to time :
Given that and , we can plug in these values:
Calculating :
so our equation becomes:
Now, simplify :
Now we have:
To solve for , divide both sides by :
Now calculating :
Thus, rounding to three decimal places, the balloon’s radius is increasing at a rate of approximately:

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