Question
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.)
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:
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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: