Calculus Problem: Blowing Up a Spherical Balloon

AI Thread Summary
The discussion focuses on a calculus problem involving the inflation of a spherical balloon. A key point is the distinction between the constant volume flow rate and the variable rate of change of the radius, dr/dt. The assumption that dr/dt is constant is challenged, emphasizing that it only holds true at a specific radius. By rearranging the expression to solve for dr/dt, one can derive the necessary answers for subsequent parts of the problem. Understanding the relationship between volume and radius is crucial for solving the overall question effectively.
Idan9988
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Homework Statement
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Relevant Equations
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IMG_20230527_195520.jpg

I'm struggling with section a. This is my calculation:
IMG20230527195328.jpg

The expression remains depend on the variable t, while in the answer is a concrete number:
Screenshot_2023-05-27-19-54-03-99_e2d5b3f32b79de1d45acd1fad96fbb0f.jpg
 
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r = r_0 + 0.9t is only valid if dr/dt is constant.

Why did you assume that dr/dt was constant? The question only tells you that dr/dt = 0.900\,\mathrm{cm}/\mathrm{s} when r = 6.50\,\mathrm{cm}.
 
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Agree,

The answer (a) has all the information. Since the volume flow rate is constant, then ##\frac {dV}{dt}## is a constant.

##\frac {dr}{dt}## is variable.

If you rearrange the expression to solve for ##\frac {dr}{dt}## and you get the answer to (b) and the behavior that explains (c).
 
Calculate ##\frac {dV} {dr}## and use this to inform your answer.
 
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