Expanding surface of sphere

[jackt]

a spherical balloon expands when it is taken from the cold outdoors to the inside of a warm house. if its surface area increases 19.4%, by what percentage does the radius of the balloon change?


I've tried this several times but I am not sure how to go about beginning the work

 

[robphy]

If I were doing this problem,
I would write down quantities with labels like:

for the volume of the ball before...
$$V_{before}=\frac{4\pi}{3}r_{before}^3$$
and, similarly, for "after".
They can be compared by taking the ratio:
$$\frac{ V_{after} }{V_{before} }$$.

How would you use this strategy with your particular problem?

What does "increases 19.4%" mean in terms of a ratio?

 

[ron_jay]

The increase in 19.4% means the difference between the new surface area and the original or we could write it as:

4(pi)R^2-4(pi)r^2 where R is the new(larger) radius and r the original radius

We also know that increase percentage is calculated on the original value which happens to be 4(pi)r^2.

Hence, {4(pi)R^2-4(pi)r^2}/4(pi)r^2 *100=19.4

Evaluate this and then find the relation only in terms of the ratio of the difference in radius to the original radius...

 

[robphy]

in terms of the difference in radius...

For this type of problem, it's more efficient to think in ratios of radii, rather than difference in radii.