# If the volume of a sphere is increased by 95 5/16% without changing the shape, what will be the percentage increase in the surface area?

Apr 26, 2018

color(blue)(56.25%)

#### Explanation:

First we observe that, for similar figures. If a similar figure is increased by a scale factor $\frac{a}{b}$, then all linear measurements will increase by $\frac{a}{b}$. Quadratic measurements will increase by a factor ${\left(\frac{a}{b}\right)}^{2}$ and cubic measurements will increase by a factor ${\left(\frac{a}{b}\right)}^{3}$.

We require an increase of 1525/16%=61/64:

$\frac{61}{64}$

Remember this will give us 1525/16%, we need to add 100% to this.,the 100% is the amount we have before adding 1525/16% to it. So:

$1 + \frac{61}{64} = \frac{125}{64}$

Now this is volume, so:

${\left(\frac{a}{b}\right)}^{3} = \frac{125}{64}$

$\frac{a}{b} = \sqrt[3]{\frac{125}{64}}$

We require the scale factor for area.So:

${\left(\frac{a}{b}\right)}^{2} = {\left(\sqrt[3]{\frac{125}{64}}\right)}^{2}$

If we now subtract 1 form this, we will get the increase in area percentage:

${\left(\sqrt[3]{\frac{125}{64}}\right)}^{2} - 1 = 0.562500000$

This is:

56.25%