Question

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?

Answer 1

`color(blue)(56.25%)`

Explanation:

First we observe that, for similar figures. If a similar figure is increased by a scale factor `a/b`, then all linear measurements will increase by `a/b`. Quadratic measurements will increase by a factor `(a/b)^2` and cubic measurements will increase by a factor `(a/b)^3`.

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

`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+61/64=125/64`

Now this is volume, so:

`(a/b)^3=125/64`

`a/b=root(3)(125/64)`

We require the scale factor for area.So:

`(a/b)^2=(root(3)(125/64))^2`

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

`(root(3)(125/64))^2-1=0.562500000`

This is:

`56.25%`