Finding radius of the sphere

**stevecowall** · January 4th 2012, 06:46 PM

Question: A sphere has the same numerical value for both its surface area and its volume. Find the radius of the sphere.

My solution: 4πr^2 = 4/3πr^3 but the question say it has to be numerical value. And that is where I got stuck at. Can you explain it to me?

**pickslides** · January 4th 2012, 06:49 PM

First, divide both sides by $r^2$

What do you get?

**Prove It** · January 4th 2012, 06:52 PM

Originally Posted by stevecowall

Question: A sphere has the same numerical value for both its surface area and its volume. Find the radius of the sphere.

My solution: 4πr^2 = 4/3πr^3 but the question say it has to be numerical value. And that is where I got stuck at. Can you explain it to me?

$\displaystyle \begin{align*} 4\pi r^2 &= \frac{4}{3}\pi r^3 \\ \pi r^2 &= \frac{1}{3}\pi r^3 \\ r^2 &= \frac{1}{3}r^3 \\ 0 &= \frac{1}{3}r^3 - r^2 \\ 0 &= r^2\left(\frac{1}{3}r - 1\right) \\ r^2 = 0 \textrm{ or }\frac{1}{3}r - 1 &= 0 \\ r = 0 \textrm{ or } r &= 3 \end{align*}$

So the radius is either 0 units or 3 units in length.

**Prove It** · January 4th 2012, 06:52 PM

Originally Posted by pickslides

First, divide both sides by $r^2$

What do you get?

What if r = 0?

**pickslides** · January 4th 2012, 06:53 PM

Originally Posted by Prove It

What if r = 0?

It wouldn't be a sphere

**Prove It** · January 4th 2012, 06:55 PM

Originally Posted by pickslides

It wouldn't be a sphere

A single point is still a sphere.

Just like a linear function is a quadratic function, just with a = 0...

Don't exclude degenerates

**pickslides** · January 4th 2012, 07:21 PM

Does a single point have a non zero surface area and non zero volume? If so can it have a zero radius?

**HallsofIvy** · January 5th 2012, 09:28 AM

Originally Posted by pickslides

Does a single point have a non zero surface area and non zero volume? If so can it have a zero radius?

Why do you insist on non zero volume and surface area?

**okokjae** · January 5th 2012, 11:16 AM

Originally Posted by stevecowall

Question: A sphere has the same numerical value for both its surface area and its volume. Find the radius of the sphere.

My solution: 4πr^2 = 4/3πr^3 but the question say it has to be numerical value. And that is where I got stuck at. Can you explain it to me?

n = pi, r = radius

4nr^2 = 4/3nr^2

Divide both sides by r^2

4n = 4/3nr

Divide by n

4 = 4/3r

Divide by 4/3

4/(4/3) = r

r = 3

The radius of the sphere is 3.

**Prove It** · January 5th 2012, 05:43 PM

Originally Posted by okokjae

n = pi, r = radius

4nr^2 = 4/3nr^2

Divide both sides by r^2

4n = 4/3nr

Divide by n

4 = 4/3r

Divide by 4/3

4/(4/3) = r

r = 3

The radius of the sphere is 3.

You can't divide by r^2 because you don't know for sure that r =/= 0.

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