# The heat experienced by a hiker at a campfire is proportional to the amount of wood on the fire, and inversely proportional to the cube of his distance from the fire. What would the equation be to set this up?

$H \propto \frac{W}{D} ^ 3$

#### Explanation:

We have some variables we need to define. There's the Heat from the fire ($H$), the amount of Wood on the fire ($W$), and the Distance from the fire ($D$).

And now let's build the relationship phrase by phrase (I'm going to use the symbology $\propto$ for proportional rather than $=$ for equals):

The heat experienced by a hiker:

$H \propto$

...is proportional to the amount of wood on the fire...

$H \propto W$

...and inversely proportional to the cube of the distance from the fire:

$H \propto \frac{W}{D} ^ 3$

Let's see how this works. Let's set some values and then make some changes. Let's set $H = 100 , W = 1 , D = 1$ so that we say that the hiker is experiencing some value of heat (100 - conveniently set so we can use it as a percentage) for 1 unit of wood at a distance of 1 unit away, so that's:

$100 \propto \frac{1}{1} ^ 3 = 1$

If we add another unit of wood, we get:

$\frac{W}{D} ^ 3 = \frac{2}{1} ^ 3 = 2 \implies H = 200$

If instead we have our hiker step back 1 unit so that s/he's at distance 2, we get:

$\frac{W}{D} ^ 3 = \frac{1}{2} ^ 3 = \frac{1}{8} \implies H = \frac{100}{8} = 12.5$

And so to compare - if the hiker wanted to sit at distance 2 from the fire but wanted to be as warm as where s/he was at distance 1, there would need to be 8 units of wood:

$\frac{W}{D} ^ 3 = \frac{8}{2} ^ 3 = \frac{8}{8} = 1 \implies H = 100$