Clark W. Griswold, the protagonist of the National Lampoon Vacation films, is attempting to decorate his home festively for the holidays. This year, he has decided to add a live nativity scene, starring various children from the neighborhood. Unfortunately, it has begun to rain very heavily, and he needs to decide how long it's safe to leave the live baby Jesus in and when he needs to go with a stunt doll stand-in.

The 4-foot-long manger has a cross section which is an equilateral triangle with side length 18 inches. If rain is accumulating in the manger at the rate of 250 cubic inches/hour, how fast is the water level rising when it is 10 inches deep?

Difficulty:

Solution

The trickiest part of related rates problems is coming up with a primary equation (that you'll eventually differentiate with respect to t). The problem gives us information about the volume increasing (dV/dt) and asks us to find dh/dt, where h is the height of the water. Therefore, we need a primary equation containing only a V and h. That's our ultimate goal, but for now, let's take one step at a time.

First of all, everything in the problem except for the length of the manger is in inches, so let's convert 4 feet to inches (48 inches, of course). Now, we need to recognize that dV/dt = 250 and eventually, h = 10. However, we have miles to go until we use that information--you only want to plug in the known values once you've differentiated the primary equation with respect to t. So, where do we start?

Well, since the manger has a cross-section that's an equilateral triangle, so will the water contained in the manger. As water begins to accumulate, it will look like this:

Each side of the water-triangle (hereafter referred to as the wangle) has side s and height h, but these values will change depending on how much water is in the manger. (Since this is an equilateral triangle, the median shown will split the larger triangle into two congruent right triangles, so I feel comfortable denoting the length above as s/2.)

According to the Pythagorean Theorem, we can find a relationship between h and s in the wangle. We'll eventually need this, so follow me here:

Let's go ahead and try to come up with our primary equation. We already know that it must contain V and h, so we might as well use the equation for volume of the water. That volume is equal to one cross-sectional area of the wangle times the length of the manger. (Any three dimensional figure's volume can be found by multiplying its cross-sectional area times its depth. For example, a right circular cylinder's volume is just the area of a circle--a cross-section--times h, how long the cylinder is.) Therefore, the volume of the water is:

It's almost time to differentiate and solve, but we've got too many variables going on here. You've got to get rid of that s, since we don't know anything about it or ds/dt, for that matter. Luckily, we already have a relationship between h and s, thanks to our previous work with the Pythagorean Theorem, so let's solve that for s in terms of h so that we can get rid of all the s's by replacing them with h's:

Now, we can rewrite that volume equation and simplify it to get:

Differentiate everything with respect to time, and then plug in the known values. Once you've done that, it's easy to solve for dh/dt: