## 2011-2012

Problem 4: B-B-B-Betty and the Jet (Related Rates)

During a taping for Circus of the Stars, beloved actress Betty White is shot out of a cannon. The firing goes completely awry and sends her on a collision course with a jet. As they converge, Betty and the jet plane at right angles to each other (see diagram below). Betty is 200 miles away from the point of impact and traveling at a constant rate of 600 mph. (Not even the laws of physics can slow Betty White!) The plane is 150 miles from impact and traveling at a constant rate of 450 mph.

At what rate is the distance *d* between Betty and the jet decreasing?

### Solution:

Consider the following diagram, which labels the legs of the right triangle as follows: *b* is the distance between Betty White and the point of impact and *p* is the distance between the plane and the point of impact.

The Pythagorean Theorem describes the relationship between the lengths of the sides of the triangle.

*b*2 + *p*2 = *d*2

Substitute *b* = 200 and *p* = 150 into the formula to solve for *d*, the distance between the two airborn objects at this moment.

You are asked to find the rate at which *d* decreases. In other words, you are calculating *dd*/*dt*. Apply implicit differentiation, with resepct to *t*.

Divide each of the terms by 2 and solve for *dd*/*dt*.

To calculate *dd*/*dt*, substitute all of the known information into the equation: *b* = 200, *db*/*dt* = –600, *p* = 150, *dp*/*dt* = –450, and *d* = 250. Note that *db*/*dt* and *dp*/*dt* are negative because the lengths of the legs of the right triangle are decreasing—the objects are on a collision course, so the distances between the objects and the point of impact are getting smaller.

The distance between Betty and the jet is decreasing at a rate of 750 mph.