Home » Related Rates » Page 2

# How quickly is the distance from a baseball to first base changing as the baseball moves along the third baseline

A baseball is moving along the third base line at a constant velocity of 100 feet per second. If a baseball diamond is a 90-foot square, then how quickly is the distance from the ball to first base changing when:

1. the ball is halfway to third base;
2. the ball is at third base.

First, we give the general set up for the problem. The following diagram illustrates the situation: The quantity we are given then is Further, ; thus, and so Now, the two requested cases:

1. If the ball is halfway to third base, this means feet (since the diamond is a square with sides of length 90 feet). So we have, 2. If the ball is at third base, this means (since is the distance from the ball to home plate), thus we have, # Compute the velocity of an airplane given the rate of change of its distance to a point on the ground

An airplane is 8 miles above the ground flying at a constant velocity, at a constant altitude. (Assume the earth is flat.) There is a point on the ground directly under the airplane’s flight path. The distance between and the airplane is decreasing at a rate of 4 miles per minute when the distance is 10 miles. Find the velocity of the airplane.

Let be the distance from the point on the ground directly beneath the plane to the point , and let be the distance from the plane to . The following diagram illustrates the situation: We are trying to compute , the velocity of the airplane. We are given that the distance from the airplane to the point is changing at a rate of 4 miles per minute when . Thus, adjusting units to mph, we have Furthermore, we can compute in terms of (by the Pythagorean identity) and then differentiate: So, at miles we have ; thus, This was computed for miles, but since we are given that the airplane is flying at a constant velocity, then this velocity is valid for all .

# Compute the rate of change of volume of a cube given the rate of change of an edge

Given a cube with edges expanding at a rate of 1 centimeter per second, what is the rate of change of the volume of the cube when the length of an edge is:

1. 5cm.
2. 10cm.
3. cm.

First, we know the volume of a cube is given by where is the length of an edge. Thus, the rate of change of the volume relative to the rate of change of an edge is given by We use this formula to find the rate of change of the volume for given values of ,

1. If then we have 2. If then we have 3. If then we have 