## 17Calculus Derivatives - Related Rates Involving Distances

If you haven't already, we recommend that you read the related rates basics page for information on how to get started on related rates problems.

Overview

This page covers related rates problems specifically involving distances. These types of problems involve
- cars, boats, airplanes and people moving in the same or different directions
- baseball questions involving runners
- unusual distance problems involving other relationships like angles and circles

When solving these types of problems, you first draw a picture and pick out the type of geometric figure involved. By far, the most used figure you will come up with is a triangle. With triangles, you will usually need the Pythagorean Theorem. Once you have a figure with all the distances labeled, you can write down the equations involved.

What To Do With Constants In Related Rates Problems

What do you do with constants that are given in the problem? First of all, you never want to just go in and plug in all your constants before you take the derivative.

Safe Answer - - Wait and plug in your constants only after you have the derivative. So, you would label all distances with variables, take the derivative with respect to t and then plug in all your given constants. This is what you need to do when you first start learning to work related rates problems. After you have some experience, you can go on to the more experienced technique.

Experienced Answer - - Once you learn the basics of related rates problems, you will have a feel for which constants you can plug in right away and which ones you can't. The difference you need to look for is
- if the variable is NOT changing, then you can substitute the constant in before taking the derivative;
- but, if the variable is changing over time, then you must wait until after you take the derivative before you can substitute the constant into the equation.

At this point, it will just confuse you more if we write down a bunch of theory on how to work these problems. You need to actually see one, then work many in order to see patterns. If you haven't already, read the main related rates page. Then, come back here, watch a video or two and try your hand at a few problems.

ladder leaning on a wall - [shape: triangle]

These problems involve a ladder (or a similar type of straight object) sliding down a wall. This type of problem is essentially a triangle that is changing shape over time and it is an extremely common type of related rates problem. We have no doubt that you will see at least one in your homework and maybe have one on an exam. [ Note: Most problems with ladders sliding down a wall involve change in distances. However, you can find at least one problem asking for an area on the basic related rates page. ]

A 41ft ladder is leaning against a vertical wall. The top of the ladder is sliding down the wall while its bottom slides along the ground away from the wall at 4ft/sec. How fast is the top of the ladder moving when it is 9ft above the ground?

Problem Statement

A 41ft ladder is leaning against a vertical wall. The top of the ladder is sliding down the wall while its bottom slides along the ground away from the wall at 4ft/sec. How fast is the top of the ladder moving when it is 9ft above the ground?

Solution

### 1108 video

video by Krista King Math

A 10ft ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at 0.5ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6ft from the wall?

Problem Statement

A 10ft ladder is leaning against a wall. If the bottom of the ladder slides away from the wall at 0.5ft/sec, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6ft from the wall?

Solution

### 1145 video

video by MathTV

We have a sliding ladder that is leaning against a wall. The foot or bottom of the ladder is moving at a rate of 0.2m/sec when it is 3m away from the wall. The top of the ladder is moving down at a rate of 0.15m/sec. What is the distance from the floor to the top of the ladder at that time?

Problem Statement

We have a sliding ladder that is leaning against a wall. The foot or bottom of the ladder is moving at a rate of 0.2m/sec when it is 3m away from the wall. The top of the ladder is moving down at a rate of 0.15m/sec. What is the distance from the floor to the top of the ladder at that time?

Hint

We do not know the length of the ladder and we do not need to know it to solve this problem. Since the length of the ladder is not changing, just use a letter, like L to represent the length of the ladder in your equations.

Problem Statement

We have a sliding ladder that is leaning against a wall. The foot or bottom of the ladder is moving at a rate of 0.2m/sec when it is 3m away from the wall. The top of the ladder is moving down at a rate of 0.15m/sec. What is the distance from the floor to the top of the ladder at that time?

4m

Problem Statement

We have a sliding ladder that is leaning against a wall. The foot or bottom of the ladder is moving at a rate of 0.2m/sec when it is 3m away from the wall. The top of the ladder is moving down at a rate of 0.15m/sec. What is the distance from the floor to the top of the ladder at that time?

Hint

We do not know the length of the ladder and we do not need to know it to solve this problem. Since the length of the ladder is not changing, just use a letter, like L to represent the length of the ladder in your equations.

Solution

### 2251 video

video by Michel vanBiezen

4m

A 12 foot ladder is leaning up against a house. The base of the ladder begins to slip away from the house at a rate of 1/2 ft/sec. How fast is the top of the ladder moving down the side of the house when the ladder is 3 feet away from the house?

Problem Statement

A 12 foot ladder is leaning up against a house. The base of the ladder begins to slip away from the house at a rate of 1/2 ft/sec. How fast is the top of the ladder moving down the side of the house when the ladder is 3 feet away from the house?

Solution

### 2460 video

video by MIP4U

A 10m ladder is leaning against a vertical wall. The top of the ladder is sliding down the wall while its bottom slides along the ground away from the wall at 4m/sec. How fast is the top of the ladder moving when the bottom is 8m from the wall?

Problem Statement

A 10m ladder is leaning against a vertical wall. The top of the ladder is sliding down the wall while its bottom slides along the ground away from the wall at 4m/sec. How fast is the top of the ladder moving when the bottom is 8m from the wall?

Solution

### 1109 video

cars, boats, planes and people moving in the same or different directions [shape: triangle]

Similar to a ladder on a wall, we have triangles in these figures but we are often given strange information about objects leaving someplace at different times that we need to handle. And we are also usually given one or more rate at which objects are moving. We usually need to find how distances are changing related to one another.
Another type of problem that is very similar to moving cars, boats and planes is when you have a fixed point of light, like a lamp post, and a person is moving away from or toward the light. You can draw a triangle and the way to solve the problem is exactly the same as with other moving objects.
We also include baseball-type questions in this group. Again, once the triangle is established, the solution method is the same. [Note: If you are not familiar with the basics of the game of baseball, this wiki page contains the basics you need to solve most problems found here. Read the first couple of paragraphs (in the middle of the page where this link positions the page) and study the picture to the right.]

Two cars leave an intersection at the same time, one headed east and the other north. The eastbound car is moving at 30mph while the northbound car is moving at 60mph. Twenty minutes later, what is the rate of change in the perimeter of the right triangle formed using the two cars and the intersection?

Problem Statement

Two cars leave an intersection at the same time, one headed east and the other north. The eastbound car is moving at 30mph while the northbound car is moving at 60mph. Twenty minutes later, what is the rate of change in the perimeter of the right triangle formed using the two cars and the intersection?

Solution

### 1110 video

video by PatrickJMT

An airplane is flying horizontally at 480 mi/hr, 3 miles above the ground as it passes over an observer on the ground. How fast is the distance from the observer to the airplane increasing 30 seconds later?

Problem Statement

An airplane is flying horizontally at 480 mi/hr, 3 miles above the ground as it passes over an observer on the ground. How fast is the distance from the observer to the airplane increasing 30 seconds later?

Solution

### 1112 video

video by Krista King Math

A 6ft tall man walks away from a 22ft street light at a speed of 8 feet per second. What is the rate of change of the length of his shadow when he is 19ft away from the light? Also, at what rate is the tip of his shadow moving?

Problem Statement

A 6ft tall man walks away from a 22ft street light at a speed of 8 feet per second. What is the rate of change of the length of his shadow when he is 19ft away from the light? Also, at what rate is the tip of his shadow moving?

Solution

### 1113 video

We have a street light 20ft high with a 5ft tall person 50ft from the light. The person is walking away from the light post at 4 ft/sec. (a) How fast is the length of the shadow changing? (b) How fast is the tip of the shadow moving?

Problem Statement

We have a street light 20ft high with a 5ft tall person 50ft from the light. The person is walking away from the light post at 4 ft/sec. (a) How fast is the length of the shadow changing? (b) How fast is the tip of the shadow moving?

(a) 4/3 ft/sec; (b) 16/3 ft/sec

Problem Statement

We have a street light 20ft high with a 5ft tall person 50ft from the light. The person is walking away from the light post at 4 ft/sec. (a) How fast is the length of the shadow changing? (b) How fast is the tip of the shadow moving?

Solution

### 2207 video

video by Michel vanBiezen

(a) 4/3 ft/sec; (b) 16/3 ft/sec

A 6ft tall man walks away from a 15ft lamppost at 5ft/sec. Find the rate at which the tip of his shadow is changing and the rate at which the length of his shadow is changing.

Problem Statement

A 6ft tall man walks away from a 15ft lamppost at 5ft/sec. Find the rate at which the tip of his shadow is changing and the rate at which the length of his shadow is changing.

Solution

### 1114 video

video by rootmath

A baseball diamond is a square with side 90ft. If a batter hits the ball and runs towards first base with a speed of 20ft/sec, at what speed is his distance from second base decreasing when he is halfway to first base?

Problem Statement

A baseball diamond is a square with side 90ft. If a batter hits the ball and runs towards first base with a speed of 20ft/sec, at what speed is his distance from second base decreasing when he is halfway to first base?

Solution

### 1115 video

video by PatrickJMT

Two cars are moving in opposite directions, one going north at 40 ft/sec and the other going south at 60 ft/sec. The second car is 3000 ft to the east of the first car. When the north-south distance between them is 4000 ft, how fast is the distance between them changing?

Problem Statement

Two cars are moving in opposite directions, one going north at 40 ft/sec and the other going south at 60 ft/sec. The second car is 3000 ft to the east of the first car. When the north-south distance between them is 4000 ft, how fast is the distance between them changing?

80 ft/sec

Problem Statement

Two cars are moving in opposite directions, one going north at 40 ft/sec and the other going south at 60 ft/sec. The second car is 3000 ft to the east of the first car. When the north-south distance between them is 4000 ft, how fast is the distance between them changing?

Solution

### 2219 video

video by Michel vanBiezen

80 ft/sec

A rocket rising vertically is tracked by radar 5 miles from the launchpad. How fast is the rocket rising when it is at an altitude of 4 miles and the distance between the rocket and the radar is increasing at 2000 mi/hr?

Problem Statement

A rocket rising vertically is tracked by radar 5 miles from the launchpad. How fast is the rocket rising when it is at an altitude of 4 miles and the distance between the rocket and the radar is increasing at 2000 mi/hr?

$$500\sqrt{41} \approx 3200$$ miles/hr

Problem Statement

A rocket rising vertically is tracked by radar 5 miles from the launchpad. How fast is the rocket rising when it is at an altitude of 4 miles and the distance between the rocket and the radar is increasing at 2000 mi/hr?

Solution

### 2245 video

video by Michel vanBiezen

$$500\sqrt{41} \approx 3200$$ miles/hr

We have two cars that are approaching the same point. The first car is traveling east at 40km/hr and is 4km from the point. The second car is traveling north at 50km/hr and is 3km from the point. How fast is the distance between them changing?

Problem Statement

We have two cars that are approaching the same point. The first car is traveling east at 40km/hr and is 4km from the point. The second car is traveling north at 50km/hr and is 3km from the point. How fast is the distance between them changing?

62km/hr

Problem Statement

We have two cars that are approaching the same point. The first car is traveling east at 40km/hr and is 4km from the point. The second car is traveling north at 50km/hr and is 3km from the point. How fast is the distance between them changing?

Solution

### 2247 video

video by Michel vanBiezen

62km/hr

At noon, ship A is 100km west of ship B. Ship A is sailing south at 35km/hr and ship B is sailing north at 25km/hr. How fast is the distance between the ships changing at 4pm?

Problem Statement

At noon, ship A is 100km west of ship B. Ship A is sailing south at 35km/hr and ship B is sailing north at 25km/hr. How fast is the distance between the ships changing at 4pm?

Solution

### 1111 video

video by PatrickJMT

unusual distance problems involving other relationships like angles and circles [shape: various]

We put the remaining types of problems into a separate category. These distance problems involve other shapes like circles and may include other parameters like angles. Once you have some experience with other types of related rates problems, you will be able to adapt what you already know to these and other unusual problems.

Basic Problems

A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 6 inches per second. What is the increase in the circumference of the ripple after 3 seconds have passed?

Problem Statement

A pebble is dropped into a pool of water, generating circular ripples. The radius of the largest ripple is increasing at a constant rate of 6 inches per second. What is the increase in the circumference of the ripple after 3 seconds have passed?

Solution

### 1117 video

video by PatrickJMT

The distance between home plate and first base on a baseball diamond is 90ft. A runner is moving towards first base at 20ft/sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 75ft away from first base?

Problem Statement

The distance between home plate and first base on a baseball diamond is 90ft. A runner is moving towards first base at 20ft/sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 75ft away from first base?

$$-100/\sqrt{61}$$ ft/sec

Problem Statement

The distance between home plate and first base on a baseball diamond is 90ft. A runner is moving towards first base at 20ft/sec. What is the rate of change in the distance between the runner and second base at the instant the runner is 75ft away from first base?

Solution

### 2084 video

video by PatrickJMT

$$-100/\sqrt{61}$$ ft/sec

You are standing still, watching a person walk straight east at 5m/sec. When he is directly north of you, he is 10 meters away. The angle $$\theta$$ is between straight north and the line from you to the person. How fast is the angle changing when $$\theta = 30^o$$?

Problem Statement

You are standing still, watching a person walk straight east at 5m/sec. When he is directly north of you, he is 10 meters away. The angle $$\theta$$ is between straight north and the line from you to the person. How fast is the angle changing when $$\theta = 30^o$$?

$$d\theta/dt = 0.375$$ rad/sec

Problem Statement

You are standing still, watching a person walk straight east at 5m/sec. When he is directly north of you, he is 10 meters away. The angle $$\theta$$ is between straight north and the line from you to the person. How fast is the angle changing when $$\theta = 30^o$$?

Solution

### 2217 video

video by Michel vanBiezen

$$d\theta/dt = 0.375$$ rad/sec

If we have a non-right triangle with sides labeled, a, b and c. Sides a and b are not changing in length. Side a has length 3m. Side b has length 4m. The angle $$\theta$$, between sides a and b, is changing at a rate of 2o/sec. When $$\theta = 60^o$$, at what rate is the opposite side, c, changing?

Problem Statement

If we have a non-right triangle with sides labeled, a, b and c. Sides a and b are not changing in length. Side a has length 3m. Side b has length 4m. The angle $$\theta$$, between sides a and b, is changing at a rate of 2o/sec. When $$\theta = 60^o$$, at what rate is the opposite side, c, changing?

Hint

1. Use the law of cosines $$c^2=a^2+b^2-2ab\cos(\theta)$$.

Problem Statement

If we have a non-right triangle with sides labeled, a, b and c. Sides a and b are not changing in length. Side a has length 3m. Side b has length 4m. The angle $$\theta$$, between sides a and b, is changing at a rate of 2o/sec. When $$\theta = 60^o$$, at what rate is the opposite side, c, changing?

$$\pi\sqrt{3}/(15\sqrt{13})$$ m/sec

Problem Statement

If we have a non-right triangle with sides labeled, a, b and c. Sides a and b are not changing in length. Side a has length 3m. Side b has length 4m. The angle $$\theta$$, between sides a and b, is changing at a rate of 2o/sec. When $$\theta = 60^o$$, at what rate is the opposite side, c, changing?

Hint

1. Use the law of cosines $$c^2=a^2+b^2-2ab\cos(\theta)$$.

Solution

### 2255 video

video by Michel vanBiezen

$$\pi\sqrt{3}/(15\sqrt{13})$$ m/sec

Intermediate Problems

In a right triangle with hypotenuse of length 10ft, one of the angles, called θ, is increasing at a constant rate of 6 radians per hour. At what rate is the side opposite θ increasing when its length is 6ft?

Problem Statement

In a right triangle with hypotenuse of length 10ft, one of the angles, called θ, is increasing at a constant rate of 6 radians per hour. At what rate is the side opposite θ increasing when its length is 6ft?

Solution

### 1116 video

video by PatrickJMT

A plane is flying at a constant altitude of 2 miles and at a constant rate of 180 mi/hr. A camera on the ground is following the plane as it flies away from the camera. How fast must the camera rotate to keep the plan in view when the camera is pointed up at an angle of π/3?

Problem Statement

A plane is flying at a constant altitude of 2 miles and at a constant rate of 180 mi/hr. A camera on the ground is following the plane as it flies away from the camera. How fast must the camera rotate to keep the plan in view when the camera is pointed up at an angle of π/3?

Solution

### 1146 video

video by MathTV

A lighthouse, 2km offshore, has a light rotating at one revolution every 10 seconds. When the light hits 1km along the shoreline, at what rate is the light moving along the shore?

Problem Statement

A lighthouse, 2km offshore, has a light rotating at one revolution every 10 seconds. When the light hits 1km along the shoreline, at what rate is the light moving along the shore?

approximately 1.57km/sec

Problem Statement

A lighthouse, 2km offshore, has a light rotating at one revolution every 10 seconds. When the light hits 1km along the shoreline, at what rate is the light moving along the shore?

Solution

### 2256 video

video by Michel vanBiezen

approximately 1.57km/sec

You CAN Ace Calculus

 derivatives chain rule implicit differentiation basics of related rates precalculus: word problems For related rates problems involving similar triangles, it may help you to review how to set up the ratios. You can find a discussion of this on the similar triangles precalculus page.

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