## 17Calculus Derivatives - Related Rates Involving Distances

##### 17Calculus

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
- shadow questions with people moving
- cars, boats and airplanes moving in the same or different directions
- baseball questions involving runners
- 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 common 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.

Triangles [ladder leaning on a wall]

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. ]

Practice

A 13ft ladder is leaning against a house. The ladder slides down the wall at a rate of 3ft/min. How fast is the ladder moving away from the base of the wall when the foot of the ladder is 5ft from the wall? How fast is the area of the triangle changing? How fast is the angle between the ladder and the ground changing?

Problem Statement

A 13ft ladder is leaning against a house. The ladder slides down the wall at a rate of 3ft/min. How fast is the ladder moving away from the base of the wall when the foot of the ladder is 5ft from the wall? How fast is the area of the triangle changing? How fast is the angle between the ladder and the ground changing?

Solution

### The Organic Chemistry Tutor - 3726 video solution

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A ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1ft/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 ladder 10ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1ft/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

### blackpenredpen - 3736 video solution

video by blackpenredpen

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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

### Krista King Math - 1108 video solution

video by Krista King Math

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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

### MathTV - 1145 video solution

video by MathTV

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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

### Michel vanBiezen - 2251 video solution

video by Michel vanBiezen

4m

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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

### MIP4U - 2460 video solution

video by MIP4U

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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

### Khan Academy - 1109 video solution

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Triangles [moving cars, boats, and planes]

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.

Practice

Two cars are moving starting at the same point. One travels north at 30mph and the other travels east at 40mph. At what rate is the distance between the two cars changing 3 hours later?

Problem Statement

Two cars are moving starting at the same point. One travels north at 30mph and the other travels east at 40mph. At what rate is the distance between the two cars changing 3 hours later?

Solution

### The Organic Chemistry Tutor - 3729 video solution

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At 1:00pm, ship B is 150 miles east of ship A. Ship A is moving 30mph north and ship B is moving 20mph south. How fast is the distance between the ships changing at 3pm?

Problem Statement

At 1:00pm, ship B is 150 miles east of ship A. Ship A is moving 30mph north and ship B is moving 20mph south. How fast is the distance between the ships changing at 3pm?

Solution

### The Organic Chemistry Tutor - 3730 video solution

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An airplane at an altitude of 3 miles travels horizontally at 400mph. It passes directly over a radar station. What is the rate at which the distance between the plane and the radar station changing when the plane is 6 miles away from the station?

Problem Statement

An airplane at an altitude of 3 miles travels horizontally at 400mph. It passes directly over a radar station. What is the rate at which the distance between the plane and the radar station changing when the plane is 6 miles away from the station?

Solution

### The Organic Chemistry Tutor - 3731 video solution

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An airplane travels horizontally at 600mph at an altitude of 3 miles toward an observer that is currently 10miles away. How fast is the angle between the ground and the observer’s line of sight to the airplane changing at this instant?

Problem Statement

An airplane travels horizontally at 600mph at an altitude of 3 miles toward an observer that is currently 10miles away. How fast is the angle between the ground and the observer’s line of sight to the airplane changing at this instant?

Solution

### The Organic Chemistry Tutor - 3732 video solution

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Car A is traveling west at 50mph and car B is traveling north at 60mph. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3mi and car B is 0.4mi from the intersection?

Problem Statement

Car A is traveling west at 50mph and car B is traveling north at 60mph. Both are headed for the intersection of the two roads. At what rate are the cars approaching each other when car A is 0.3mi and car B is 0.4mi from the intersection?

Solution

### blackpenredpen - 3737 video solution

video by blackpenredpen

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At noon, ship A is 100km west of ship B. Ship A is sailing south at 40km/hr is ship B is sailing north at 20km/hr. How fast is the distance between the ships changing at 4:00pm?

Problem Statement

At noon, ship A is 100km west of ship B. Ship A is sailing south at 40km/hr is ship B is sailing north at 20km/hr. How fast is the distance between the ships changing at 4:00pm?

Solution

### blackpenredpen - 3738 video solution

video by blackpenredpen

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A balloon leaves the ground 500feet away from an observer and rises vertically at the rate of 140ft/min. At what rate is the angle of inclination of the observer's line of sight increasing at the instant when the balloon is 500ft above the observer's eye?

Problem Statement

A balloon leaves the ground 500feet away from an observer and rises vertically at the rate of 140ft/min. At what rate is the angle of inclination of the observer's line of sight increasing at the instant when the balloon is 500ft above the observer's eye?

Solution

### blackpenredpen - 3740 video solution

video by blackpenredpen

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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

### PatrickJMT - 1110 video solution

video by PatrickJMT

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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

### Krista King Math - 1112 video solution

video by Krista King Math

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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

### Michel vanBiezen - 2219 video solution

video by Michel vanBiezen

80 ft/sec

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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 22245 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 22245 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 22245 mi/hr?

Solution

### Michel vanBiezen - 2245 video solution

video by Michel vanBiezen

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

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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

### Michel vanBiezen - 2247 video solution

video by Michel vanBiezen

62km/hr

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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

### PatrickJMT - 1111 video solution

video by PatrickJMT

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You will very likely be asked to solve questions involving light casting a shadow of a person as the person moves toward or away from the light source. This situation involves solving a problem with a right triangle.

Practice

A 6ft man walks at a rate of 3ft/sec away from a street light that is 21 feet above the ground. At what rate is the length of his shadow changing when he is 8ft from the light? At what rate is the tip of his shadow moving when he is 10ft from the light?

Problem Statement

A 6ft man walks at a rate of 3ft/sec away from a street light that is 21 feet above the ground. At what rate is the length of his shadow changing when he is 8ft from the light? At what rate is the tip of his shadow moving when he is 10ft from the light?

Solution

### The Organic Chemistry Tutor - 3719 video solution

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A street light is mounted on a pole 24ft tall. A main 6ft tall walks away from the pole at a rate of 4ft/sec. How fast is the tip of his shadow moving when he is 20ft away from the pole? How fast is the length of his shadow changing at this instant?

Problem Statement

A street light is mounted on a pole 24ft tall. A main 6ft tall walks away from the pole at a rate of 4ft/sec. How fast is the tip of his shadow moving when he is 20ft away from the pole? How fast is the length of his shadow changing at this instant?

Solution

### The Organic Chemistry Tutor - 3727 video solution

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A spotlight on the ground shines on a wall 18m away. If a 2m tall man walks toward the building at a speed of 2m/sec, how fast is the length of his shadow on the building changing when he is 8m from the building?

Problem Statement

A spotlight on the ground shines on a wall 18m away. If a 2m tall man walks toward the building at a speed of 2m/sec, how fast is the length of his shadow on the building changing when he is 8m from the building?

Solution

### The Organic Chemistry Tutor - 3728 video solution

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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

### TheInfiniteLooper - 1113 video solution

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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

### Michel vanBiezen - 2207 video solution

video by Michel vanBiezen

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

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A 6ft tall man walks away from a 15ft lamp post 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 lamp post 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

### rootmath - 1114 video solution

video by rootmath

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Triangles [baseball]

An American game, baseball, provides some interesting related rates problems. 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.

Practice

A baseball diamond is a square with side 100ft. A batter hits the ball and runs toward first base with a speed of 20ft/sec. At what rate is his distance from second base changing when he is 30ft from first base? How fast is his distance changing from home plate if he is halfway from first base running toward second base?

Problem Statement

A baseball diamond is a square with side 100ft. A batter hits the ball and runs toward first base with a speed of 20ft/sec. At what rate is his distance from second base changing when he is 30ft from first base? How fast is his distance changing from home plate if he is halfway from first base running toward second base?

Solution

### The Organic Chemistry Tutor - 3716 video solution

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A baseball diamond is a square with 90ft sides. 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 90ft sides. 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

### PatrickJMT - 1115 video solution

video by PatrickJMT

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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

### PatrickJMT - 2084 video solution

video by PatrickJMT

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

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Various [other relationships like angles and circles]

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

Practice

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

### PatrickJMT - 1117 video solution

video by PatrickJMT

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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

### Michel vanBiezen - 2217 video solution

video by Michel vanBiezen

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

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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

### Michel vanBiezen - 2255 video solution

video by Michel vanBiezen

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

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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

### PatrickJMT - 1116 video solution

video by PatrickJMT

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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 $$\pi/3$$?

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 $$\pi/3$$?

Solution

### MathTV - 1146 video solution

video by MathTV

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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

### Michel vanBiezen - 2256 video solution

video by Michel vanBiezen

approximately 1.57km/sec

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