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17calculus > derivatives > related rates > distances |
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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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How to Ace the Rest of Calculus: The Streetwise Guide, Including MultiVariable Calculus |
Related Rates Involving Distances |
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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. |
This page covers related rates problems specifically involving distances. These types of problems involve
- ladders sliding down walls
- 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 |
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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 ] |
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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 1 |
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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? |
solution |
Practice 2 |
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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? |
solution |
Practice 3 |
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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? |
solution |
cars, boats, planes and people moving in the same or different directions [ shape: triangle ] |
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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 walking ( or 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 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 this link positions the page) and study the picture to the right. ]
Basic Problems |
Practice 4 |
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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? |
solution |
Practice 5 |
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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? |
solution |
Practice 6 |
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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? |
solution |
Practice 7 |
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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 |
Intermediate Problems |
Practice 8 |
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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? |
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Practice 9 |
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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 |
unusual distance problems involving other relationships like angles and circles [ shape: various ] |
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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 |
Practice 10 |
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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 |
Practice 11 | |
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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? | |
answer |
solution |
Intermediate Problems |
Practice 12 |
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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? |
solution |
Practice 13 |
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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 π/3? |
solution |