The best examples of examples of example problems on related rates

Let’s start where most calculus textbooks love to start: geometry. These are the best examples of examples of example problems on related rates because they’re visual, repeatable, and easy to adapt.

1. The expanding circle (and why area and radius don’t grow at the same pace)

Imagine a circular oil spill on water. The radius of the spill is increasing at a steady rate of 2 feet per minute. When the radius is 10 feet, how fast is the area growing?

This is a textbook example of a related rates setup:

• You’re given how fast the radius changes: \(dr/dt\).
• You’re asked how fast the area changes: \(dA/dt\).
• The link between them is the geometry formula \(A = \pi r^2\).

Different textbooks give slightly different numbers, but the pattern stays the same. If you’re collecting examples of examples of example problems on related rates for practice, this “expanding circle” should be near the top of your list. It forces you to:

• Write a known formula.
• Differentiate both sides with respect to time.
• Plug in the instant you care about.

2. The filling cone: volume, height, and radius all changing

Picture a cone-shaped tank being filled with water. Suppose:

• The tank is 12 feet tall with a radius of 6 feet at the top.
• Water is poured in at 3 cubic feet per minute (that’s \(dV/dt\)).
• You’re asked how fast the water level is rising when the water is 4 feet deep.

This is one of the best examples because:

• It connects volume and height.
• It often forces you to relate radius and height using similar triangles.
• It’s very realistic: conical tanks are common in engineering and industry.

Here’s the structure you see repeated in many examples of examples of example problems on related rates:

• Start with \(V = \frac{1}{3}\pi r^2 h\).
• Use the geometry of the tank to express \(r\) in terms of \(h\).
• Differentiate with respect to time to connect \(dV/dt\) and \(dh/dt\).

In 2024–2025, you’ll still see this cone example in AP Calculus, university calculus, and online resources like MIT OpenCourseWare (ocw.mit.edu). The numbers may change, but the structure is the same.

3. The sliding ladder: a classic right triangle in motion

Another staple in lists of examples of examples of example problems on related rates is the sliding ladder:

A 15-foot ladder leans against a vertical wall. The bottom is being pulled away from the wall at 2 feet per second. How fast is the top sliding down the wall when the bottom is 9 feet from the wall?

Here’s why this one shows up everywhere:

• It uses the Pythagorean theorem: \(x^2 + y^2 = 15^2\).
• You relate horizontal speed \(dx/dt\) to vertical speed \(dy/dt\).
• It’s a perfect example of how a fixed length can still create changing sides.

If you’re looking for the best examples to prepare for exams, a ladder problem is almost guaranteed to appear in some form.

Motion and distance: real examples with cars, planes, and people

Geometry is great, but many students connect better with motion: cars, planes, runners, and so on. These are real examples of how related rates show up in the physical world.

4. Two cars at a right angle: distance between them changing

Consider two cars approaching an intersection at right angles:

• Car A is heading north at 40 miles per hour.
• Car B is heading east at 30 miles per hour.
• At a certain instant, Car A is 0.5 miles south of the intersection and Car B is 0.8 miles west of it.

How fast is the distance between the cars changing at that instant?

This setup is one of the best examples of examples of example problems on related rates involving motion because:

• You build a right triangle with legs as the distances of the cars from the intersection.
• The hypotenuse is the distance between the cars.
• You relate \(dx/dt\), \(dy/dt\), and \(dz/dt\) using the Pythagorean theorem.

This pattern appears in physics courses, too. Many calculus-based physics classes, like those at major universities (for example, Harvard’s introductory calculus-based physics courses), use nearly identical related rates problems to connect motion and geometry.

5. The airplane and the observer on the ground

An airplane flies horizontally at an altitude of 3 miles and a speed of 500 miles per hour. It passes directly over an observer on the ground. How fast is the straight-line distance between the plane and the observer increasing when the plane is 4 miles horizontally from the observer?

This is a real example of how navigation and tracking problems can be modeled:

• The altitude is fixed, so one leg of your right triangle is constant.
• The horizontal distance is changing at a known rate.
• You’re asked about the rate of change of the slant distance.

This is a clean, modern-feeling twist on the classic related rates triangle.

6. The runner and the spotlight: angles and shadows

A runner is moving along a straight track at 15 feet per second. A spotlight is fixed 20 feet from the track, shining on the runner. How fast is the angle of the spotlight changing when the runner is 25 feet from the closest point to the light?

This is one of the more interesting examples of examples of example problems on related rates because:

• It introduces an angle \(\theta\) into the picture.
• You use tangent: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).
• You end up relating \(dx/dt\) to \(d\theta/dt\).

Problems like this show up in engineering and robotics, where you care about how quickly a sensor or camera must rotate to track a moving object.

Growth, decay, and modern applications: 2024–2025 flavored examples

Related rates aren’t just about ladders and cars. They also describe how quantities change together in more modern contexts: population growth, medical dosing, environmental changes, and more.

7. A balloon rising and a camera tracking from the ground

A weather balloon is rising vertically at 10 feet per second. A camera is placed on the ground 50 feet from the launch point. How fast is the distance between the camera and the balloon increasing when the balloon is 30 feet above the ground?

This is structurally similar to the airplane example, but with a vertical motion instead of horizontal. Weather balloons and atmospheric measurements are still used widely; agencies like the National Weather Service (weather.gov) rely on such data. The math behind tracking devices and sensors often looks just like this.

Again, you:

• Build a right triangle with a fixed horizontal leg (50 feet).
• Let the vertical leg be the balloon’s height, changing at 10 ft/s.
• Use the Pythagorean theorem to relate distance and height.

8. Water draining from a tank: volume decreasing, height falling

Consider a cylindrical tank draining through a small hole at the bottom. Suppose:

• The tank has a radius of 5 feet.
• Water is leaving at 2 cubic feet per minute.
• You’re asked how fast the water level is falling when the water depth is 6 feet.

This is a practical engineering example of examples of example problems on related rates. You’ll see similar problems in fluid dynamics and environmental engineering courses.

You connect:

• Volume of a cylinder: \(V = \pi r^2 h\).
• Rate of change of volume, \(dV/dt\), is negative because water is leaving.
• Rate of change of height, \(dh/dt\), is what you want.

This type of problem shows up in real-world modeling, for example when estimating how quickly a storage tank will empty in emergency planning.

9. Population and resource use: a conceptual 2024–2025 example

Even though real demographic modeling is more complex, you can still use related rates ideas to think about linked growth.

Imagine a city where:

• Population is increasing at 5,000 people per year.
• The number of hospital beds per person is being kept constant by building new facilities.

If the city has 200,000 people and 2.5 beds per 1,000 people, how fast must the number of beds increase to maintain the same ratio?

You’d set up:

• A relationship between population and beds.
• A rate of change of population from demographic data.
• A required rate of change of beds.

While this is more of a conceptual illustration than a strict geometry problem, it’s still one of the best examples to show students how related rates thinking extends beyond triangles and tanks. Public health organizations like the CDC (cdc.gov) and NIH (nih.gov) use far more detailed models, but the core idea of linked changing quantities is the same.

How to recognize patterns across different related rates examples

When you look across all these examples of examples of example problems on related rates, some patterns jump out. If you can spot these, you can handle almost any new problem that shows up on homework or an exam.

Common structure across the best examples

Most related rates problems quietly follow the same script:

You start with a picture in your head (or on paper). It might be:

• A right triangle for cars, planes, ladders, cameras, or spotlights.
• A cone or cylinder for tanks, cups, or pools.
• A circle for oil spills or expanding ripples.

Then you find a relationship between the quantities:

• Geometry formulas like \(A = \pi r^2\), \(V = \pi r^2 h\), or \(x^2 + y^2 = c^2\).
• Trig relationships like \(\tan \theta = x/y\).
• Ratios or proportionality (for population and resources, for example).

Next, you differentiate with respect to time. This is where the related part of “related rates” lives. You use the chain rule and implicit differentiation to connect the rates of change.

Finally, you plug in the instant you care about. You’re not solving for all time, just a snapshot.

Why these examples keep showing up in 2024–2025

Even as textbooks move online and new resources appear, the core examples haven’t changed much. You’ll still see the same ladder, cone, car, and balloon problems on:

• AP Calculus exams.
• University calculus courses.
• Online learning platforms and open courseware.

What has changed is how they’re presented. Instructors now often tie them to:

• Sensors and robotics (spotlights, cameras, drones).
• Environmental modeling (spills, draining tanks, rainfall).
• Health and population models (resource planning, spread of information or disease).

So when you’re studying, don’t just memorize one example of a ladder problem. Instead, notice the pattern so you can adapt it to any new scenario that looks even vaguely similar.

FAQ: Common questions about related rates examples

Q: What are some of the best examples of related rates problems to practice first?
Start with the sliding ladder, the expanding circle, and the conical tank filling with water. These are the best examples of examples of example problems on related rates because they show you triangles, circles, and volumes—the main geometric building blocks.

Q: Can you give an example of a real-world related rates problem outside of pure geometry?
Yes. A strong example of a real-world problem is tracking a drone with a ground-based camera: the drone’s horizontal position changes at a known rate, and you want to know how fast the camera angle must change to keep it centered. This is a modern cousin of the spotlight-and-runner example.

Q: How many different types of examples of related rates problems should I know for an exam?
You don’t need hundreds. If you understand a handful of patterns—right triangles, circular areas, cylindrical and conical volumes, and angle-based problems—you can handle most exam questions. Many examples of examples of example problems on related rates are just variations on those themes.

Q: Where can I find more reliable practice problems and explanations?
Look for calculus materials from universities and trusted organizations. Sites like MIT OpenCourseWare (ocw.mit.edu), major university math departments, and educational non-profits like Khan Academy offer large collections of worked examples.

Q: How do I know which quantities to differentiate in a related rates question?
Focus on what’s changing with time. If a quantity is described as moving, growing, shrinking, or being filled or drained, it has a rate of change. The problem will usually give you some rates and ask for another. Your job is to write an equation that connects all those changing quantities, then differentiate both sides with respect to time.

If you treat each of these as templates rather than one-off puzzles, you’ll start to see that most new related rates questions are just familiar stories with different numbers and names. That’s the real power of working through these examples of examples of example problems on related rates: they train your eye to see the same structure hiding underneath very different word problems.