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Given an equation, find a rate (related rates problem)

Calculus Related Rates Problem: Given an equation, find the rate. Not all Related Rates problems are word problems. Here’s a different type that came from a student. The problem gives you an equation, and then asks you to find a rate: If $y = x^3 + 2x$ and $\dfrac{dx}{dt} = 6$, find $\dfrac{dy}{dt}$ when $x=5.$ […]

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snowball melts, area decreases at given rate (related rates problem)

Calculus Related Rates Problem: As a snowball melts, its area decreases at a given rate. How fast does its radius change? A spherical snowball melts symmetrically such that it is always a sphere. Its surface area decreases at the rate of $\pi$ in$^2$/min. How fast is its radius changing at the instant when $r = […]

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shadow lamp post (related rates problem)

Calculus Related Rates Problem: Lamp post casts a shadow of a man walking. A 1.8-meter tall man walks away from a 6.0-meter lamp post at the rate of 1.5 m/s. The light at the top of the post casts a shadow in front of the man. How fast is the “head” of his shadow moving […]

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how fast is the ladder’s top sliding (related rates problem)

Calculus Related Rates Problem: How fast is the ladder’s top sliding? A 10-ft ladder is leaning against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house. When the base has slid to 8 ft from the house, it […]

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water drains from a cone (related rates problem)

Calculus Related Rates Problem: How fast is the water level falling as water drains from the cone? An inverted cone is 20 cm tall, has an opening radius of 8 cm, and was initially full of water. The water now drains from the cone at the constant rate of 15 cm$^3$ each second. The water’s […]

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Angle change as a ladder slides (related rates problem)

Calculus Related Rates Problem: At what rate does the angle change as a ladder slides away from a house? A 10-ft ladder leans against a house on flat ground. The house is to the left of the ladder. The base of the ladder starts to slide away from the house at 2 ft/s. At what […]

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snowball melts (related rates problem)

Calculus Related Rates Problem: As a snowball melts, how fast is its radius changing? A spherical snowball melts at the rate of $2 \pi$ cm$^3$/hr. It melts symmetrically such that it is always a sphere. How fast is its radius changing at the instant $r = 10$ cm? Hint: The volume of a sphere is […]

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Related Rates

Related Rates: Problems and Solutions We will solve every Related Rates problem using the same Problem Solving Strategy time and again. You can see an overview of that strategy here (link will open in a new tab). As stated in the Problem Solving Strategy, nearly every Related Rates problem will fall into one of four […]

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Related Rates Problem – Cylinder Drains Water

A student recently wrote to ask if we’d help solve a common related rates problem about water draining from a cylindrical tank. The problem was something like this: Cylinder Drains Water. A cylinder filled with water has a 3.0-foot radius and 10-foot height. It is drained such that the depth of the water is decreasing […]

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4 Steps to Solve Any Related Rates Problem – Part 2

In our last post, we developed four steps to solve any related rates problem. We introduced three examples to illustrate the basic ideas, and solved two of them there. As promised, we’ll solve the third here. Water Leaving a Cone Example Here’s the problem statement, now with some additional details about the cone itself and […]

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