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Questions tagged [related-rates]

For questions about related rates, computing a rate of change by relating it to other quantities whose rates are known.

In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. Typically, the rate of change is with respect to time.

Related rates have some broad applications in science and engineering. Related rates is often first introduced as an application of implicit differentiation. Also, note the use of the chain rule when differentiated with respect to time or any other variable.

See also: Wikipedia

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Visualizing derivatives w.r.t another function

I came across some derivatives w.r.t functions such as, $\frac{d(7^x)}{d(x^7)}$, I tried plotting their graphs and seeing how we can relate the change in the value of $7^x$ as the function $x^7$ is changing. Can someone please provide visualization…
heerboi
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Is this related rates solution correct, or a crazy coincidence?

One of my students arrived at a correct solution to a straightforward related rates problem, but I can't understand their method... Problem: an airplane flies at a constant altitude of 10km and speed 0.2km/s in the direction of a stationary camera…
Michael Megliola
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To find rate of change of area of triangle when rate of change and value of length of base and height are 3cm/min, 5cm/min and 8cm,10cm respectively.

I am trying understand very simple related rates problem (area of triangle on youtube): The base of a right triangle is increasing at 3cm/min while the height of the triangle is increasing at a rate of 5cm/min. How fast is the area of the triangle…
Michal
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Help Finding $\frac{dV}{dA}$ in terms of $r$

The volume of a sphere, $V$ cm³, of radius $r$ is given by the formula $V = \frac{4}{3} \pi r^3$. The surface area of a sphere $A$ cm² of radius $r$ cm is given by the formula $A=4\pi r^2$. Find $\frac{dV}{dA}$ in terms of $r$. Here's my workings to…
Anna
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How fast is the car traveling along the highway? MIT 18.01 OCW final exam

The following question appeared in the MIT 18.01 single variable calculus OCW final exam: A highway patrol plane is flying 1 mile above a long, straight road, with constant ground speed of 120 m.p.h. Using radar, the pilot detects a car whose…
Kais Hasan
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Related rates ---not use chain rules?

Here is the question:Two people start from the same point at the same time. One walks north at 2 mi/h and the other walks west at 4 mi/h. How fast is the distance between them changing after 30 minutes?. I understand here I should use…
elainehxw
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What did I do wrong in this simple related rates problem?

The question is A street light is mounted at the top of a 15-ft-tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ftys along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? I drew a…
iwjueph94rgytbhr
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Trigonometric related rates

I need help with the relationship between the variables and the derivatives In the below question, I thought it would be something like $\tan(\theta)=\frac{y}{x}$ where $y=40t$ and $x=25-30t$. And then, at the maximum, there would be no change in…
Ally
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Differential equation: mixing salt into water problem where there is also leakage. Is my differential equation correct, and how to solve?

Initially, a tank contains $50$ litres of water and the tank water contains no salt. Salt is added to the water at time $t=0,$ at a constant rate of $5$ grams per minute. The salt does not change the volume of the water in the tank. The water in…
Adam Rubinson
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The rate of change in the area of a circle sector

a circular clock of radius 5 inches .At time t minutes past noon, how fast is the area of the sector of the circle between the hour and minute hand increasing? I used implicit differentiation on the equation $A=\frac{(r²\theta)}{2}$ with r as a…
Saif DOGHRI
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Calculus for the Practical Man: Chapter 4, Problem 16

A rope 28 feet long is attached to a block on level ground and runs over a pulley 12 feet above the ground. The rope is stretched taut and the free end is drawn directly away from the block and pulley at the rate of 13 ft. per sec. How fast will…
Rafael Vergnaud
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Related Rates: Shadow of Ball Problem

A light is at the top of a pole $50$ft high. A ball is dropped from the same height from a point $30$ft away from the light. How fast is the shadow of the ball moving along the ground $1\over 2$ seconds later? (Assume the ball falls a distance $s =…
Camelot823
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Using related rates, why can we ignore dimensions and consider rates of change, when in seemingly identical situations, we must consider both?

So I was preparing a lesson on related rates for the calc 1 class I am a TA for and I realized that the two problems below in the photo are basically identical: Given a right triangle, x, x', y, y' are known, Find z' (or s'). Problem #1 and $4 are…
Haphazardly Proven
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Calculus - Related Rates : Shadow Problem

Question state is : "A person 6 ft tall walks at 5 ft/s along one edge of a road 30 ft wide. On the other edge of the road is a light atop a pole 18 ft high. How fast is the length of the person’s shadow (on the horizontal ground) increasing when…
user896498
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What is the error in my work for this related rates problem?

I have no clue why my math isn't working out and it is very frustrating. I feel as if I have the concepts correct; however, I just keep getting the wrong answer. Help?
Marlon Henderson
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