# Calculus - How fast is this distance changing?

• erik05
In summary, the conversation discusses a calculus problem where two people are walking at different speeds and the question asks for the rate of change of the distance between them after 15 minutes. The formula used is the cosine law, which is a generalized form of the Pythagorean theorem. The answer given is 2.125 km/h but there is some discussion about whether or not it is correct.
erik05
Hey guys. Just a quick calculus question for you all.

Two people start from the same point. One walks east at 3 km/h and the other walks northeast at 2 km/h. How fast is the distance between them changing after 15 min? Ans: 2.125 km/h

Here's what I did:

dz/dt= 3 km/h
dx/dt= 2 km/h
dy/dt= ?

2 km/h * 0.25h= 0.5 km = x
3 km/h * 0.25h= 0.75 km= z
y= 0.5590
x^2 + y^2 = z^2

2x (dx/dt) + 2y (dy/dt)= 2z (dz/dt)
dy/dt= (z (dz/dt) - x (dx/dt))/ y

So I put all the numbers in and I get 2.236. ...and that's kind of close to the answer. Anyways, if anyone could point out what I'm doing wrong, it would be much appreciated. Thanks.

I think those 2 answers (yours & the one given) are wrong...

After 1/4 hr

$$a=0.75 Km;b=0.5 Km;\alpha=45 \ \mbox{deg}$$

The distance is

$$D=\sqrt{a^{2}+b^{2}-2ab\cos\alpha}\simeq 0.53 Km$$

Then

$$\frac{dD}{dt}=\frac{\partial D}{\partial a}\frac{da}{dt}+\frac{\partial D}{\partial b}\frac{db}{dt}$$

$$\frac{dD}{dt}=\frac{2a-b\sqrt{2}}{2D}\cdot 3+\frac{2b-a\sqrt{2}}{2D}\cdot 2 \simeq 2.36 \frac{\mbox{Km}}{\mbox{hr}}$$

Daniel.

Yeah, I thought the answer was wrong. A question though, why is the formula $$D=\sqrt{a^{2}+b^{2}-2ab\cos$$ used and not pythagoras?

Because the triangle is not rectangular.That formula gives u the distance at any time.

Daniel.

Because this is not a right triangle. The Pythagorean theorem works only for right triangles. What dextercioby used was the "cosine law", a generalized form of the Pythagorean theorem: c2= a2+ b2- 2ab cos C where C is the angle opposite side c.

Ah, I see. Sorry,I have another question. Would you apply pythagoras to the question if it told you that the person that is walking northeast remains north of the person walking east at all times?

You apply the cosine law in every possible case,because you'd still have triangle at any moment of time and you'd need to find one of its sides...

Daniel.

## 1. What is calculus?

Calculus is a branch of mathematics that deals with the study of continuous change. It is used to find the rate of change of a quantity, the slope of a curve, and the area under a curve.

## 2. What is the difference between differential and integral calculus?

Differential calculus deals with the instantaneous rate of change of a function, while integral calculus deals with finding the area under a curve or the accumulation of a quantity over a certain interval.

## 3. How is calculus used in real life?

Calculus is used in many fields, such as physics, engineering, economics, and statistics. It is used to model and analyze real-world situations involving continuous change, such as motion, growth, and decay.

## 4. How is the concept of "how fast is this distance changing" applied in calculus?

In calculus, this concept is known as the derivative. It represents the rate of change of a function at a specific point and can be used to determine how fast a quantity is changing over time.

## 5. Can calculus be used to solve real-life problems?

Yes, calculus is a powerful tool in solving real-life problems. It allows us to find optimal solutions, make predictions, and understand the behavior of complex systems. Many real-world phenomena, such as population growth and stock market trends, can be modeled and analyzed using calculus.

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