Ship A is currently 85 km south of ship B. Ship A travels north at 30 km/h and ship B travels east at 20 km/h. How fast is the distance between them changing in 1.5 hours?

I have established the givens but I'm not completely sure how to proceed with this related rates problem. Do I use Pythagoras or what exactly should I do...if I understand the process I can do more practice


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You start with Pythagorean Theorem,

$$a^2 + b^2 = c^2$$

Then you take the derivative of both sides, and divide both sides by 2,

$$2a\frac{da}{dt} + 2b\frac{db}{dt} = 2c\frac{dc}{dt}$$ $$a\frac{da}{dt} + b\frac{db}{dt} = c\frac{dc}{dt}$$

Now you solve for the rate at which the distance between them is changing,

$$\frac{a\frac{da}{dt} + b\frac{db}{dt}}{c} = \frac{dc}{dt}$$

Then, you substitute your values into this equation. Hope this helps!

  • 1
    $\begingroup$ Couldn't have said it better. The only thing is the OP needs to make sure he makes $\frac{da}{dt}$ negative since ship A is going toward ship B initially. $\endgroup$ – Lanier Freeman Dec 1 '15 at 1:53

I would recommend drawing a picture of where the ships are in the beginning and where they are 1.5 hours later. Your second picture should be a right triangle with side lengths:

$$a=85 \ km-\bigg(30 \ \frac{km}{h}*1.5 \ h\bigg)=40 \ km,$$ $$b=20 \ \frac{km}{h}*1.5 \ h=30 \ km.$$

Then use Pythagorean Theorem to find the hypotenuse.

$$c^2=a^2+b^2$$ $$c^2=40^2+30^2$$ $$c^2=2,500$$ $$c=50 \ km$$

You also know that $\frac{da}{dt}=-30\frac{km}{h}$ because ship A is moving $30\frac{km}{h}$ and making side $a$ shorter as it moves. Similarly, $\frac{db}{dt}=20\frac{km}{h}$ because side $b$ of your triangle is growing at that rate.

Now all you need to do is differentiate the Pythagorean Theorem with respect to time, solve for $\frac{dc}{dt}$, and plug in the other values.

$$c^2=a^2+b^2$$ $$2c\frac{dc}{dt}=2a\frac{da}{dt}+2b\frac{db}{dt}$$ $$c\frac{dc}{dt}=a\frac{da}{dt}+b\frac{db}{dt}$$ $$\frac{dc}{dt}=\frac{a\frac{da}{dt}+b\frac{db}{dt}}{c}$$ $$\frac{dc}{dt}=\frac{40*(-30)+30*20}{50}=-12 \ \frac{km}{h}$$

Here's a link to a site with a much more detailed explanation of how to solve a problem like this. This site has a good explanation of how to draw your picture and go through the problem.



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