# Math Help - Distance

1. ## Distance

I dont quite get the picture of this question....

Aruneem travels from Patna to Calcutta, a distance of 200 km at the speed of 40 km/hr . At the same time, Nilanjan starts from Calcutta at a speed of 20 km/hr along a road, which is perpendicular to the road on which Aruneem is travelling. When will Aruneem and Nilanjan be closest to each other?

ans: In 4 hr

2. Hello, MathLearner!

Aruneem travels from Patna to Calcutta, a distance of 200 km at the speed of 40 km/hr.
At the same time, Nilanjan starts from Calcutta at a speed of 20 km/hr along a road
which is perpendicular to the road on which Aruneem is travelling.
When will Aruneem and Nilanjan be closest to each other?

Code:
           40t    A     200-40t
P * - - - - - * - - - - - - - - * C
*              |
*           |
*        | 20t
*     |
*  |
* B

Aruneem starts at $P$ and travels toward $C$ at 40 km/hr.
. . After $t$ hours, he has travelled $40t$ km to point $A.$
Note that: . $AC \:=\:200-40t$

At the same time, Nilanjan starts at $C$ and travels south at 20 km/hr.
. . After $t$ hours, he has travelled $20t$ km to point $B.$

Their distance is $AB$, the hypotenuse of right triangle $ACB.$

Hence: . $D \;=\;AB^2 \:=\:(200-40t)^2 + (20t)^2 \:=\:2000t^2 - 16,000t + 40,000$

Then: . $D' \;=\;4000t - 16,000 \:=\:0 \quad\Rightarrow\quad t \,=\,4$

Therefore, minimum distance occurs in 4 hours.

3. soroban i understood everything except where you got the D' equation.

4. Hello, Jonboy!

We have right triangle $ABC$ with $C = 90^o,\;AC \:=\:200-40t,\;BC \:=\:20t$
. . and we want to minimize $AB.$

Pythagorus says: . $AB^2 \;=\;AC^2 + BC^2 \;=\;(200-40t)^2 + (20t)^2$

See it?

5. hello again Sorobon !!!

Brilliant answer.. now i understood what exactly the question meant...

Thanks

6. Thanks for trying to explain Soroban, I see what the picture means. But as far as getting the D prime equation, that's not something I've learned and I don't see how you got there.

But if I wanted to find the minimum here, I would've just done:

$\frac{ - b}{2a} = \frac{16000}{4000} = 4$

7. Originally Posted by Jonboy
soroban i understood everything except where you got the D' equation.
He differentiated.

8. Originally Posted by Isomorphism
He differentiated.
So wish I knew how to do that. I will next year when I take Calc. Maybe sooner if I can find a simple explanation of Calc at my library.

9. Originally Posted by Jonboy
So wish I knew how to do that. I will next year when I take Calc. Maybe sooner if I can find a simple explanation of Calc at my library.
Differentiation is just finding the equation that will give slope at a point. It is defined as

Let $f(x)$ be a differentiable function.

Then $\frac{dy}{dx}=f'(x)=\lim_{\Delta{x}\to{0}}\frac{f\ left(x+\Delta{x}\right)-f(x)}{\Delta{x}}$

10. calculus has such a beautiful notation. thanks for explaining Mathstud28 to this lost person.