Optimizing the proximity.

Calculus Level 3

A helicopter is flying along the curve given by $y=x^{2}+7$ . A soldier placed at $(3,7)$ wants to shoot down the helicopter. He would find it feasible to do so when It is closest to him. Find this closest distance. If your answer is $a$ , enter the value of $a^{2}$ in the answer box.

The answer is 5.

2 solutions

Agnishom Chattopadhyay
Jul 17, 2014

Let f(x) be the square of the distance from (3,7) to (x,y).

We have: $f(x)=(3-x)^2+(7-y)^2$

Substitue y with $x^2+7$ and simplify to get:

$f(x)=x^4+x^2-6x+9$

We'll have to minimise f(x)

So,

$f'(x)=0$

The derivative of f'(x) is $4x^3+2x-6$ which we must equate to 0. Fortunately, there is one solution of x which is real and it is equal to 1.

So, x=1 minimises the distance. Plug back 1 in the equation to get the answer $\boxed{a^2=f(1)=5}$

I do not have any non-calculus way but I think something like that will be more tedious

Nice solution!!!!

By the way, @Krishna Ar, I just love your tagline, your trademark or whatever you say, your pet note!!!!! ;) Try a different one in your next question or else this will become quite boring. Nice problem and the topic, even!!!!

Kartik Sharma - 6 years, 11 months ago

Ok..I guess you mean the "Kindly.....Blah Blah Blah". Right? Was this problem too easy? How did you solve it? Extrema or any other way? Are you very good at chain rule, Implicit differentiation and the like? How far have you done calculus? @Kartik Sharma

Krishna Ar - 6 years, 11 months ago

Yeah, I was pointing to that only.

No, as I have already written, this was a good problem, not easy actually. I first used Hit and trial method and re-checked or verified it using Agnishom's method(of course I got it after some time only).

To be very clean, I suck at Calculus. Although I know derivatives(the chain rule, implicit etc.) but I haven't started to learn integration(though I know the basics). By the way, you are a level 4- 5 in 5 subjects, now. Oh, you dropped a level in Calculus!!

Can you please give me some tips on how to improve my problem-solving and some resources? Hey, I would love if you have a look at my new problems and help me in one question- HELP!!!!HELP!!!, please.

Kartik Sharma - 6 years, 11 months ago

@Kartik Sharma – Well, Honestly @Kartik Sharma -You're much better than me in calculus. You solved this problem which was taken form CBSE-Class 12 2001 paper!!! Thus- an inferior like me can't help you with pro-sol :P But I would try to solve that question u posted. BTW, From which book did u learn sine rule ? And where did u take that order of 2...question from...? I see you have a lot of awesome resources@!!!!

Krishna Ar - 6 years, 11 months ago

@Krishna Ar – Sine-cosine rule?Hmmm Actually, I don't remember. Order of 2 was taken from one AoPS's SatoNT paper, I guess. And, yeah you solved it as I supposed. Awesome resources? I myself am looking for some resources and I am just proving the idiom right that boy under armpit, chaos in the city. And yes, that is pointed to you. Please, I don't think I can be helped any better than by a level 5 person!!

Kartik Sharma - 6 years, 10 months ago

@Kartik Sharma – LOL XD That was a dialogue from Abhishek Bachchan's movie. It's not at all an idiom!!!! LOL LOL LOL @Kartik Sharma

Krishna Ar - 6 years, 10 months ago

@Krishna Ar – LOL, you got that joke. I knew it wasn't an idiom but as it is said by Mr. Ajay Devgn and it does make sense and so, it should be included as an idiom. By the way, only this idiom came in my mind(actually hyper city noise pollution was edited just for the sake of making the joke hard to find) and I still don't know a replacement for it. ;)

Coming to the topic, please!!

Kartik Sharma - 6 years, 10 months ago

Rajen Kapur
Jul 17, 2014

Any point X on the parabola is (t, 7 + t^2) which if at a minimum distance from point Y given as (3, 7) requires that line XY is to be a normal to the parabola. t = 1 meets this requirement. Squared distance between (1, 8) and (3, 7) is 5. ANSWER = 5

Optimizing the proximity.

The answer is 5.

2 solutions

0 pending reports