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Two straight roads intersect at right angles. Two men, A and B, are 100 km from the intersection, one on each road. They drive towards the intersection at 30km^1 and 40 khmh^1 respectively. Find the distance of each driver from the intersection as a function of t, the time in hours for which they are driving. Hence find their distance apart, d(t), at any time. For what value of t is the distance apart least?
 2 years ago
 2 years ago
Two straight roads intersect at right angles. Two men, A and B, are 100 km from the intersection, one on each road. They drive towards the intersection at 30km^1 and 40 khmh^1 respectively. Find the distance of each driver from the intersection as a function of t, the time in hours for which they are driving. Hence find their distance apart, d(t), at any time. For what value of t is the distance apart least?
 2 years ago
 2 years ago

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Ishaan94Best ResponseYou've already chosen the best response.0
ah you posted it again
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Yup, i was staring it for hours i didn't even know how to do it _
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
I tried working backwards but it didn't work. Or using the pythagorus theorem *sigh*
 2 years ago

saifoo.khanBest ResponseYou've already chosen the best response.0
that looks soo scary.
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
LOL, I know its killing me =/
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
I know, I already have the diagram, but how can I apply the pythagorus theorem ? I tried it nd I got huge numbers
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Like this : 3999.23 / dt = 2000a + 2000b / dt ?
 2 years ago

IanTBest ResponseYou've already chosen the best response.0
dw:1317388825437:dw this is the graph of time against distance. Very badly drawn.
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.0
\[H^2 = (100  30t)^2 +(10040t)^2\]
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Don't you find H as well ? That is 141.42 ?
 2 years ago

SimonSBest ResponseYou've already chosen the best response.2
Let A be the location of the driver driving at 40 km/hr and B be the location of the driver driving at 30 km/hr A and B are measured by distance from the intersection. Then the distance you want is \[d = \sqrt{A^2 + B^2}\] Now initially of course, i.e., when t = 0, A = B = 100 km and hence d= 100.sqrt(2) which is approximately 141 km. Now, what about for times t > 0. Well, A drives towards the intersection so if t is in hours: A(t) = 100  40t Making sense so far? If so. What is B(t) ?
 2 years ago

IanTBest ResponseYou've already chosen the best response.0
from the graph, the slope is 0.333... The equation of the line is t = (0.0333...)d + 3.333...
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
SimonS is trying to explain what Ishaan is saying and exactly right. So Mimi, what is B(t)?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
and for the record what Ian is saying is unfortunately wrong.
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
B(t) is the distance B from the intersection ?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
The distance from the intersection of the driver driving at 30 km/hr
 2 years ago

IanTBest ResponseYou've already chosen the best response.0
For my sake, can you point out the error. Reading the other posts, I realize I'm not helping with the problem, but can you show me where I went wrong?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
@mimi, do you see where the formula for A(t) came from?
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Yup, do i expand (10030t)^2 ?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
Or not yet. So A(t) = 100  40t and hence B(t) = ... ?
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Then the pythagoras theorem xD
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
so d(t)^2 = A(t)^2 + B(t)^2 = .... Now expand.
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
@Ian: you'll see that the equation we ultimately arrive at for d is not linear in t. Where you went wrong was writing down an equation without having really thought through the problem.
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
H^2 = 2500t^2  14000t + 20000 What do i do next ? differentiate it ?
 2 years ago

phiBest ResponseYou've already chosen the best response.0
you want H(t)= sqrt(stuff) then differentiate, set to zero and solve for t
 2 years ago

phiBest ResponseYou've already chosen the best response.0
And, if you are comfortable with the chain rule, it wasn't necessary to expand all that stuff. But you already did that work...
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
Yes. So now you follow the usual procedure to find the minimum of a function. By the way, H is at a minimum exactly when H^2 is at a minimum. So you can take the square root, but it's going to make the differentiation a little more difficult for no more insight. If I was doing this problem I would analyze H^2(t), the equation you have written down.
 2 years ago

IanTBest ResponseYou've already chosen the best response.0
My graph is actually for t and the distance of A from the intersection. Not the final function. Am I correct, unhelpful as it may be?
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
ah, in which case, the graph was right.
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Can I differentiate it keeping h^2 ?
 2 years ago

phiBest ResponseYou've already chosen the best response.0
@JamesJ True, but the question asks for h(t)
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
Let f(t) = h^2(t). Differentiate this function.
 2 years ago

Ishaan94Best ResponseYou've already chosen the best response.0
won't make any difference differentiate with h^2 so it is easy
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
I'm just saying it's easier to differentiate the function f(t) = h^2(t). For the ultimate answer you need h(T) for whatever T gives the minimum. But it's easier to deal with the differentiation if you keep it in the h^2 form.
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
But if this causes anyone confusion, differentiate the function h(t)
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
h^2 = 5000t1400 h = squareroot ( 5000t1400) Is that right >
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
No. As you write above: H^2 = 2500t^2  14000t + 20000
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Don't I differentiate it >
 2 years ago

SimonSBest ResponseYou've already chosen the best response.2
Yes, but you made a few mistakes and your notation is wrong. (h^2) ' = 5000t  14000
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Okay, the its h' = squareroot (5000t1400)
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
No, no, definitely not. (h^2) ' does not equal (h')^2
 2 years ago

phiBest ResponseYou've already chosen the best response.0
you are dropping a zero on 14000
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
Just set (h^2) ' = 0 and solve for t
 2 years ago

JamesJBest ResponseYou've already chosen the best response.1
i.e., (h^2) ' = 5000t  14000 = 0 means t = .... Now for that t, h(t) = ....
 2 years ago

Mimi_x3Best ResponseYou've already chosen the best response.6
Oh I got it 2.8 thank you (:
 2 years ago
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