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18.01 Single Variable Calculus, Fall 2007
Transcript – Lecture 13
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PROFESSOR: Today we're going to keep on going with related rates. And you may
recall that last time we were in the middle of a problem with this geometry. There
was a right triangle. There was a road. Which was going this way, from right to left.
And the police were up here, monitoring the situation. 30 feet from the road. And
you're here. And you're heading this way. Maybe it's a two lane highway, but anyway
it's only going this direction. And this distance was 50 feet. So, because you're
moving, this distance is varying and so we gave it a letter. And, similarly, your
distance to the foot of the perpendicular with the road is also varying. At this instant
it's 40, because this is a 3, 4, 5 right triangle. So this was the situation that we were
in last time. And we're going to pick up where we left off.
The question is, are you speeding if the rate of change of D with respect to t is 80
feet per second. Now, technically that would be - 80, because you're going towards
the policemen. Alright, so D is shrinking at a rate of - 80 feet per second. And I
remind you that 95 feet per second is approximately the speed limit. Which is 65
miles per hour. So, again, this is where we were last time. And, got a little question
mark there. And so let's solve this problem.
So, this is the setup. There's a right triangle. So there's a relationship between these
lengths. And the relationship is that x ^2 + 30 ^2 = D ^2. So that's the first
relationship that we have. And the second relationship that we have, we've already
written down. Which is dx / dt - oops, sorry. dD / dt = minus 80. Now, the idea here
is relatively straightforward. We just want to use differentiation. Now, you could
solve for x. Alright, x is the square root of D^2 - 30 squared. That's one possibility.
But this is basically a waste of time. It's a waste of your time. So it's easier, or