Question

Calculus I Related Rates:
An airplane is flying at an altitude of 5 miles and passes directly over a radar antenna (see figure). When the plane is 10 miles away (s = 10), the radar detects that the distance s is changing at a rate of 240 miles per hour. What is the speed of the plane?

Answer 1

Given: \[\frac{ds}{dt} = 240mph\]
Find: \[\frac{dx}{dt}\] when x = 10

Answer 2

I know that the Pythagorean theorem is the equation I need; since it relates the two variables in question; 'x' and 's'

Answer 3

From the Pythagorean theorem:
\[s^2 = x^2 + 25\]
\[s = \sqrt{x^2+25}\]

Answer 4

After taking the derivative with respect to time:
\[\frac{ds}{dt} = \frac{1}{2}(x^2+25)^{-\frac{1}{2}} * (2x)\]
\[\frac{ds}{dt} = \frac{x}{\sqrt{x^2+25}}\]

Answer 5

Now I'm lost... please help

Answer 6

What you calculated is not ds/dt but ds/dx

Answer 7

besides, it's more useful if you calculate dx/ds

Answer 8

because then you can say that dx/ds *ds/dt = dx/dt

Answer 9

First, it might be easier to stay with
\[ s^2 = 25+x^2\]
take the derivative with respect to t:
\[ 2 s \frac{ds}{dt} = 2 x \frac{dx}{dt}\]
s= 10, ds/dt= 240, x = sqrt(100-25)