A plane flying horizontally at an altitude of 1 mile and a...

A plane flying horizontally at an altitude of 1 mile and a speed of 400 miles per hour passes directly over a radar station. Let θ(t) be the angle between the ground and the line connecting the radar station to the plane at time t. Draw a clear diagram to illustrate this. How fast is θ(t) decreasing when θ = (π/4) radians? You must state the units of your answer. [HINT: In your diagram, let x(t) denote the horizontal distance traveled by the plane, relative to the radar station. Find a formula for x(t) in terms of θ(t), then differentiate both sides with respect to t.]

Transcript

00:01 A plane flying horizontally at an altitude of one mile and a speed of 400 miles per hour passes directly over a radar station.

00:07 Let theta of t be the angle between the ground and the line connecting the radar station to the plane at time t.

00:13 How fast is theta of t decreasing when theta equals pi over four radiance? so we're going to need to draw a diagram here.

00:20 We have the radar station on the ground and then the plane passes directly above it.

00:27 So we'll say the plane is right here.

00:30 And it's moving horizontally and we know the altitude of the plane is one mile so it's one mile above the ground or above the radar station and then we have this distance that the plane is traveling that's changing constantly so we'll just call that x because we don't know exactly where the plane is we just know it's moving horizontally and then if we go from the radar station to wherever the plane is at any time we will get this triangle and we have this angle theta here that shows the angle described between the ground and the line connecting the radar station to the plane so this angle would change as time goes on as x gets bigger and bigger bigger this theta would decrease if you need to see that say we had the x really really really really really big that angle would be significantly smaller it's going to get smaller and smaller smaller the bigger x gets so this would be a right triangle and we can get a few more lines here so we make this right triangle here since theta is in this right triangle the lower one we know this is one mile and we know this would just be x.

01:52 So if we look at x as a function what is x with respect to t but we know x is changing at 400 miles every hour so it will just be 400 times the hour t that would tell us how x has traveled that would be in miles 400 times t miles so if we start from trigonometry here we can see that the tangent of angle theta is the opposite side which is one over the adjacent side which is x and then if we further simplify this we know x and x will be found with respect to time and this angle is changing also so we could use that using function notation we can say theta t so we can further simplify this and call that of theta t equals 1, the x of t is 400t.

02:49 That's the equation for x...

Question

Please give Ace some feedback