Finding the Rate

Question

A plane flying horizontally at an altitude of 2 miles and a speed of 460 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it has a total distance of 5 miles away from the station. (Round your answer to the nearest whole number.)

Answer 1

Since you are being asked to "find the rate at which the distance from the plane to the station is increasing" then you will need to write an equation for the DISTANCE of the plane and then take the derivative of it, with respect to time, to get the rate.

So, what is the equation for the distance between the plane and the radar station? Draw a picture:

Let the distance between the plane and the radar tower be "s" as shown. The horizontal distance is the speed multiplied by the time. So "460t" where "t" is in hours. The vertical distance is always 2 miles.

So use the Pythagorean Theorem to write a distance equation:

s = √(2² + (460t)²)

s = √(4 + 211600t²)

s = (4 + 211600t²)^0.5

Now take the derivative with respect to time using the power rule and chain rule:

ds/dt = 0.5(4 + 211600t²)^-0.5(423200t)

ds/dt = 211600t(4 + 211600t²)^-0.5

To determine the rate (ds/dt) when the plane is 5 miles away, you must calculate the time in order to use the equation.

5 = √(2² + (460t)²)

square both sides to get:

25 = 4 + 211600t²

21 = 211600t²

9.9243856 x 10^-5 = t²

t = √(9.9243856 x 10^-5)

t = 0.009962 hours (which is not much more than half a minute)

Plug in t = 0.009962 hours into the rate equation:

ds/dt = 211600t(4 + 211600t²)^-0.5

ds/dt at t = 0.009962 = 211600(0.009962)(4 + 211600(0.009962)²)^-0.5

ds/dt = 421.597 mi/hr

Rounding to the nearest whole number, ds/dt = 422 mi/hr