Question

A plane flying horizontally at an altitude of 1 mi and a speed of 450 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 3 mi away from the station. (Round your answer to the nearest whole number.)

Answer 1

Given

The plane horizontal altitude is 1 mi and speed is 450 mi/h.

The plane distance from the station is 3 mi.

Step-by-step explanation

Draw the figure,

P is the plane's position

R is the radar station's position

V is the point located vertically of the radar station at the plane's height.

h is the plane's height

d is the distance between the plane and the radar station

x is the distance between the plane and the V point

Since the plane flies horizontally, we can conclude that PVR is a right triangle. Therefore, the pythagoreas theorem is used to find x,

`d=√(h^2+x^2)`

Substitute the values,

`\begin{gathered} x=√(3^2-1^2) \\ x=√(9-1) \\ x=√(8) \\ x=2√(2) \end{gathered}`

When d=3 mi, and, since the plane flies horizontally, we know that h=1mi regardless of the situation. We are looking for

`(dd)/(dt)=d`
`\begin{gathered} d^2=h^2+x^2 \\ 2d(dd)/(dt)=2h(dh)/(dt)+2x(dx)/(dt) \\ d(dd)/(dt)=x(dx)/(dt) \end{gathered}`

Substitute the values.

`\begin{gathered} (dd)/(dt)=(x)/(d)(dx)/(dt) \\ (dd)/(dt)=(2√(2))/(3)*450 \\ (dd)/(dt)=150*2*√(2) \\ (dd)/(dt)=300√(2) \end{gathered}`

Answer

Hence the rate at which the distance from the plane to the station is increasing when it has a total distance of 3 mi away from the station is

`424.6mi\text{ /h}`