An airplane is flying parallel to the ground at an altitude of 2 km and at a speed of 4(1/2km)/minf the plane flies directly over the Statue of Liberty,

Question

An airplane is flying parallel to the ground at an altitude of 2 km and at a speed of 4(1/2km)/min. If the plane flies directly over the Statue of Liberty, at what rate is the line- of- sight distance between the plane and the statue changing 20 sec later? Show your complete solutions.

Answer 1

Draw the right triangle from the ground to the plane (2km) and from the ground to the SOL (Statue...) which will be 20s * 4.5 km/60s = 1.5 km) and the hypotenuse is the LOS (line of sight, let's say z). We'll call the horizontal distance x.

Equation relating x to hypotenuse:

x² + 2² = z² which is always true

We need to make the velocities or rates of change explicit by differentiating by t:

2x dx/dt = 2z dz/dt

which we can rearrange in order to solve for dz/dt given dx/dt x, and z

dz/dt = (x/z) dx/dt and you know that x = 1.5 km, dx/dt = 4.5 km/min, and z = sqrt( 1.5² + 2²) at the point of interest. Plug in and solve.