Question

A plane flying horizontally at an altitude of 5 mi and a speed of 470 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 is 7 mi away from the station. For full credit I expect to see a well-labeled picture. Show all work. Round your final answer to the nearest whole number. Do not round intermediate values. Include units of measure in your answer.

Answer 1

Step 1)

Lets' assume that,

y = height of the plane over the radar station

P = point located vertically above radar station

y = distance between point p and plane

z = distance between plane and radar station

Step 2)

As given plane is flying horizontally at an altitude of 5 mi hence we have y = 5

speed of the plane is 470 mi/h hence we have dx/dt = 470

we have to find rate at which the distance from the plane to the station is increasing when it is 7 mi away from the station

it means we have to find rate at which the distance from the plane to the station is increasing when z = 7

according Pythagorean theorem we can write,

we have y = 5 and z = 7 hence,

but distance between point P and plane cannot be negative hence,

Hence we can say that when y = 5, z = 7 we have x = 2sqrt(6)

Step 3)

we have,

differentiate both the side with respect to t we can say that,

Height of the plane is constant hence we can say that,

Hence we can write,

we have,

Hence,

rounding to nearest whole number we can say that,

we can write,

Hence we can say that the distance from the plane to the station is increasing at a rate of 329 mi/h when it is 7 mi away from the station