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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Flight Paths An air traffic controller spots two planes at the same altitude flying toward each other (see figure). Their flight paths are 20° and 315°. One plane is 150 miles from point P with a speed of 375 miles per hour. The other is 190 miles from point P with a speed of 450 miles per hour.

Chapter 10, Problem 4PS, Flight Paths An air traffic controller spots two planes at the same altitude flying toward each

(a) Find parametric equations for the path of each plane where t is the time in hours, with t = 0 corresponding to the time at which the air traffic controller spots the planes.

(b) Use the result of part (a) to write the distance between the planes as a function of t.

(c) Use a graphing utility to graph the function in part (b). When will the distance between the planes be minimum? If the planes must keep a separation of at least 3 miles, is the requirement met?

(a)

To determine

To calculate: The parametric equations for the path of each plane if

The flight paths are 20° and 315°.

One plane is 150 miles from P with a speed of 375 miles per hour.

The other plane is 190 miles from point P with a speed of 450 miles per hour.

The figure is given below.

Calculus, Chapter 10, Problem 4PS

Explanation

Given:

The flight paths are 20° and 315°.

One plane is 150 miles from P with a speed of 375 miles per hour.

The other plane is 190 miles from point P with a speed of 450 miles per hour.

The given figure is:

Formula used:

Speed formula:

speed=distancetime

Calculation:

Observe from the given figure, that the angle made by one plane with positive x-axis is

90°20°=70°

Distance of the flight from the point P is 150 miles.

Since t=0 it corresponds to the time when the air traffic controller spots the plane. Hence, the distance covered by the plane with a speed of 375 miles per hour is 375t

(b)

To determine

To calculate: The distance between the planes as a function of t if

The parametric equation of one of the planes:

x1=(150375t)cos70°y1=(150375t)sin70°

The parametric equation of another plane:

x2=(190450t)cos135°y2=(190450t)sin45°

(c)

To determine

To graph:

The function y=[(190450t)cos135°(150375t)cos70°]2+[(190450t)sin45°(150375t)sin70°]2 and calculate the minimum distance between the planes and find if the planes keep a separation of at least 3miles.

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