Chapter 4.6, Problem 32E

To determine

To calculate: Car speed along the way of plane.

Answer to Problem 32E

The car speed along the way of plane is $80\text{mph}$ .

Explanation of Solution

Given information:

Patrol airplane flies $3miles$ above a level

Speed $120mph$

Car sight distance from plane $5miles$

Line-of-sight distance speed $160mph$

Calculation:

  $s$ is the line of sight distance $A$ is the point on the ground directly below the plane.

Know from the Pythagorean Theorem that the distance from $A$ to the car is

  $4.\left(3^2+4^2=5^2\right)$

  $\frac{ds}{dt}=-160\text{mph}$

  $\frac{dx}{dt}=?$

Can take the Pythagorean Theorem and differentiate with respect to $t$ . The plane's altitude is constant.

Solve for $\frac{dx}{dt}$ . then plug in all the known values.

  

  $\begin{array}{c}{x}^2+3^2=s^2\\ 2x\frac{dx}{dt}+0=2s\frac{ds}{dt}\\ \frac{dx}{dt}=\frac{2s}{2x}\frac{ds}{dt}\\ =\frac{s}{x}\frac{ds}{dt}\\ =\frac{5}{4}(-160)\\ =-200\text{mph}\end{array}$

  $\frac{dx}{dt}$ is negative since the car is traveling to the right and $x$ is decreasing. Note that this is how fast the car appears to be traveling relative to the plane, not a stationary point on the ground. To get how fast the car is traveling relative to the ground, subtract the plane's speed from $200$

  $200-120=80\text{mph}$

The car speed along the way of plane is $80\text{mph}$ .