Chapter 6.2, Problem 57E

Determining a Distance If both cars in Exercise 62 are on one side of the plane and if the angle of depression to one car is 38° and that to the other car is 52°, how far apart are the cars?

To determine

To find: The distance between the two cars if the two cars are on the same plane.

Answer to Problem 57E

The distance between the two cars if the two cars are on the same plane is $2570\text{ft}$.

Explanation of Solution

Given:

The elevation of an airplane flying straightaway is 5150 ft. Two motorists are driving cars on the same sides on the highway with angle of depression $38°$ and $52°$.

Formula used:

The formula to calculate $\theta$ with tangent is,

$\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$ (1)

Calculation:

Assume the distance between two cars is $x$.

Figure (1)

Figure (1) shows the angle of depression are $38°$ and $52°$ with the elevation of 5150 ft.

As $AB\perp AE$ the sum of the angles at point $A$ is $90°$.

$\begin{array}{c}52°+a=90°\\ a=90°-52°\\ a=38°\end{array}$

From triangle $ABC$, substitute $38°$ for $\theta$, $y$ for $\text{opposite}$ and 5150 for $\text{adjacent}$ in equation (1).

$\begin{array}{c}\tan 38°=\frac{y}{5150}\\ y=\tan 38°\times 5150\\ y=0.781\times 5150\\ y\approx 4022\end{array}$

From triangle $ADE$, substitute $52°$ for $\theta$, $d$ for $\text{opposite}$ and 5150 for $\text{adjacent}$ in equation (1).

$\begin{array}{c}\tan 52°=\frac{d}{5150}\\ d=\tan 52°\times 5150\\ d=1.28\times 5150\\ d=6592\end{array}$

But $x=d-y$

Substitute $4022$ for $y$ and $6592$ for $d$ in above equation to find the value of x.

$\begin{array}{c}x=6592-4022\\ =2570\end{array}$

Hence, the distance between the two cars if the two cars are on the same plane is $2570\text{ft}$.