An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway, both on one side of the plane. If the angle of depression to one car is 34° and to the other is 51°, how far apart are the cars? (Round your answer to the nearest foot.)
done , check your previous post
Didn't notice the small variation from the previous post.
Easy to make the adjustment.
To find the distance between the two cars on the highway, we can use trigonometry and the concept of similar triangles.
Let's start by visualizing the problem. We have an airplane flying directly above a straight highway, and two cars on one side of the plane. The angles of depression from the plane to each car are given as 34° and 51°.
The angles of depression give us the angle between the line of sight from the plane to each car and a horizontal line parallel to the ground. We can consider this horizontal line as the base of a right triangle formed by the plane, the car, and the ground.
Let's label some points in our diagram:
- Let A be the position of the airplane.
- Let B be the position of the car with a 34° angle of depression.
- Let C be the position of the car with a 51° angle of depression.
- Let D be the point on the ground directly below the airplane.
We are asked to find the distance BC, which represents the distance between the two cars.
Now, consider the triangles ABD and ACD. These triangles are similar because they share an angle at A (angle ABD = angle ACD) and both have a right angle (angle ADB = angle ADC = 90°). By the property of similar triangles, their corresponding sides are proportional.
We can use the tangent function to find the lengths of AB and AC. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, the opposite side is the elevation, and the adjacent side is the horizontal distance.
For triangle ABD:
Tangent(34°) = AB / BD
AB = BD * Tangent(34°)
For triangle ACD:
Tangent(51°) = AC / CD
AC = CD * Tangent(51°)
We are given the elevation of the plane, which is the length of BD. The goal is to find the length of CD and BC.
To find CD, we can use the fact that the sum of the two distances (CD + BD) is equal to the elevation. So:
CD + BD = 5150 ft
CD = 5150 ft - BD
Now, let's substitute these values into our equations:
AB = BD * Tangent(34°)
AC = (5150 ft - BD) * Tangent(51°)
Since we want to find the distance between the two cars, BC, which is equal to (AB + AC), we can substitute the equations for AB and AC:
BC = AB + AC
BC = BD * Tangent(34°) + (5150 ft - BD) * Tangent(51°)
To find the value of BC, we need to use the given values and calculate:
BC = BD * Tangent(34°) + (5150 ft - BD) * Tangent(51°)
The result will give us the distance between the two cars.