An airplane is flying at an elevation of 5150 ft, directly above a straight highway. Two motorists are driving cars on the highway on opposite sides of the plane, and the angle of depression to one car is 35° and to the other is 55°. How far apart are the cars? (Round your answer to the nearest foot.)
Well, that's quite a high-flying situation you've got there! Let's see if I can help.
First, we can imagine a right triangle formed by the plane, one of the motorists, and the point on the highway directly beneath the plane. The angle of depression to that motorist is 35°.
Similarly, we can imagine another right triangle formed by the plane, the other motorist, and the point on the highway directly beneath the plane. The angle of depression to that motorist is 55°.
Now, we can use some trigonometry magic! The tangent of an angle is equal to the opposite side over the adjacent side. In this case, the opposite side is the elevation of the plane (5150 ft) and the adjacent side is the distance between the plane and the motorist.
For the first motorist (35° angle of depression), we have:
tan(35°) = 5150 ft / x
And for the second motorist (55° angle of depression), we have:
tan(55°) = 5150 ft / (x + d)
where d is the distance between the two motorists.
Now, to find the distance between the cars, we need to solve for d.
Let's do some math... (Warning: Math ahead!!)
First, solving the first equation for x:
x = 5150 ft / tan(35°)
x ≈ 7510 ft
Now, plug this value of x into the second equation:
tan(55°) = 5150 ft / (7510 ft + d)
Rearranging the equation:
5150 ft = tan(55°) * (7510 ft + d)
Solving for d:
d = (5150 ft / tan(55°)) - 7510 ft
d ≈ 248.04 ft
Voila! The distance between the cars is approximately 248 feet. (Round to the nearest foot)
I hope that answers your question with a touch of laughter along the way!
To solve this problem, we can use trigonometry and the concept of angles of depression.
We know that the angle of depression is the angle between the line of sight from an observer to an object below the horizontal level and the horizontal level itself.
Let's assume the distance between the airplane and one car is 'x' units.
Using the tangent function, we can determine the distances from the airplane to each car:
In the first case, the angle of depression is 35°, so we have:
Tan(35°) = (5150 ft) / x
Rearranging the equation to solve for x:
x = (5150 ft) / Tan(35°)
Using a calculator, we find that x ≈ 7516.63 ft.
In the second case, the angle of depression is 55°, so we have:
Tan(55°) = (5150 ft) / (x + d)
where 'd' is the distance between the two cars.
Solving for d:
d = (5150 ft) / Tan(55°) - x
Again, using a calculator, we find that d ≈ 4861.11 ft.
To find the distance between the two cars, we can add the distances from the airplane to each car:
Distance = x + d ≈ 7516.63 ft + 4861.11 ft
Therefore, the distance between the two cars is approximately 12,377.74 ft or round to the nearest foot: 12,378 ft.
To find the distance between the two cars on the highway, we can use trigonometry and the concept of angles of depression.
Let's denote the distance between the plane and the car with the angle of depression of 35° as "x", and the distance between the plane and the car with the angle of depression of 55° as "y".
Using trigonometry, we can set up the following equations:
1) tan(35°) = x / 5150 ft
2) tan(55°) = y / 5150 ft
We need to find the individual distances "x" and "y", which represent the distances between the plane and each car. To do that, we can rearrange the equations to solve for "x" and "y":
1) x = tan(35°) * 5150 ft
2) y = tan(55°) * 5150 ft
Let's evaluate these equations:
1) x = tan(35°) * 5150 ft
x ≈ 0.7002 * 5150 ft
x ≈ 3604.13 ft
2) y = tan(55°) * 5150 ft
y ≈ 1.4281 * 5150 ft
y ≈ 7358.84 ft
Therefore, the two cars are approximately 3604.13 feet and 7358.84 feet away from the plane, respectively. To find the total distance between the two cars, we can simply add these distances:
Total distance = x + y
Total distance ≈ 3604.13 ft + 7358.84 ft
Total distance ≈ 10962.97 ft
Rounded to the nearest foot, the two cars are approximately 10,963 feet apart.
Did you make a sketch?
According to mine
distance of first car:
tan 35 = 5150/x
x = 5150/tan35
distance for 2nd car:
tan 55 = 5150/y
y = 5150/tan55
total distance between two cars
= x + y
= 5150/tan35 + 5150/tan55 = appr 10961 feet