If two runners determine the angle of elevation of a hot air ballon is 24.5 degrees and 22 degrees. The hot air ballon is 1500 feet directily above a point on the highway east of both runners. How far apart are they? A trigonometry question.
distance= 1500/tan22-1500/tan24.5
figure out how I did that in my head.
171505.302
this answer looks bigt but i did it and this is the right answer
To find the distance between the two runners, we can use the concept of trigonometry and specifically the tangent function.
Let's assume that the two runners are located at points A and B, and the point on the highway directly below the hot air balloon is point C.
From the given information, we can form a right-angled triangle ABC where ∠BAC is the angle of elevation of 24.5 degrees, and ∠ABC is the angle of elevation of 22 degrees. The altitude of the hot air balloon from point C is 1500 feet.
Now, let's consider the angle of elevation ∠BAC. By using the tangent function, we have:
tan(∠BAC) = height of balloon / distance AB
Substituting the given values, we get:
tan(24.5°) = 1500 / AB
Similarly, for the angle of elevation ∠ABC, we have:
tan(∠ABC) = height of balloon / distance AC
tan(22°) = 1500 / AC
Since AB and AC are distances on the highway, and the runners are directly east of each other, we can assume AB = AC + x, where x is the distance between the two runners.
Now, we can rewrite the equation for tan(∠ABC) using the distance AC + x:
tan(22°) = 1500 / (AC + x)
Now, we have two equations:
tan(24.5°) = 1500 / AB
tan(22°) = 1500 / (AC + x)
We can solve these two equations simultaneously to find the value of x, which represents the distance between the two runners.
Note: To solve this, you can use algebraic methods, such as substitution or elimination, or you can use a scientific calculator that has trigonometric functions to find the values of AB and AC + x.