A woman on the top of a 448 foot high building spots a small plane. As she views the plane, its angle of elevation is 62 degrees. At the same instant a man at the ground-level entrance to the entrance to the building sees the plane and notes that its an angle of elevation is 65 degrees. How far id the woman from the plane? How far is the man from the plane? How high is the plane?
Chloe, the tutors are not here to do YOUR homework.
You have to try to solve these yourself.
Post your work for each problem and someone will help if you are stuck.
It helps if you draw a picture.
The problems I've done for you today all involve right triangles.
So read each problem and try to make a right triangle out of the info given.
2044
A. 3617.6
B. 4018.7
C. 3642.12
To solve this problem, we can use trigonometry and the concept of similar triangles.
Let's assign some variables:
- Let the distance between the woman and the plane be "x".
- Let the distance between the man and the plane be "y".
- Let the height of the plane be "h".
From the woman's perspective, when she looks at the plane, she forms a right triangle with the plane at the top vertex, herself at the bottom vertex, and the line connecting them as the hypotenuse. The angle of elevation of 62 degrees is the angle between the hypotenuse and the horizontal line.
We can use the tangent function to relate the angle and the sides of the triangle:
tan(62 degrees) = h / x
Rearranging the equation, we get:
x = h / tan(62 degrees)
Similarly, from the man's perspective, he forms a right triangle with the plane at the top vertex, himself at the bottom vertex, and the line connecting them as the hypotenuse. The angle of elevation of 65 degrees is the angle between the hypotenuse and the horizontal line.
Again, we can use the tangent function:
tan(65 degrees) = h / y
Rearranging the equation, we get:
y = h / tan(65 degrees)
To find the height of the plane, we can set up a proportion using similar triangles. Since the woman and the man are at the same known height of 448 feet, and we know their respective distances from the plane, we can create the following ratio:
h / 448 = x / y
Rearranging the equation, we get:
h = (448 * x) / y
Now we have three equations to solve. Here's how to find the values:
1. Substitute the values of the angles into the tangent functions to find x and y:
x = h / tan(62 degrees) = h / tan(1.082104136 deg)
y = h / tan(65 degrees) = h / tan(1.130953743 deg)
2. Use the ratio equation to solve for h:
h = (448 * x) / y
Substitute the values of x and y into this equation and solve for h.
3. Substitute the value of h back into the equations for x and y to find their values.
Now you have the values of x, y, and h to answer the question.