When melonie looks at the top of a tower, the angle of elevation is 27 degrees. when melonie walks 20m further away from the tower on level ground, the angle of elevation decreases to 20 degrees. what is the height of the tower?
Draw the diagram, and review your basic trig functions. Then it should be clear that the height h can be found using
h cot20° - h cot27° = 20
so,
h = 20/(cot20° - cot27°)
To find the height of the tower, we can use trigonometry. Let's break down the problem:
We have two right triangles. In the first triangle, the angle of elevation is 27 degrees, and in the second triangle, the angle of elevation is 20 degrees. The height of the tower will remain the same in both triangles.
Let's define the variables:
- In the first triangle, let the distance of Melonie from the tower be d1 (the side adjacent to the angle of elevation).
- In the second triangle, let the distance of Melonie from the tower be d2 (also the side adjacent to the angle of elevation). We know that d2 = d1 + 20m, as Melonie walks 20m further away from the tower.
Using trigonometry, we can set up the following equations based on the tangent function:
In the first triangle:
tan(27 degrees) = height of the tower / d1
In the second triangle:
tan(20 degrees) = height of the tower / d2
Now we can solve these equations simultaneously to find the height of the tower. Let's do some algebraic manipulation:
Rearrange the first equation:
height of the tower = d1 * tan(27 degrees)
Substitute the value of d2 in terms of d1 in the second equation:
tan(20 degrees) = height of the tower / (d1 + 20m)
Now substitute the value of the height of the tower from the rearranged first equation into the second equation:
tan(20 degrees) = (d1 * tan(27 degrees)) / (d1 + 20m)
We can solve this equation for d1, which will give us the distance from Melonie to the tower in the first triangle. Once we have d1, we can substitute it into the first equation to find the height of the tower.
Let's calculate it step-by-step:
1. Convert the angles from degrees to radians:
27 degrees = 27 * π/180 radians
20 degrees = 20 * π/180 radians
2. Substitute the values and solve the equation for d1:
tan(20 * π/180) = (d1 * tan(27 * π/180)) / (d1 + 20m)
Using a scientific calculator or trigonometric identity, we can find the value of tan(20 * π/180).
3. Simplify the equation and solve for d1:
tan(20 * π/180) = (d1 * tan(27 * π/180)) / (d1 + 20m)
After finding the value of d1 using the above equation, substitute it into the first equation:
height of the tower = d1 * tan(27 degrees)
This will give you the height of the tower.