a tower standing on level ground is due north of piont N and due west of point W, a distance 1,500 feet from N. If the angles of elevation of the top of the tower as measured from N and W are 8 and 11, respectively, find the height of the tower.
(H/tan8º)^2 + (H/tan11º)^2 = 1500^2
To find the height of the tower, we'll use trigonometric functions and create a right triangle.
Let's label the distance from N to the tower as x, the height of the tower as h, and the distance from W to the tower as y.
We can use the tangent function to find x:
tan(8 degrees) = h/x
Similarly, we can use the tangent function to find y:
tan(11 degrees) = h/y
Using these equations, we can solve for x and y:
x = h/tan(8 degrees)
y = h/tan(11 degrees)
Since the tower is due north of point N and due west of point W, we can form a right-angle triangle with x and y as the two short sides.
Using Pythagoras' theorem:
x^2 + y^2 = 1500^2
Substituting the values of x and y in terms of h:
(h/tan(8 degrees))^2 + (h/tan(11 degrees))^2 = 1500^2
Simplifying the equation:
h^2 * ((1/tan^2(8 degrees)) + (1/tan^2(11 degrees))) = 1500^2
Now, we can solve for h:
h^2 = (1500^2) / ((1/tan^2(8 degrees)) + (1/tan^2(11 degrees)))
Finally, taking the square root of both sides:
h = sqrt((1500^2) / ((1/tan^2(8 degrees)) + (1/tan^2(11 degrees))))
Calculating this expression will give us the height of the tower.
To find the height of the tower, we will use trigonometric functions and create a right-angled triangle. Let's go step by step:
1. Draw a diagram: Sketch a diagram with a tower (T), point N (due south of the tower), and point W (due east of the tower). Label the distance from N to the tower as 1,500 feet.
2. Define the angles: Label the angle of elevation from N as angle A and angle of elevation from W as angle B. We know that angle A is 8 degrees and angle B is 11 degrees.
3. Set up the right-angled triangle: Extend lines from N to T, and from W to T, creating a right-angled triangle NWT. Mark the height of the tower as h (the side opposite angle A and adjacent to angle B).
4. Apply trigonometric functions:
- For angle A: The side opposite angle A is h, and the side adjacent to angle A is 1,500 feet. Therefore, we can use the tangent function:
tan(A) = opposite / adjacent
tan(8) = h / 1,500
- For angle B: The side opposite angle B is h, and the side adjacent to angle B is 1,500 feet (same as angle A). Thus, we can also use the tangent function:
tan(B) = opposite / adjacent
tan(11) = h / 1,500
5. Solve the equations:
We can rearrange the equations to solve for h:
a) tan(8) = h / 1,500
h = 1,500 * tan(8)
b) tan(11) = h / 1,500
h = 1,500 * tan(11)
6. Calculate the height of the tower:
Using a calculator or software to compute the tangent of the given angles:
a) h = 1,500 * tan(8) ≈ 198.55 feet
b) h = 1,500 * tan(11) ≈ 307.22 feet
Therefore, the height of the tower is approximately 198.55 feet when measured from point N and 307.22 feet when measured from point W.