A television antenna is situated atop a hill. From a point 200 m from the base of the hill, the angle of elevation of the top of the antenna is 80°. The angle of elevation of the antenna from the same point is 75°. How tall is the antenna?
if the hill has height h and the antenna has height a,
h/200 = tan 75°
(h+a)/200 = tan 80°
eliminate h, and then solve for a
To find the height of the antenna, we can use trigonometric ratios and the concept of similar triangles.
Let's label the height of the antenna as 'h' and the distance from the base of the hill to the point as 'x'.
From the given information, we know that the angle of elevation from the point to the top of the antenna is 80° and from the same point to the antenna is 75°.
Using the tangent ratio, we can set up the equation:
tan(80°) = h / x ...(1)
and
tan(75°) = h / (x + 200) ...(2)
Now, let's solve these equations to find the value of 'h'.
From equation (1), we have:
h = x * tan(80°)
From equation (2), we have:
h = (x + 200) * tan(75°)
Since both expressions are equal to 'h', we can set them equal to each other:
x * tan(80°) = (x + 200) * tan(75°)
Now, let's solve for 'x'.
tan(80°) / tan(75°) = (x + 200) / x
We can rearrange the equation:
x / (x + 200) = tan(75°) / tan(80°)
Now, let's calculate the value of 'x'.
tan(75°) ≈ 3.732
tan(80°) ≈ 5.671
x / (x + 200) = 3.732 / 5.671
Cross-multiplying the equation:
5.671x = 3.732x + 746.4
5.671x - 3.732x = 746.4
x ≈ 372.3
Now that we have found the value of 'x', we can substitute it back into one of the equations (1 or 2) to find the height 'h'.
Using equation (1):
h = x * tan(80°)
h ≈ 372.3 * tan(80°)
h ≈ 372.3 * 5.671
h ≈ 2112.7
Therefore, the antenna is approximately 2112.7 meters tall.