Explanation:
Let's first draw a diagram to better visualize the problem:
* T (top of incomplete tower)
/|
/ |
/ |
/ | h (height of incomplete tower)
/ |
/ |
/θ1 |
/___ | M (man's position, height = 6 feet)
d
We can see that we have a right triangle with the tower's height as the opposite side, the distance between the man and the tower as the adjacent side, and the angle of elevation θ1 as 30°. We can use trigonometry to find the height of the incomplete tower:
tan(30°) = h / d
h = d * tan(30°)
We don't know the value of d, but we can use the fact that the man's height plus the height of the incomplete tower equals the distance from the man to the top of the incomplete tower:
d = h / tan(30°) + 6
Now we can use trigonometry again to find the height of the complete tower. Let's call this height H and the new angle of elevation θ2:
* T (top of complete tower)
/|
/ |
/ |
/ | H (height of complete tower)
/ |
/ |
/θ2 |
/___ | M (man's position, height = 6 feet)
d
We have another right triangle, this time with the height of the complete tower as the opposite side, the same distance between the man and the tower as the adjacent side, and the new angle of elevation θ2 as 60°. We can use the tangent function again:
tan(60°) = H / d
H = d * tan(60°)
We can substitute the value of d we found earlier:
H = (h / tan(30°) + 6) * tan(60°)
Simplifying:
H = h * sqrt(3) + 6 * sqrt(3)
(a) To find how high the tower must be raised, we subtract the height of the incomplete tower from the height of the complete tower:
raise = H - h
raise = h * (sqrt(3) - 1) + 6 * sqrt(3)
Substituting the value of h we found earlier:
raise = 24 * (sqrt(3) - 1) + 6 * sqrt(3)
raise ≈ 38.8 feet
(b) The height of the completed tower is simply the height of the incomplete tower plus the raise we found:
height = h + raise
height = 24 + 38.8
height ≈ 62.8 feet
Therefore, the height of the tower after completing its construction work is approximately 62.8 feet.