An engineer sees that the angle of elevation to a vertical tower is 45 degrees. When he moves 40 feet closer to the tower on the same straight, level path, he sees that the angle of elevation becomes 55 degrees. How tall is the tower?
Tan 45 = h/d1
h = d1*Tan45
Tan55 = h/(d1-40)
h = (d1-40)*Tan55
h = d1*Tan45 = (d1-40)*Tan55
d1*1 = (d1-40)*Tan55
Divide both sides by Tan55:
0.7d1 = d1-40
0.3d1 = 40
d1 = 133.3 Ft.
h = d1*Tan45 = 133.3 * 1 = 133.3 Ft.
To solve this problem, we can use trigonometric ratios, specifically tangent.
Let's define some variables:
- Let h be the height of the tower.
- Let x be the distance from the engineer to the base of the tower when he first sees it at a 45-degree angle of elevation.
- Let x - 40 be the distance from the engineer to the base of the tower when he sees it at a 55-degree angle of elevation.
Now, we can set up two equations using the tangent function:
Equation 1: When the angle of elevation is 45 degrees
tan(45 degrees) = h / x
Equation 2: When the angle of elevation becomes 55 degrees
tan(55 degrees) = h / (x - 40)
Using the tangent function, we know that:
tan(45 degrees) = 1
tan(55 degrees) = 1.428
Plugging these values into our equations, we have:
1 = h / x
1.428 = h / (x - 40)
From equation 1, we can solve for h:
h = x
Substituting this value into equation 2, we get:
1.428 = x / (x - 40)
Now, we can solve this equation for x:
1.428(x - 40) = x
Expanding and rearranging the equation:
1.428x - 57.12 = x
0.428x = 57.12
x = 133.95
Therefore, the distance from the engineer to the base of the tower when he first sees it at a 45-degree angle of elevation is approximately 133.95 feet.
Finally, substituting this value back into equation 1 to solve for h:
h = x
h = 133.95
So, the height of the tower is approximately 133.95 feet.