To find the angle of elevation of the ladder, we can use the trigonometric function tangent (tan).
The tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the height the ladder reaches on the building, and the adjacent side is the distance from the base of the ladder to the base of the building.
So, we have:
tan(angle) = opposite/adjacent
tan(angle) = height/7
To find the angle, we can take the inverse tangent (arctan) of both sides:
angle = arctan(height/7)
To find the height the ladder reaches on the building, we can use the Pythagorean theorem. The ladder, the base of the ladder, and the height on the building form a right triangle. The sum of the squares of the lengths of the two shorter sides (legs) of a right triangle is equal to the square of the length of the longest side (hypotenuse).
Using the Pythagorean theorem, we have:
hypotenuse^2 = height^2 + 7^2
Since we know the length of the hypotenuse (20 ft), we can solve for the height:
20^2 = height^2 + 7^2
400 = height^2 + 49
Subtracting 49 from both sides:
400 - 49 = height^2
351 = height^2
Taking the square root of both sides:
height = √351 ≈ 18.7 ft
So, the angle of elevation of the ladder is arctan(height/7) and the height the ladder reaches on the building is approximately 18.7 ft.