A damsel is in distress and is being held captive in a tower. Her knight in shining armor is on the ground below with a ladder. When the knight stands 10 feet from the base of the tower and looks up at his precious damsel, the angle of elevation to her window is 60°. How long does the ladder have to be? And how high is the window to reach the damsel?
10/x = cos60°
x = 20 ft
h/20 = sin60°
To determine the length of the ladder and the height of the window, we can use basic trigonometry, specifically the concept of right triangles. Recall that the tangent function relates the angle of elevation to the opposite and adjacent sides of a right triangle.
Let's label the length of the ladder as 'L' and the height of the window as 'h.' Now, we can set up the following equations based on the given information and trigonometric relationships:
1. The tangent of the angle of elevation can be written as the ratio of the opposite side (height of the window 'h') to the adjacent side (distance from the knight to the base of the tower '10 feet') of the right triangle:
tan(60°) = h / 10
2. The length of the ladder 'L' can be found using the Pythagorean theorem, which states that the square of the hypotenuse (L) is equal to the sum of the squares of the other two sides:
L^2 = h^2 + 10^2
Now, we can solve these equations to find the length of the ladder (L) and the height of the window (h).
1. Solving equation (1) for 'h':
tan(60°) = h / 10
h = 10 * tan(60°)
h ≈ 10 * 1.732
h ≈ 17.32 feet
Therefore, the height of the window is approximately 17.32 feet.
2. Plugging the calculated value of 'h' into equation (2) to find 'L':
L^2 = h^2 + 10^2
L^2 = (17.32)^2 + 10^2
L^2 = 299.7024 + 100
L^2 ≈ 399.7024
L ≈ √399.7024
L ≈ 19.99 feet
Therefore, the length of the ladder needs to be approximately 19.99 feet.
So, to rescue the damsel in distress, the knight needs a ladder that is approximately 19.99 feet long, and the window is located at a height of approximately 17.32 feet from the base of the tower.