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 15 feet from the base of the tower and looks up at his precious damsel, the angle of elevation to her window is 60 degrees. How long does the ladder have to be?
To find out how long the ladder needs to be, we can use trigonometry.
Let's call the distance from the base of the tower to the window "x," and the length of the ladder "l." We can use the tangent function to solve for x.
In a right triangle, the tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is x, and the adjacent side is 15 feet.
So, we have:
tan(60 degrees) = x / 15
To solve for x, we can multiply both sides of the equation by 15:
15 * tan(60 degrees) = x
Now, we can use a calculator to find the value of tan(60 degrees).
tan(60 degrees) ≈ 1.732
Substituting this value into the equation, we have:
15 * 1.732 = x
x ≈ 25.98
Therefore, the distance from the base of the tower to the window is approximately 25.98 feet.
To find the length of the ladder, we can use the Pythagorean theorem. In the right triangle formed by the ladder, the base of the tower, and the distance from the base to the window, the ladder is the hypotenuse.
Using the formula:
l^2 = x^2 + 15^2
l^2 = 25.98^2 + 15^2
l^2 ≈ 674.9004 + 225
l^2 ≈ 899.9004
Taking the square root of both sides to solve for l:
l ≈ √899.9004
l ≈ 29.99
Therefore, the ladder needs to be approximately 29.99 feet long.
To determine the length of the ladder, we can use trigonometry. In this case, we can use the tangent function since we have the angle of elevation (60 degrees) and the opposite side length (the height of the tower).
Let's break down the problem:
1. The knight is standing 15 feet from the base of the tower.
2. The angle of elevation from the knight's position to the window of the tower is 60 degrees.
3. We need to find the length of the ladder, which is the hypotenuse of the triangle formed by the ladder, the height of the tower, and the distance between the knight and the base of the tower.
First, let's label the lengths in the triangle:
- The height of the tower (opposite side) is the distance between the window and the base of the tower.
- The distance from the base of the tower to the knight (adjacent side) is 15 feet.
- The length of the ladder (hypotenuse) is what we are trying to find.
Now, we can use the tangent function to calculate the length of the ladder:
tan(angle) = opposite/adjacent
Substituting the values we have:
tan(60 degrees) = height of the tower / 15 feet
To isolate the height of the tower, we can rearrange the equation:
height of the tower = tan(60 degrees) * 15 feet
Using a calculator, we can find the value of tan(60 degrees) ≈ 1.732.
height of the tower ≈ 1.732 * 15 feet
Simplifying the calculation:
height of the tower ≈ 25.98 feet
Therefore, the ladder needs to be approximately 25.98 feet long to reach the window and rescue the damsel in distress.