Camila needs to fix the window on the front of her house. The bottom of the window is 16 ft. from the ground. She only has a 20 ft. ladder. How far from the base of the house should she set the ladder so that the top of the ladder is exactly at the bottom of the window?
Using the Pythagorean theorem, we can solve this problem.
Let x be the distance from the base of the house to the ladder.
Using the Pythagorean theorem, we have:
x^2 + 16^2 = 20^2
Simplifying, we get:
x^2 + 256 = 400
Subtracting 256 from both sides, we get:
x^2 = 144
Taking the square root of both sides, we get:
x = 12 ft
Therefore, Camila should set the ladder 12 ft from the base of the house.
To solve this problem, we can use the concept of similar triangles.
Here's how we can approach it:
1. Draw a diagram to visualize the situation. Draw the house, the window, and the ladder. Label the distance from the base of the house to the ladder as "x".
2. Create a right triangle using the ladder as the hypotenuse, the distance from the base of the house to the ladder as one leg, and the distance from the top of the ladder to the bottom of the window as the other leg.
3. Since the ladder is 20 ft long and the bottom of the window is 16 ft from the ground, the remaining distance from the top of the ladder to the window is 20 ft - 16 ft = 4 ft.
4. Now we have two similar triangles - the larger triangle formed by the ladder, the ground, and the distance from the base of the house to the ladder, and the smaller triangle formed by the ladder, the ground, and the distance from the top of the ladder to the window.
5. Using the concept of similar triangles, we can set up the following proportion:
(distance from the base of the house to the ladder) / (distance from the top of the ladder to the window) = (length of the larger triangle) / (length of the smaller triangle).
Mathematically, we can write it as:
x / 4 = 20 / 16.
6. Now, we solve for x by cross multiplying:
16x = 4 * 20.
7. Simplify and solve for x:
16x = 80.
x = 80 / 16.
x = 5.
Therefore, Camila should set the ladder 5 ft away from the base of the house to ensure that the top of the ladder is exactly at the bottom of the window.
To determine the distance from the base of the house where Camila should set the ladder, we can use the Pythagorean theorem.
The theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the ladder forms the hypotenuse, and the distance from the base of the house to the ladder forms one side.
Let's call the distance from the base of the house to the ladder "x." The ladder is 20 ft., and the bottom of the window is 16 ft., so the height from the ground to the top of the ladder would be 20 - 16 = 4 ft.
Using the Pythagorean theorem, we have the equation:
x^2 + 4^2 = 20^2
Simplifying,
x^2 + 16 = 400
Subtracting 16 from both sides,
x^2 = 384
Taking the square root of both sides,
x ≈ 19.60 ft.
Therefore, Camila should set the ladder approximately 19.60 ft. from the base of the house to have the top of the ladder exactly at the bottom of the window.