sam needs to clean the window on his house. he has a 25-foot ladder and places the base of the ladder 10 feet from the wall of the house.
A) how high up the wall will the ladder reach?
B) if his windows are 20 feet above the ground, what is the farthest distance the baes of the ladder can be from the wall?
Use the Pythagorean Theorem.
a^2 + b^2 = c^2
10^2 + b^2 = 25^2
100 + b^2 = 625
b^2 = 525
b = 22.9 feet
Do the second problem using the Pythagorean Theorem.
A) To find out how high up the wall the ladder will reach, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right 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.
Let's denote the height the ladder reaches as 'h'. We have the ladder, the base distance from the wall, and the height as three sides of a right triangle. The ladder is the hypotenuse, the distance from the base to the wall is one of the other sides, and the height reached is the remaining side.
Using the Pythagorean theorem:
ladder^2 = base^2 + height^2
(25 ft)^2 = (10 ft)^2 + h^2
625 ft^2 = 100 ft^2 + h^2
625 ft^2 - 100 ft^2 = h^2
525 ft^2 = h^2
To find the value of 'h', take the square root of both sides:
sqrt(525 ft^2) = sqrt(h^2)
h ≈ 22.91 ft
Therefore, the ladder will reach approximately 22.91 feet up the wall.
B) If the windows are 20 feet above the ground, we need to calculate the maximum distance the base of the ladder can be from the wall. Here, the ladder and the distance from the base to the wall are two sides of a right triangle, and the height is the remaining side.
Using the Pythagorean theorem:
ladder^2 = base^2 + height^2
(25 ft)^2 = base^2 + (20 ft)^2
625 ft^2 = base^2 + 400 ft^2
225 ft^2 = base^2
To find the value of 'base', take the square root of both sides:
sqrt(225 ft^2) = sqrt(base^2)
base = 15 ft
Therefore, the farthest distance the base of the ladder can be from the wall is 15 feet.
To find the height the ladder will reach on the wall, we can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the ladder) is equal to the sum of the squares of the other two sides.
Let's calculate the height the ladder will reach on the wall:
A) Using the Pythagorean theorem:
- The base of the ladder (one side of the triangle) is 10 feet.
- The ladder itself (the hypotenuse) is 25 feet.
- Let's assume the height the ladder reaches on the wall is represented by 'h'.
The Pythagorean theorem formula is: c^2 = a^2 + b^2
Using this formula, we have: 25^2 = 10^2 + h^2
Simplifying the equation: 625 = 100 + h^2
Subtracting 100 from both sides: 525 = h^2
Now, we can take the square root of both sides to find 'h':
√525 = √(h^2)
22.91 ≈ h
Therefore, the ladder will reach approximately 22.91 feet up the wall.
B) To find the farthest distance the base of the ladder can be from the wall, we'll use the same approach and rearrange the Pythagorean theorem formula:
Let d represent the farthest distance the base of the ladder can be from the wall.
Using the formula, where a = d, b = 20, and c = 25:
25^2 = d^2 + 20^2
625 = d^2 + 400
225 = d^2
Taking the square root of both sides:
√225 = √(d^2)
15 = d
Therefore, the farthest distance the base of the ladder can be from the wall is 15 feet.