A painter leans a ladder against a vertical wall. The top of the ladder is 7 meters above the ground. When the bottom of the ladder is moved 1 meter farther away from the wall, the top of the ladder is 5 meters above the ground. What is the length of the ladder? Round to the nearest thousandth.
To find the length of the ladder, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's call the length of the ladder "x", the distance of the bottom of the ladder from the wall "a", and the height above the ground "b".
Using the given information, we know that when the ladder is leaning against the wall, with the bottom of the ladder "a" meters away from the wall, the height above the ground is "b = 7 meters".
When the bottom of the ladder is moved 1 meter farther away from the wall, the height above the ground becomes "b = 5 meters".
We can set up two equations using the Pythagorean theorem:
Equation 1: x^2 = a^2 + 7^2
Equation 2: x^2 = (a+1)^2 + 5^2
Now, we can solve these equations to find the length of the ladder.
From Equation 1, we have:
x^2 = a^2 + 49
From Equation 2, we have:
x^2 = (a+1)^2 + 25
= a^2 + 2a + 1 + 25
= a^2 + 2a + 26
Since both equations are equal to x^2, we can set them equal to each other:
a^2 + 49 = a^2 + 2a + 26
Simplifying the equation by canceling out the common terms:
49 = 2a + 26
Subtracting 26 from both sides:
23 = 2a
Dividing both sides by 2:
a = 11.5
Now that we have the value of "a", we can substitute it into either Equation 1 or 2 to find the value of "x" (the length of the ladder).
Using Equation 1:
x^2 = a^2 + 49
x^2 = 11.5^2 + 49
x^2 = 132.25 + 49
x^2 = 181.25
To find the square root of both sides:
x = √181.25
Rounding to the nearest thousandth:
x ≈ 13.477
Therefore, the length of the ladder, rounded to the nearest thousandth, is approximately 13.477 meters.
Let's assume that the length of the ladder is represented by 'L' meters and the distance of the bottom of the ladder to the wall is represented by 'x' meters.
According to the problem, we have the following information:
- When the ladder is leaning against the wall, the top of the ladder is 7 meters above the ground.
- When the bottom of the ladder is moved 1 meter farther away from the wall, the top of the ladder is 5 meters above the ground.
We can create a right triangle with the ladder as the hypotenuse, the vertical wall as one side, and the ground as the other side.
Using the Pythagorean theorem, we have:
L^2 = x^2 + 7^2 ----(1) (since the top of the ladder is 7 meters above the ground)
(L+1)^2 = x^2 + 5^2 ----(2) (when the bottom of the ladder is moved 1 meter farther away from the wall, the top of the ladder is 5 meters above the ground)
Expanding equation (2), we get:
L^2 + 2L + 1 = x^2 + 25 ----(3)
Subtracting equation (1) from equation (3), we get:
2L + 1 - 49 = 0
2L - 48 = 0
2L = 48
L = 48/2
L = 24
Therefore, the length of the ladder is approximately 24 meters.