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.
Let's represent the length of the ladder as 'x' meters.
Using the Pythagorean theorem, we can set up the following equation:
x^2 = (x + 1)^2 - 7^2
Expanding the equation:
x^2 = x^2 + 2x + 1 - 49
Combining like terms:
0 = 2x - 48
Now, solve for 'x':
2x = 48
x = 48/2
x = 24
Therefore, the length of the ladder is approximately 24 meters (rounded 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 sum of the squares of the lengths of the two legs (sides adjacent to the right angle) is equal to the square of the length of the hypotenuse (longest side).
Let's denote the length of the ladder as "x". We have a right triangle formed by the ground, the wall, and the ladder.
The length of the vertical side (height) of the triangle is 7 meters when the bottom of the ladder is against the wall. When the bottom is moved 1 meter farther away from the wall, the height decreases to 5 meters.
Using the Pythagorean theorem, we have the equation:
x^2 = (7^2 - 5^2)
x^2 = (49 - 25)
x^2 = 24
To find the length of the ladder "x", we take the square root of both sides:
x = sqrt(24)
x ≈ 4.899
Rounding to the nearest thousandth, the length of the ladder is approximately 4.899 meters.