If an 8-foot ladder is leaned against a wall with its foot braced 4 feet from the wall’s base, how far up the wall will the top of the ladder reach? Round to the nearest hundredth.
Use the Pythagorean Theorem.
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To find out how far up the wall the top of the ladder will reach, we can use the Pythagorean theorem.
According to the theorem, in a right triangle, the square of the length 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 is the hypotenuse, and the distance from the wall's base to the foot of the ladder is one side, while the distance from the top of the ladder to the top of the wall is the other side.
So, we have:
H^2 = A^2 + B^2
Where:
H is the length of the ladder (8 feet)
A is the distance from the wall's base to the foot of the ladder (4 feet)
B is the distance from the top of the ladder to the top of the wall (unknown)
Substituting the given values into the equation, we get:
8^2 = 4^2 + B^2
Simplifying:
64 = 16 + B^2
Rearranging the equation to solve for B:
B^2 = 64 - 16
B^2 = 48
Now, taking the square root of both sides:
B = sqrt(48)
Calculating the square root of 48, we find:
B ≈ 6.93
Therefore, the top of the ladder will reach approximately 6.93 feet up the wall.
To determine how far up the wall the top of the ladder will reach, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the ladder acts as the hypotenuse, and the distance from the foot of the ladder to the wall's base and the height of the wall act as the other two sides.
Let's represent the distance up the wall with 'x'. Applying the Pythagorean theorem, we have:
(4^2) + (x^2) = (8^2)
Simplifying, we get:
16 + x^2 = 64
Now, subtracting 16 from both sides, we have:
x^2 = 48
To find the value of 'x', we take the square root of both sides:
x = √48
Now, evaluating the square root of 48, we get:
x ≈ 6.93
Therefore, the top of the ladder will reach approximately 6.93 feet up the wall.