Sylvia is replacing a piece of siding on her house. To make the 12 ft. ladder stable, the bottom of the ladder needs to be 6 ft. from the base of her house. Applying the Pythagorean Theorem and rounding to the nearest tenth, how high will the ladder reach so that Sylvia can replace the siding?(1 point)
Responses
13.4 ft.
13.4 ft.
180 ft.
180 ft.
108 ft.
108 ft.
10.4 ft.
To find the height that the ladder will reach, 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.
In this case, the ladder forms a right triangle with the house wall and the ground. Let h represent the height the ladder reaches.
Given that the bottom of the ladder is 6 ft from the base of the house and the ladder is 12 ft long (hypotenuse), we can set up the equation:
6^2 + h^2 = 12^2
Simplifying, we get:
36 + h^2 = 144
Subtracting 36 from both sides, we have:
h^2 = 108
Taking the square root of both sides, we find:
h = √108
Rounding to the nearest tenth, the height the ladder will reach is approximately 10.4 ft.
Therefore, the correct answer is 10.4 ft.
To solve this problem, we can use the Pythagorean Theorem, which states that 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 forms a right triangle with the ground and the height of the house. The ladder acts as the hypotenuse, the ground distance is 6 ft, and we need to find the height of the ladder.
Using the Pythagorean Theorem, we can set up the equation as follows:
hypotenuse^2 = side1^2 + side2^2
Ladder^2 = 6^2 + height^2
We need to solve for the height, so we rearrange the equation:
height^2 = ladder^2 - 6^2
height^2 = ladder^2 - 36
Now, we can substitute the given ladder length into the equation:
height^2 = 12^2 - 36
height^2 = 144 - 36
height^2 = 108.
Taking the square root of both sides to solve for the height, we find the height to be approximately 10.4 ft.
Therefore, the ladder will reach approximately 10.4 ft in height, rounded to the nearest tenth.
To solve this problem using the Pythagorean theorem, we need to recall the formula:
c^2 = a^2 + b^2
Where c is the length of the ladder, and a and b are the distances from the base and the height respectively.
In this case, we have the following information:
- The distance from the base of the ladder to the house is 6 ft.
- The length of the ladder is given as 12 ft.
Let's substitute these values into the formula and solve for the height:
c^2 = a^2 + b^2
12^2 = 6^2 + b^2
144 = 36 + b^2
b^2 = 108
Now we can take the square root of both sides to find the height (rounded to the nearest tenth):
b = √108
b ≈ 10.4 ft
Therefore, the ladder will reach approximately 10.4 ft in height so that Sylvia can replace the siding.