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?
Use the Pythagorean theorem quick check answers
1. 14.7
2. 16.97
3. 10.4
4. 22.4
5. 4.7
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Let h be the height of the ladder. Using the Pythagorean theorem, we can write the equation (6)^2 + h^2 = (12)^2.
Expanding the equation, we get 36 + h^2 = 144.
Subtracting 36 from both sides, we get h^2 = 108.
Taking the square root of both sides, we get h = ±√108.
Since the ladder cannot have a negative height, we take the positive value.
Therefore, h = √108 ≈ 10.4.
So, the ladder will reach a height of approximately 10.4 ft. Answer: \boxed{10.4}.
To find the height the ladder will reach, 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 wall. The bottom of the ladder is 6 ft. from the base of the house, which is one of the legs of the triangle. The length of the ladder itself is the hypotenuse, and the height it reaches is the other leg we want to find.
Let's say the height the ladder reaches is represented by 'h'. According to the Pythagorean Theorem:
h^2 = (length of the hypotenuse)^2 - (length of the other leg)^2
In this case, the hypotenuse is the length of the ladder, which is 12 ft. So we have:
h^2 = 12^2 - 6^2
Simplifying:
h^2 = 144 - 36
h^2 = 108
To find the value of 'h', we need to take the square root of both sides of the equation:
h ≈ sqrt(108)
Using a calculator, the square root of 108 is approximately 10.4 ft.
Therefore, rounding to the nearest tenth, the ladder will reach a height of approximately 10.4 ft.
To find the height the ladder will reach, we can use the Pythagorean Theorem, which 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.
In this case, the ladder acts as the hypotenuse, the distance from the base of the house to the ladder as one side, and the height the ladder will reach as the other side. Let's call the height of the ladder h.
According to the given information, the distance from the base of the house to the ladder is 6 ft. We can represent this as one side of a right triangle. Let's call this side "a".
The ladder is 12 ft. long, which can be represented as the hypotenuse. We will call this side "c".
Using the Pythagorean Theorem, we can write the equation: a^2 + b^2 = c^2.
Substituting the given values, we have:
6^2 + h^2 = 12^2
Simplifying the equation, we have:
36 + h^2 = 144
Now, we can solve for h:
h^2 = 144 - 36
h^2 = 108
Taking the square root of both sides, we have:
h ≈ √108
Approximating the square root of 108 to the nearest tenth, we get:
h ≈ 10.4
Therefore, the ladder will reach a height of approximately 10.4 ft.
An electric pole has a support cable that is attached to the pole 20 ft. from the ground. The cable is anchored on the ground 10 ft. from the base of the pole. How long is the cable? Round to the nearest tenth.
To find the length of the cable, we can use the Pythagorean Theorem.
Let 'c' be the length of the cable.
According to the given information, the height of the pole is 20 ft and the distance from the base of the pole to the anchor point on the ground is 10 ft. These two dimensions form a right-angled triangle, with the cable acting as the hypotenuse.
Using the Pythagorean theorem, we can write the equation:
c^2 = (length of one side)^2 + (length of the other side)^2
In this case, the two sides are the height of the pole and the distance from the base of the pole to the anchor point on the ground.
Substituting the given values, we have:
c^2 = 20^2 + 10^2
Simplifying the equation, we have:
c^2 = 400 + 100
c^2 = 500
To find the value of 'c', we need to take the square root of both sides of the equation:
c ≈ sqrt(500)
Using a calculator, the square root of 500 is approximately 22.4 ft.
Therefore, rounding to the nearest tenth, the length of the cable is approximately 22.4 ft.
A streetlight post needs some extra support for potential bad weather and high winds. The city wants to install metal support brackets on the light post. Each bracket will be braced into the ground and will then attach to a point on the pole that is 4.5 ft. from the ground. Each bracket is 6.5 ft. long. How far from the base of the pole should each bracket be braced in the ground? Round to the nearest tenth.
To find the distance from the base of the pole where each bracket should be braced into the ground, we can use the Pythagorean Theorem.
Let 'd' be the distance from the base of the pole where each bracket should be braced into the ground.
According to the given information, each bracket is 6.5 ft long and attached to a point on the pole that is 4.5 ft from the ground. These two dimensions form a right-angled triangle, with the distance from the base of the pole to where each bracket is braced into the ground acting as one side.
Using the Pythagorean theorem, we can write the equation:
d^2 = (length of one side)^2 + (length of the other side)^2
In this case, the two sides are the length of each bracket and the distance from the point on the pole to the ground.
Substituting the given values, we have:
d^2 = 6.5^2 + 4.5^2
Simplifying the equation, we have:
d^2 = 42.25 + 20.25
d^2 = 62.5
To find the value of 'd', we need to take the square root of both sides of the equation:
d ≈ sqrt(62.5)
Using a calculator, the square root of 62.5 is approximately 7.9 ft.
Therefore, rounding to the nearest tenth, each bracket should be braced in the ground approximately 7.9 ft from the base of the pole.