A supporting wire is to be attached to the top of a 32-foot antenna. If the wire must be anchored 32 feet from the base of the antenna, what length of wire is required? Please explain
Use the Pythagorean theorem because you want the hypotenuse of this right triangle.
a^2 + b^2 = c^2
32^2 + 32^2 = c^2
1024 + 1024 = c^2
2048 = c^2
45.454833 = c
Well, isn't this a "wire"d problem? Let's untangle it together!
We have a 32-foot antenna, and we need to attach a supporting wire from the top of the antenna to a point 32 feet away from the base. Essentially, we are forming a right triangle with the antenna acting as the height and the wire as the hypotenuse.
According to Pythagoras, the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In our case, the hypotenuse is the wire, and the other two sides are the antenna (32 feet) and the distance from the base (32 feet).
So, using the Pythagorean theorem, we have:
wire^2 = antenna^2 + distance^2
Let's plug in the values we have:
wire^2 = 32^2 + 32^2
Simplifying:
wire^2 = 1024 + 1024
wire^2 = 2048
Now let's solve for the wire:
wire = √2048
*waving a magical clown wand* and poof!
wire ≈ 45.25 feet
So, approximately, you'll need a wire that is about 45.25 feet long to anchor it 32 feet from the base of the antenna.
To find the length of wire required, you can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this scenario, the antenna acts as the hypotenuse, and the wire and the distance from the base of the antenna to the anchor point form the other two sides of the right triangle.
Step 1: Draw a diagram to represent the problem. The antenna is the hypotenuse, and the wire along with the distance from the base of the antenna to the anchor point forms two sides of the right triangle.
Step 2: Label the given values:
- Hypotenuse (the antenna) = 32 feet
- One of the other two sides (distance from the base to anchor point) = 32 feet
Step 3: Use the Pythagorean theorem to find the length of the wire (the remaining side). The formula is as follows:
a² + b² = c²
Let "a" represent the wire, "b" represent the distance from the base to the anchor point, and "c" represent the antenna.
32² + 32² = c²
Step 4: Calculate the square of the given values:
32² = 1024
Step 5: Simplify the equation:
1024 + 1024 = c²
2048 = c²
Step 6: Take the square root of both sides to find the value of c:
√2048 = √(c²)
c ≈ 45.25
Therefore, approximately 45.25 feet of wire is required.
To find the length of wire required, we can use the Pythagorean theorem. In this case, the antenna forms a right-angled triangle with the supporting wire and the ground.
The height of the antenna (the side opposite the right angle) is given as 32 feet. The horizontal distance from the base of the antenna to the anchor point (the side adjacent to the right angle) is also given as 32 feet.
Using the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides, we can find the length of the wire.
Let's represent the length of the wire as 'x'. According to the Pythagorean theorem:
x^2 = 32^2 + 32^2
Simplifying the equation:
x^2 = 1024 + 1024
x^2 = 2048
To solve for 'x', we take the square root of both sides:
x = √2048
Using a calculator to find the square root:
x ≈ 45.25 feet
Therefore, approximately 45.25 feet of wire is required to support the 32-foot antenna when anchored 32 feet from its base.