A wire is stretched from the top of a two feet tall anchor to the top of a 45 foot tall antena. If the wire is straight and has a slope of 3/5, then what is the length of the wire in feet?
I made a sketch on a grid, I labeled the point at the anchor (0,2) and the point of atachment at the pole as (x,45)
(45-2)/(x-0) = 3/5
3x = 215
x = 215/3
Now , use the formula for the distance between 2 points
To find the length of the wire, we can use the Pythagorean theorem. The Pythagorean theorem 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 wire forms the hypotenuse of a right triangle, and the two sides represent the height of the anchor and the height of the antenna.
Let's represent the height of the anchor as side A and the height of the antenna as side B. We know that the slope of the wire is 3/5 which means that for every 3 units it goes up vertically, it goes 5 units horizontally.
Using this information, we can set up a proportion to find the length of side A. The height of the antenna (B) is 45 feet, and the corresponding horizontal distance is 5 units. So, if side A has a height of x feet, the horizontal distance would be (5/3)*x feet.
Now we can apply the Pythagorean theorem:
A^2 + B^2 = C^2
Substituting the values, we get:
(x^2) + (45^2) = ((5/3)*x)^2
Simplifying the equation:
x^2 + 2025 = (25/9) * x^2
(8/9) * x^2 = 2025
Dividing both sides by (8/9):
x^2 = 2025 * (9/8)
x^2 = 45^2 * 9
Taking the square root of both sides, we get:
x = 45 * 3
Therefore, the height of the anchor is 45*3 = 135 feet, and the length of the wire (hypotenuse) is the square root of (135^2 + 45^2).