Two poles are connected by a wire that is also connected to the ground. The first pole is 60 ft tall and the second pole is 30 ft tall. There is a distance of 90 ft between the two poles. Where should the wire be anchored to the ground (in ft, from the shorter pole) to minimize the amount of wire needed?
Hmmm. at the base of the short pole? That is 30 ft down. Anywhere else, it will be longer. maybe I am not seeing the picture here.
What i found out from working this wrong a few times, it is 30 because that is the minimum spot it would take to minimize the amount of wire needed for the project.
To minimize the amount of wire needed, we can use the concept of the Pythagorean theorem. Let's assume that the wire is anchored to the ground, a distance of 'x' feet from the shorter pole.
By forming a right triangle with the wire as the hypotenuse, we can use the Pythagorean theorem to find the length of the wire:
x^2 + (60-30)^2 = (90-x)^2
Simplifying the equation, we get:
x^2 + 30^2 = (90-x)^2
x^2 + 900 = (90-x)^2
Expanding the right side of the equation, we get:
x^2 + 900 = 8100 - 180x + x^2
Rearranging the equation, we get:
180x = 7200
Dividing both sides by 180, we get:
x = 40
Therefore, the wire should be anchored to the ground, 40 ft away from the shorter pole, to minimize the amount of wire needed.
To solve this problem, we can use the concept of similar triangles.
Let's assume that the wire is anchored at a distance 'x' feet from the shorter pole.
When we draw a diagram, we can see that the wire, the height of the poles, and the distance between the poles form two similar triangles.
The ratio of corresponding sides of similar triangles is equal. So, we can set up the following proportion:
(height of the shorter pole) / (distance from the shorter pole to the anchor point) = (height of the taller pole) / (distance from the taller pole to the anchor point)
Using the given information, we can substitute the values into the proportion:
30 ft / x ft = 60 ft / (90 ft - x ft)
Now we can solve this proportion for 'x'.
Cross multiplying, we get:
30 ft * (90 ft - x ft) = 60 ft * x ft
Expanding and rearranging terms, we have:
2700 ft - 30 ft * x ft = 60 ft * x ft
Adding 30 ft * x ft to both sides, we get:
2700 ft = 90 ft * x ft
Dividing both sides by 90 ft, we find:
2700 ft / 90 ft = x ft
Simplifying the equation, we get:
30 ft = x ft
Therefore, the wire should be anchored 30 ft away from the shorter pole to minimize the amount of wire needed.