There are $2$ towers that are each $10$m high. A rope that is $15$m long is strung between the tops of the towers. At its lowest point the rope sags $2.5$m about the ground (see schematic diagram). How far apart are the towers?
I started attempting the question but found nothing relevant, and now I'm confused from where even to start with the question.
Plz someone help Me in solving this!
P.S-I don’t have a ton of knowledge in mathematics. An answer using elementary math would be appreciated.
On
The answer is zero!
It is very easy, just consider a right triangle with hypotenuse adjacent to the half of the rope length (for example left half of the rope). In this right angle we have one side 7.5m (10-2.5= 7.5). Because the length of the rope is 15m, then half of the length will be 7.5m, so the hypotenuse adjacent to it will be less than 7.5m. (the straight line is shorter than the curve line). Now we have a right triangle that it's hypotenuse is shorter than one side and this is wrong per geometric & math principles (every body knows that in right triangles the hypotenuse is longer than the other sides). So our consideration is wrong and there is no triangle between two towers (using same logic for the right half of the rope) and the distance between towers is zero.
Answer: zero meters.
The image is deceptive here. If you were to string a 15 meter rope between two 10 meter tall towers and then imagine scooting the two towers closer and closer to each other, what would be the lowest point the rope would reach? That would occur when the rope went only straight down and straight back up again without any horizontal breadth to it. In that case, exactly half of the rope's length would be spent going down, and the other half spent going back up. In other words: the lowest the rope could reach would be 15/2 = 7.5 meters down from the top. For 10 meter tall towers, that would leave exactly 2.5 meters of air space at the bottom which is exactly what we have! Therefore the two towers must be right next to each other, since otherwise the lowest point in the rope would have to be higher than 2.5 meters.