There are 2 towers that are each 10m high. A rope that is 15m long is strung between the tops of the towers. At its lowest point the rope sags 2.5m above the ground (see schematic diagram). How far apart are the towers?
7.5m
0 meter
what is the shape of the rope?
straight lines?
parabola?
catenary?
12.5
Zero
To find out how far apart the towers are, we need to use the concept of right triangles and the Pythagorean theorem.
Let's label the distance between the towers as "x." We can form a right triangle with one side being the distance between the ground and the lowest point of the sagging rope (2.5m), the second side being half the distance between the towers (x/2), and the hypotenuse being the length of the rope (15m).
Using the Pythagorean theorem, we can find the value of x by applying the formula:
(2.5)^2 + (x/2)^2 = (15)^2
Simplifying the equation:
6.25 + (x^2/4) = 225
Multiplying both sides by 4 to eliminate the fraction:
25 + x^2/4 = 900
Subtracting 25 from both sides:
x^2/4 = 875
Multiplying both sides by 4:
x^2 = 3500
Taking the square root of both sides:
x = √3500
Approximately,
x = 59.16
Therefore, the two towers are approximately 59.16 meters apart.