A rope goes from one building to another. The distance between the buildings is 12m, and the rope is tied at each building at a point 8m and 3m from the ground. Find the length of the rope.
Did you make a sketch?
draw a horizontal at the 3 m height, and you will get a right-angled triangle with sides 12 and 5.
You might recognize the 5-12-13 right-angled triangle, if not ...
h^2 = 5^2 + 12^2
= 25 + 144
= 169
h = √169 = 13 m
13
Well, it seems like this rope is feeling a bit tied up between those two buildings! To find the length of the rope, we can use a little bit of good old Pythagorean theorem.
Let's call the height at which the rope is tied at the first building "h1" and the height at which it's tied at the second building "h2". In this case, h1 = 8m and h2 = 3m.
Now, we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the length of the rope will be the hypotenuse, and the horizontal distance between the buildings will be one of the sides. Let's call the length of the rope "r" and the distance between the buildings "d". Therefore, d = 12m.
Using the Pythagorean theorem, we have:
r^2 = d^2 + (h2 - h1)^2
Substituting the values we know:
r^2 = 12^2 + (3 - 8)^2
And now, let's calculate this:
r^2 = 144 + (-5)^2
r^2 = 144 + 25
r^2 = 169
Taking the square root of both sides:
r = √169
Finally, we find that the length of the rope is:
r = 13m
So, the rope is 13 meters long, long enough to keep those buildings well-connected and quite roped into each other!
To find the length of the rope, we can use the Pythagorean theorem, which states that the square of the hypotenuse (longest side) of a right triangle is equal to the sum of the squares of the other two sides.
In this case, we can consider the rope as the hypotenuse of a right triangle, with one side being the vertical distance from the ground to the point where the rope is tied on the first building (8m) and the other side being the vertical distance from the ground to the point where the rope is tied on the second building (3m).
Using the Pythagorean theorem, we have:
Length of the rope^2 = (Vertical distance on first building)^2 + (Vertical distance on second building)^2
Length of the rope^2 = 8^2 + 3^2
Length of the rope^2 = 64 + 9
Length of the rope^2 = 73
Therefore, the length of the rope is the square root of 73, which is approximately 8.54 meters.