In a large level rectangular field a tall wooden tower was built for training parachute jumpers. The tower is in the shape of a truncated rectangular pyramid. That is, the base is a rectangle, and the top is a smaller rectangle parallel to the base. The longer sides of the field, the base of the tower and the top of the tower are all oriented east to west.
After several years of use, a strong gale tilted the tower, so that the western edge of the top was higher than the eastern edge. (The top was still a rectangle with its eastern and western edges parallel to the eastern and western edges of the field.) The engineers determined that the tower was still fit for use, but to prevent further tilting they stretched cables tightly from the corners of the field to the nearest corners of the top of the tower. The lengths of these cables, going clockwise are 95, 109, 125, and X meters.
What is X?
To simplify I imagined the top of the tower was just a rectangular magic carpet floating in the air, as the rest of the tower seemed irrelevant. Let the distance (along the ground) from the south edge of the field to the south edge of the carpet be a, west edge to west edge be b, north to north be c and east to east be d. Additionally let the height above the ground of the west edge of the carpet be g and the east edge be h.
Then we have four equations for the lengths of the ropes (call them R1, R2, R3, and R4), and given the three we know can use substituion to find the fourth.
a^2 + b^2 + g^2 = R1^2
b^2 + c^2 + g^2 = R2^2
c^2 + d^2 + h^2 = R3^2
d^2 + a^2 + h^2 = R4^2
If I've done my math right, plugging in the three known values into any three clockwise-consecutive corners produces a length of 113 meters for the fourth.
|
|
Posted by tomarken
on 2024-02-19 16:52:06 |
Solution
