Hi,
I’d really appreciate if you can help me out with the following question:
Nick bought a table for his basement. Unfortunately, he discovered that his table rocked back and forth on its four legs. After measuring the table legs and finding that they were all exactly the same length, he decided his problem was that the basement concrete floor was uneven. Assuming the surface of the floor is continuous, show that if Nick rotates the table it will stabilize in less than a 90 degree turn.
Thank you for your time.
To show that rotating the table will stabilize it in less than a 90 degree turn, we can use the Intermediate Value Theorem from calculus.
The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on two different values, then it must also take on every value in between.
In this case, the function we are interested in is the height of the table as a function of its rotation angle. Let's assume that the table is initially placed flush against the uneven floor, so its height is zero. As Nick rotates the table, the height of the table legs will change due to the unevenness of the floor.
Now, let's consider rotating the table by 90 degrees. Since the floor is uneven, the height of each leg will change as the table rotates. Let's say that at some point during this 90-degree rotation, the height of one of the legs drops to a negative value, meaning that the leg is temporarily off the ground.
According to the Intermediate Value Theorem, since the height function is continuous, it must have taken on every value between zero (initial height) and the negative value during this rotation. This means that at some point during this rotation, the height of the table leg must have been exactly zero again, indicating that the table has stabilized on the uneven floor.
Therefore, we have shown that if Nick rotates the table by less than 90 degrees, the table will stabilize on the uneven floor.