Hi, one Professor gave me a very hard question, He has a little Disk (used in machines) that has a big hole in the middle, he measured the thickness every 90°, and he got the following measurements:
0.95 mm
1.02 mm
1.02 mm
0.96 mm
He tells me that with 3 of these measurements, I can know which is the maximum thickness and the minimum thickness even though he didn't measure the thickness everywhere, he wants to know the maximum and minimum thickness to know if the Disk has a big slope and may fail in the machine.
I would be really happy if you could help me with this one :)
Of course, I'd be happy to help you with this question!
To determine the maximum and minimum thickness of the disk using only three measurements, you can employ a concept called the Extreme Value Theorem. According to this theorem, if a given function is continuous on a closed interval, then it must attain both its maximum and minimum values within that interval.
Here's how you can apply this concept to the problem:
1. Based on the given measurements, you have four data points: 0°, 90°, 180°, and 270°, with corresponding thicknesses of 0.95 mm, 1.02 mm, 1.02 mm, and 0.96 mm, respectively.
2. To find the maximum thickness, compare the thicknesses between adjacent measurements. Start with the first two measurements: 0.95 mm and 1.02 mm.
- If the first measurement (0.95 mm) is greater than or equal to the second measurement (1.02 mm), then the second measurement cannot be the maximum thickness.
- If the first measurement is less than the second measurement, then the second measurement might potentially be the maximum thickness.
3. Next, compare the second and third measurements: 1.02 mm and 1.02 mm.
- If the second measurement (1.02 mm) is greater than or equal to the third measurement (1.02 mm), then the third measurement cannot be the maximum thickness.
- If the second measurement is less than the third measurement, then the third measurement might potentially be the maximum thickness.
4. Finally, compare the third and fourth measurements: 1.02 mm and 0.96 mm.
- If the third measurement (1.02 mm) is greater than or equal to the fourth measurement (0.96 mm), then the fourth measurement cannot be the maximum thickness.
- If the third measurement is less than the fourth measurement, then the fourth measurement might potentially be the maximum thickness.
By applying this process, you can identify the potential candidate(s) for the maximum thickness. Similarly, you can determine the potential candidate(s) for the minimum thickness using the same approach.
In this case, the maximum thickness candidate would be 1.02 mm as it is the only measurement that is not ruled out as a potential maximum thickness.
To find the minimum thickness, you can use the same process but reverse the comparison logic. In this case, the minimum thickness candidate would be 0.95 mm as it is the only measurement that is not ruled out as a potential minimum thickness.
So, based on the given measurements, the maximum thickness is 1.02 mm, and the minimum thickness is 0.95 mm.