Metal Heat Treatment: How to create a sinusoidal function that increases in frequency

[babygotjack5]

Alright, this is a hypothetical problem for my math class, (but it seemed to fit here better than calculus help) and though I am allowed to site sources, I don't necessarily want the answer just given to me (I mean, I'd really like to figure it out).

Homework Statement

• Treatment lasts 15 hours.
• Oven starts at 300 degrees
• Oven ends at 100 degrees
• Max temperature: 500
• Min temperature: 50
• Temperature change cannot be more than 4 degrees C per minute over the course of 10 min
• It must increase gradually for the first few hours, then fluctuate rapidly as it approaches 100 degrees.

Homework Equations

This is what I'm looking for

The Attempt at a Solution

My attempts thus far consist of drawings of what the function should look like. From what I can tell, this is a cos (or sin) function that decreases in altitude as it increases in frequency (the function of a wave?) and has a slight downward slope.

So, my question is, how does one "make" such a function? I am inclined to think that it must be like an "linear-exponential-sinusoidal" function, but I'm not sure how to plug it in all together.

 

[AvroArrow]

Any advice or resources that could help me figure out how to create this function would be appreciated.

 

[kerimek]

As a scientist, your approach to this problem should be analytical and systematic. Here are some steps you can follow to create a sinusoidal function that increases in frequency for the given heat treatment scenario:

1. Determine the time interval for the treatment (15 hours) and the temperature range (50-500 degrees).
2. Divide the time interval into smaller segments, for example, 15 segments of 1 hour each.
3. Determine the temperature change for each segment, keeping in mind that it cannot exceed 4 degrees C per minute over the course of 10 minutes.
4. Start with a simple sinusoidal function, such as y = sin(x), and adjust it to fit the given criteria.
5. To increase the frequency, you can multiply the x-value by a constant, such as 2 or 3.
6. To ensure a gradual increase in temperature, you can add a linear function to the sinusoidal function, such as y = sin(x) + 0.5x.
7. To create the fluctuation in temperature as it approaches 100 degrees, you can add a small amplitude to the sinusoidal function, such as y = sin(2x) + 0.1x.
8. Adjust the amplitude and the frequency until the function fits the given temperature range and change criteria.
9. Once you have the function, plot it on a graph and make any necessary adjustments.
10. Test the function by plugging in different values for x (time) and checking if the temperature change stays within the given criteria.
11. Use the function to create a table of values for the temperature at different time intervals.
12. Use the table of values to create a graph of the temperature over time.
13. Check the graph to see if it follows the given criteria.
14. If necessary, make any final adjustments to the function until it accurately represents the heat treatment scenario.

Remember to document your process and any sources you use to create the function. Good luck!