What type of mathematical function is this? Thanks :)

AI Thread Summary
The discussion revolves around modeling the temperature increase of an oven using mathematical functions. It suggests that the temperature change could be modeled by an exponential function, specifically one derived from Newton's Law of Cooling. The proposed formula is f(t) = -280e^(-kt) + 350, which approximates the temperature at various time intervals. However, it is noted that a quadratic polynomial could also fit the data points more accurately, as any set of three points can be represented by a quadratic equation. The conversation emphasizes the flexibility of mathematical modeling in representing real-world scenarios.
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John lives in Dallas and his kitchen has a room temperature of about 70 degrees fahrenheit. He wanted to make her family some cookies for dessert, so he preheated her oven to 350 degrees fahrenheit. In 1 minute, the oven was 135 degrees fahrenheit. In 2 minutes, the oven was about 200 degrees fahrenheit. In 4 minutes, it went up to about 300 degrees fahrenheit. John's cookie dough was ready to go in the oven about 10 minutes after he turned it on.

I believe that this is an exponential function. Not sure, please double check.

Include example of formula.

Thank you
 
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We could likely model this with a function of the form (which comes from Newton's Law Of Cooling):

$$f(t)=c_1e^{-kt}+350$$

Since we know $f(0)=70$, we then have:

$$f(t)=-280e^{-kt}+350$$

And we know $f(1)=135$, so we have:

$$f(t)=-280\left(\frac{43}{56}\right)^t+350=70\left(5-4\left(\frac{43}{56}\right)^t\right)$$

This doesn't fit the remaining points exactly, but it is the type of function we would expect. :D
 
If you want to exactly fit all points, given any n points, there exist a polynomial of degree n-1 that passes through each of those points. Here, there are three points, (1, 135), (2, 200), and (4, 300) so there exist a quadratic polynomial that passes through the three points.

Any quadratic polynomial can be written in the form y= ax^2+ bx+ c. The data above gives 135= a+ b+ c, 200= 4a+ 2b+ c, and 300= 16a+ 4b+ c. Solve those three equations for a, b, and c.
 
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