What type of mathematical function is this? Thanks :)

[Hawaiianboi808]

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

 

[MarkFL]

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

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

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

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

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

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

This doesn't fit the remaining points exactly, but it is the type of function we would expect. :D

 

[HallsofIvy]

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.