# A student doing a Science Fair experiment put a hot bowl of soup in the refrigerator and checked the temperature of the soup every 2 minutes: a) Creat

A student doing a Science Fair experiment put a hot bowl of soup in the refrigerator and checked the temperature of the soup every 2 minutes: a) Create a model describing how the soup cools. Be sure to look at the residuals to verify that your model is appropriate.
b) Explain what the two values in the equation suggest about the soup.
c) Estimate the temperature of the soup after 3 minutes.
d) Estimate the temperature of the soup after 25 minutes.
e) How much confidence do you place in those estimates? Why?

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a) Input the times under L1 in your graphing calculator and the corresponding temperatures under L2. The types of regressions we can use to write a model are linear, quadratic, cubic, quartic, natural log, exponential, power, logistic, and sinusoidal.
The graph is in the shape of half of a U so it is likely an exponential model. Use the ExpReg feature on your calculator to find the model and the value of R2. This gives $$\displaystyle{a}≈{71}{\quad\text{and}\quad}{b}≈{0.8658}$$ for $$\displaystyle{y}={a}⋅{b}^{{x}}$$ which gives a model of $$\displaystyle{y}={71}{\left({0.8658}\right)}^{{x}}$$. Since $$\displaystyle{R}^{{2}}≈{0.9965}$$ which is very close to 1, then this model is a very good fit for the data.
b) For an exponential function, aa is the initial value and bb is the growth or decay factor. Since $$\displaystyle{a}≈{71}$$, then the initial temperature is about $$71^{\circ}C$$. Since $$b \approx 0.8658$$ and $$1-0.8658=0.1342=13.42\%$$, then the temperature is decreasing by about 13.42% each minute the temperature is decreasing by about 13.42% each minute.
c) After $$x=3$$ minutes, the temperature is about $$\displaystyle{y}={71}{\left({0.8658}\right)}^{{3}}≈{46.1}^{\circ}{C}$$.
d) After $$x=25$$ minutes, the temperature is about $$\displaystyle{y}={71}{\left({0.8658}\right)}^{{25}}≈{1.9}^{\circ}{C}$$.
e) Since $$x=3$$ lies between two points that were used to create the model, I have high confidence in the estimate of $$46.1^{\circ}C$$. Since $$x=25$$ lies outside of the points used to create the model but is close to the largest point with $$x=20$$, I am moderately confident in the estimate of $$1.9^{\circ}C$$.