A can of soda is initially at room temperature, $72\mathrm{.}3$

$$ Fahrenheit, and at time t $=0$

is placed in a refrigerator set at $37\mathrm{.}7$

$$

$.$ In addition, we know that after $30$ minutes, the soda$$s temperature has

dropped to $59\mathrm{.}5$

$$

$.$ Let F$($t$)$ represent the temperature of the soda in degrees Fahrenheit at time in minutes.

$($a$)$ Use algebraic reasoning and your understanding of the physical situation to determine the exact values

of a$,$ c$,$ and k in the model F$($t$)=$ ae$$kt $+$ c$.$ Show your steps, with sentences to explain your thinking.

$($Hint: This is similar to Activity $3\mathrm{.}2\mathrm{.}3$ that we did in class. The long term behavior tells you the value

of c$,$ and the initial temperature tells you F$(0)$ which is a $+$ c$.$ Use the last point to find k$.)$

$($b$)$ Determine the exact time the object$$s temperature is $42\mathrm{.}4$

$$

$.$ Clearly show your algebraic work and

thinking. $($Hint: In part a you should have gotten F$($t$)=34\mathrm{.}6$e

$0\mathrm{.}015398$t $+37\mathrm{.}7.$ If you didn$$t$,$ go back!

To solve this problem, a natural logarithm to get a variable out of the exponent.$)$

$($c$)$ In Desmos, enter the function F $($using the numbers you found in part $($a$)).$ Use Desmos to find the

average rate of change of F on the interval $[30,35].$ What is the meaning

$($with units$)$ of these values?

$($d$)$ Select two from the following list that are true about the function F$.$

i$.$ decreasing at a decreasing rate

ii$.$ decreasing at an increasing rate

iii. increasing at a decreasing rate

iv$.$ increasing at an increasing rate

v$.$ concave up

vi$.$ concave down

Explain how you know using your answers in part c$.$