Question

after sitting on a shelf for a while, a can of soda at a room temperature (67°f) is placed inside a refrigerator and slowly cools. the temperature of the refrigerator is 35°f. newtons law of cooling explains that the temperature of the can of soda will decrease proportionally to the difference between the temperature of the can of soda and the temperature of the refrigerator, as given by the formula below: $t = t_a+(t_0 - t_a)e^{-kt}$ $t_a=$ the temperature surrounding the object $t_0=$ the initial temperature of the object $t =$ the time in minutes $t=$ the temperature of the object after $t$ minutes $k =$ decay constant the can of soda reaches the temperature of 51°f after 15 minutes. using this information, find the value of $k$, to the nearest thousandth. use the resulting equation to determine the fahrenheit temperature of the can of soda, to the nearest degree, after 70 minutes. enter only the final temperature into the input box.

Explanation:

Step1: Identify given values

$T_a = 35$, $T_0=67$, $t = 15$, $T = 51$

Step2: Substitute values into formula

$51=35+(67 - 35)e^{-15k}$

Step3: Simplify the equation

$51 - 35=(67 - 35)e^{-15k}$
$16 = 32e^{-15k}$
$\frac{16}{32}=e^{-15k}$
$0.5=e^{-15k}$

Step4: Take natural - logarithm of both sides

$\ln(0.5)=\ln(e^{-15k})$
Since $\ln(e^x)=x$, we have $\ln(0.5)=-15k$
$k=-\frac{\ln(0.5)}{15}$
$k\approx0.046$

Step5: Find the temperature after 70 minutes

Now that $k\approx0.046$, $T_a = 35$, $T_0 = 67$, $t = 70$
$T=35+(67 - 35)e^{-0.046\times70}$
$T=35 + 32e^{-3.22}$
$T=35+32\times0.0398$
$T=35 + 1.2736$
$T\approx36$

Answer:

36