Question

After sitting on a shelf for a while, a can of soda at room temperature (73 F) is placed inside a refrigerator and slowly cools. The temperature of the refrigerator is 35 F.
The can of soda reaches the temperature of 53 F after 35 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.

Answer 1

Answer:

Step-by-step explanation:

Use Newton's Law of Cooling here:

$T(t)=T_s+(T_0-T_s)^{-kt$

where T(t) is the temp of something after a certain amount of time, t, has gone by; Ts is the surrounding temp, and T0 is the initial temp. k is the constant of cooling. We need to first solve for this using the information given. Filling in what we know:

$53=35+(73-35)^{-35k$ which can simplify a bit to

$53=35+(38)^{-35k$ and $18=38^{-35k$

Take the natural log of both sides:

$ln(18)=ln(38)^{-35k$

Taking the natural log allows us to pull that exponent down out front:

$ln(18)=-35k*ln(38)$

and now we can divide both sides by ln(38) to get

-35k = .794585 so

k = -.023

Now that have the value for k, we can go on to solve the rest of the problem which is asking us the temp of the soda after 70 minutes. Filling in using the k value and the new time of 70 minutes:

$T(t)=35+(38)^{70(-.023)$ and

$T(t)=35+(38)^{-1.61$ and

$T(t)=35+.0028612013$ and

T(t) = 35 F, basically the temp of the fridge, which is not surprising!