Question
After sitting on a shelf for a while, a can of soda at a room temperature is placed inside
refrigerator and slowly cools. The temperature of the refrigerator is . Newton’s 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, a
given by the formula below:

the initial temperature of the object
the time in minutes
the temperature of the object after minutes
The can of soda reaches the temperature of after 15 minutes. Using this information,
find the value of , to the nearest thousandth. Use the resulting equation to determine the
Fahrenheit temperature of the can of soda, to the nearest degree, after 115 minutes.
Enter only the final temperature into the input box.
Answer Aumpt out ots

Ask by Powell Haynes. in the United States

Feb 28,2025

Question

Question
After sitting on a shelf for a while, a can of soda at a room temperature is placed inside
refrigerator and slowly cools. The temperature of the refrigerator is . Newton’s 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, a
given by the formula below:

the initial temperature of the object
the time in minutes
the temperature of the object after minutes
The can of soda reaches the temperature of after 15 minutes. Using this information,
find the value of , to the nearest thousandth. Use the resulting equation to determine the
Fahrenheit temperature of the can of soda, to the nearest degree, after 115 minutes.
Enter only the final temperature into the input box.
Answer Aumpt out ots

Ask by Powell Haynes. in the United States

Feb 28,2025

UpStudy AI · Accepted Answer

To find the value of $$ k $$ and then determine the temperature of the can of soda after 115 minutes, we can use the given information and Newton's Law of Cooling formula.

Given:
- Initial temperature of the can of soda, $$ T_{0} = 73^{\circ} \mathrm{F} $$
- Temperature of the refrigerator, $$ T_{a} = 39^{\circ} \mathrm{F} $$
- Temperature of the can of soda after 15 minutes, $$ T = 61^{\circ} \mathrm{F} $$
- Time, $$ t = 15 $$ minutes

Newton's Law of Cooling formula:
$$ T = T_{a} + (T_{0} - T_{a}) e^{-kt} $$

Substitute the given values into the formula:
$$ 61 = 39 + (73 - 39) e^{-k \cdot 15} $$

Simplify the equation:
$$ 61 = 39 + 34 e^{-15k} $$

Now, we need to solve for $$ k $$ using the given information. Let's solve the equation for $$ k $$ first.
Solve the equation by following steps:
- step0: Solve for $$k$$:
  $$61=39+34e^{-15k}$$
- step1: Swap the sides:
  $$39+34e^{-15k}=61$$
- step2: Move the expression to the left side:
  $$39+34e^{-15k}-61=0$$
- step3: Subtract the numbers:
  $$-22+34e^{-15k}=0$$
- step4: Factor the expression:
  $$-2\left(11e^{15k}-17\right)\left(e^{k}\right)^{-15}=0$$
- step5: Rewrite the expression:
  $$\frac{-22e^{15k}+34}{\left(e^{k}\right)^{15}}=0$$
- step6: Cross multiply:
  $$-22e^{15k}+34=\left(e^{k}\right)^{15}\times 0$$
- step7: Simplify the equation:
  $$-22e^{15k}+34=0$$
- step8: Rewrite the expression:
  $$-22e^{15k}=-34$$
- step9: Change the sign:
  $$22e^{15k}=34$$
- step10: Divide both sides:
  $$\frac{22e^{15k}}{22}=\frac{34}{22}$$
- step11: Divide the numbers:
  $$e^{15k}=\frac{17}{11}$$
- step12: Take the logarithm of both sides:
  $$\ln{\left(e^{15k}\right)}=\ln{\left(\frac{17}{11}\right)}$$
- step13: Evaluate the logarithm:
  $$15k=\ln{\left(\frac{17}{11}\right)}$$
- step14: Divide both sides:
  $$\frac{15k}{15}=\frac{\ln{\left(\frac{17}{11}\right)}}{15}$$
- step15: Divide the numbers:
  $$k=\frac{\ln{\left(\frac{17}{11}\right)}}{15}$$
- step16: Simplify:
  $$k=\frac{\ln{\left(17\right)}-\ln{\left(11\right)}}{15}$$
The value of $$ k $$ is $$ \frac{\ln{(17)}-\ln{(11)}}{15} $$.

Now, we can use this value of $$ k $$ to determine the temperature of the can of soda after 115 minutes. Substitute $$ k $$ back into the formula and calculate the temperature.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\ln{\left(17\right)}-\ln{\left(11\right)}}{15}$$
The value of $$ k $$ is approximately 0.029021.

Now, we can substitute this value of $$ k $$ back into the formula to determine the temperature of the can of soda after 115 minutes.
Calculate the value by following steps:
- step0: Calculate:
  $$39+\left(73-39\right)e^{-0.029021\times 115}$$
- step1: Subtract the numbers:
  $$39+34e^{-0.029021\times 115}$$
- step2: Multiply the numbers:
  $$39+34e^{-3.337415}$$
- step3: Convert the expressions:
  $$39+34e^{-\frac{667483}{200000}}$$
- step4: Multiply the terms:
  $$39+\frac{34e^{\frac{132517}{200000}}}{e^{4}}$$
- step5: Reduce fractions to a common denominator:
  $$\frac{39e^{4}}{e^{4}}+\frac{34e^{\frac{132517}{200000}}}{e^{4}}$$
- step6: Transform the expression:
  $$\frac{39e^{4}+34e^{\frac{132517}{200000}}}{e^{4}}$$
- step7: Simplify:
  $$\frac{39e^{4}+34\sqrt[200000]{e^{132517}}}{e^{4}}$$
The temperature of the can of soda after 115 minutes is approximately 40.207975 degrees Fahrenheit.
After 115 minutes, the temperature of the can of soda is approximately $$ 40^{\circ} \mathrm{F} $$.

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