Math / Calculus

QuestionA bottle of seltzer water initially has a temperature of $79^{\circ} \mathrm{F}$ . It is left to cool in a refrigerator that has a temperature of $40^{\circ} \mathrm{F}$ . After 10 minutes the temperature of the seltzer water is $66^{\circ} \mathrm{F}$ . Complete parts a through C . a. Use Newton's Law of Cooling, $\mathrm{T}=\mathrm{C}+\left(\mathrm{T}_{0}-\mathrm{C}\right) e^{\mathrm{kt}}$ , to find a model for the temperature of the seltzer water, T , after t minutes. $T=40+(39) e^{-0.0405 t}$ (Simplify your answer. Use integers or decimals for any numbers in the equation. Round to four decimal places as needed.) b. What is the temperature of the seltzer water after 15 minutes? $\square$ ${ }^{\circ} \mathrm{F}$ (Round to nearest degree as needed.)

Studdy Solution

STEP 1

1. The initial temperature of the seltzer water is $79^{\circ} \mathrm{F}$ .
2. The refrigerator's temperature is $40^{\circ} \mathrm{F}$ .
3. After 10 minutes, the seltzer water's temperature is $66^{\circ} \mathrm{F}$ .
4. We use Newton's Law of Cooling: $T = C + (T_0 - C) e^{kt}$ .

STEP 2

1. Verify and understand the given model for the temperature.
2. Use the model to calculate the temperature after 15 minutes.

STEP 3

Verify the given model $T = 40 + (39) e^{-0.0405 t}$ using the initial conditions.
The initial temperature $T_0$ is $79^{\circ} \mathrm{F}$ , and the surrounding temperature $C$ is $40^{\circ} \mathrm{F}$ .
The equation simplifies to $T = 40 + (79 - 40) e^{kt} = 40 + 39 e^{kt}$ .
Given $T = 66^{\circ} \mathrm{F}$ after 10 minutes, we solve for $k$ :
$66 = 40 + 39 e^{10k}$
$26 = 39 e^{10k}$
$\frac{26}{39} = e^{10k}$
$\ln\left(\frac{26}{39}\right) = 10k$
$k = \frac{\ln\left(\frac{26}{39}\right)}{10} \approx -0.0405$
The model $T = 40 + 39 e^{-0.0405 t}$ is verified.

STEP 4

Calculate the temperature after 15 minutes using the model:
$T = 40 + 39 e^{-0.0405 \times 15}$
$T \approx 40 + 39 e^{-0.6075}$
$T \approx 40 + 39 \times 0.5442$
$T \approx 40 + 21.222$
$T \approx 61.222$
Round to the nearest degree:
The temperature after 15 minutes is approximately $61^{\circ} \mathrm{F}$ .
The temperature of the seltzer water after 15 minutes is:
$\boxed{61^{\circ} \mathrm{F}}$

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