Math  /  Calculus

QuestionA bottle of seltzer water initially has a temperature of 79F79^{\circ} \mathrm{F}. It is left to cool in a refrigerator that has a temperature of 40F40^{\circ} \mathrm{F}. After 10 minutes the temperature of the seltzer water is 66F66^{\circ} \mathrm{F}. Complete parts a through C . a. Use Newton's Law of Cooling, T=C+(T0C)ekt\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)e0.0405tT=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 F{ }^{\circ} \mathrm{F} (Round to nearest degree as needed.)

Studdy Solution
Calculate the temperature after 15 minutes using the model:
T=40+39e0.0405×15 T = 40 + 39 e^{-0.0405 \times 15}
T40+39e0.6075 T \approx 40 + 39 e^{-0.6075}
T40+39×0.5442 T \approx 40 + 39 \times 0.5442
T40+21.222 T \approx 40 + 21.222
T61.222 T \approx 61.222
Round to the nearest degree:
The temperature after 15 minutes is approximately 61F 61^{\circ} \mathrm{F} .
The temperature of the seltzer water after 15 minutes is:
61F \boxed{61^{\circ} \mathrm{F}}

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