Question

question 7 · 1 point
a cooler has a temperature of 32 degrees fahrenheit. a bottled drink is placed in the cooler with an initial temperature of 70 degrees fahrenheit. the function $f(t) = ce^{(-kt)} + 32$ represents the situation, where $t$ is time in minutes, $c$ is a constant, and $k$ is a constant.
after 3 minutes the bottle has a temperature of 42 degrees. what is the approximate value of $k$?
select the correct answer below:
○ 0.445
○ 0.465
○ 0.497
○ 0.512
○ 0.541

Explanation:

Step1: Find constant C

At \(t=0\), \(f(0)=70\). Substitute into \(f(t)=Ce^{-kt}+32\):
\(70 = C + 32 \implies C=38\).

Step2: Substitute known values for k

At \(t=3\), \(f(3)=42\). Use \(C=38\):
\(42 = 38e^{-3k} + 32\).

Step3: Solve for \(e^{-3k}\)

Subtract 32: \(10 = 38e^{-3k} \implies e^{-3k}=\frac{10}{38}\approx0.2632\).

Step4: Take natural log of both sides

\(\ln(e^{-3k})=\ln(0.2632) \implies -3k\approx-1.337\).

Step5: Calculate k

\(k\approx\frac{1.337}{3}\approx0.4457\).

Answer:

0.445