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Goryan [66]
3 years ago
9

A cooler has a temperature of 32 degrees Fahrenheit. A bottled drink is placed in the cooler with an initial temperature of

Mathematics
1 answer:
FinnZ [79.3K]3 years ago
8 0

Answer:

k \approx 0.44

Step-by-step explanation:

Given function:

f(t) = (ce)^{-kt}+32

As per question statement:

Initial temperature of bottle is 70 ^\circ F.

i.e. when time = 0 minutes, f(t) = 70 ^\circ F

70 = ce^{-k\times 0}+32\\\Rightarrow 38 = ce^{0}\\\Rightarrow c = 38

After t = 3, f(t) = 42^\circ F

42 = 38 \times e^{-k\times 3}+32\\\Rightarrow 42-32 = 38 \times e^{-3k} \\\Rightarrow 10  = 38 \times e^{-3k} \\\Rightarrow e^{3k} = \dfrac{38}{10}\\\Rightarrow e^{3k} = 3.8\\\\\text{Taking } log_e \text{both the sides:}\\\\\Rightarrow log_e {e^3k} = log_e {3.8}\\\Rightarrow 3k \times log_ee=log_e {3.8}   (\because log_pq^r=r \times log_pq)\\\Rightarrow 3k \times 1=log_e {3.8}\\\Rightarrow 3k = 1.34\\\Rightarrow k = \dfrac{1.34}{3}\\\Rightarrow k \approx 0.44

Hence, the value is:

k \approx 0.44

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