Answer:
b) 35 °F
c) 3.5 minutes
Step-by-step explanation:
The temperature of the liquid is modeled using the following equation:
[tex]T(m)=74-39(0.7)^{m}[/tex]
Here, m is the time in minutes and T(m) represents the temperature in Fahrenheit of the liquid after m minutes.
b) Initial Temperature of the Liquid.
When the liquid is just taken out of the refrigerator the time m is equal to 0. So, substituting m = 0 in given equation will give us the Initial temperature of the Liquid.
[tex]T(0)=74-39(0.87)^{0}\\\\ T(0)=35[/tex]
This means, the initial temperature of the liquid was 35 °F.
c) Time taken to reach 50 °F
In order to find the time, in minutes, it will take to reach 50°F, we replace T(m) by 50 and find the corresponding value of m.
[tex]50=74-39(0.87)^{m}\\\\ 39(0.87)^{m}=74-50\\\\ 39(0.87)^{m}=24\\\\ (0.87)^{m}=\frac{24}{39}[/tex]
Taking log of both sides, we get:
[tex]log (0.87)^{m}=log(\frac{24}{39})\\\\ m \times log(0.87)=log(\frac{24}{39})\\\\m=log(\frac{24}{39}) \times \frac{1}{log(0.87)}\\\\ m=3.5[/tex]
Thus, it will take 3.5 minutes to reach 50°F.
(A) In the given equation, m is the time in minutes and T(m) represents the temperature in Fahrenheit of the liquid.
(b) The initial value of temperature is 35 Degrees Fahrenheit.
(c) The required time at the nearest tenth, to take for the temperature to reach 50°F is 3.50 minutes.
(A)
The equation of temperature model is, [tex]T(m)=74-39(0.87)^{m}[/tex].
Here m is the time in minutes and T(m) represents the temperature in Fahrenheit of the liquid after m minutes.
(b)
When the liquid is just taken out of the refrigerator the time m is equal to 0. So, substituting m = 0 in given equation will give us the Initial temperature of the Liquid. Then the equation becomes,
[tex]T(0) = 74 -39(0.87)^{0}\\T(0)= 74-39\\T(0)=35 \;\rm ^{\circ}F[/tex]
Thus, we can conclude that the initial value of temperature is 35 Degrees Fahrenheit.
(c)
In order to find the time, in minutes, it will take to reach 50°F, we replace T(m) by 50 and find the corresponding value of m.
[tex]T(m)=74-39(0.87)^{m}\\\\50=74-39(0.87)^{m}\\\\39(0.87)^{m}=74-50\\\\ln(0.87)^{m}=ln((74-50)/39)\\\\m=\dfrac{ln(0.615)}{ln(0.87)} \\\\m=3.50 \;\rm min[/tex]
Thus, we can conclude that the required time at the nearest tenth, to take for the temperature to reach 50°F is 3.50 minutes.
Learn more about the linear model here:
the nearest tenth, does it take for the temperature to reach 50°F
