Question

A pizza pan is removed at 7:00 PM from an oven whose temperature is fixed at 450°F into a room that is a constant 71°F. After 5 minutes, the pizza pan is at (a) At what time is the temperature of the pan 130°F? (b) Determine the time that needs to elapse before the pan is 230° (c) What do you notice about the temperature as time passes? (a) At what time is the temperature of the pan 130°F? PM (Do not round until the final answer. Then round to the nearest minute as needed.) (b) What is the time that needs to elapse before the pan is 230°F? minutes (Do not round until the final answer. Then round to the nearest minute as needed.) (c) What do you notice about the temperature as time passes? The temperature of the pan is approaching 0°F The temperature of the pan is approaching 71°F. The temnerate of the nani anmachine 379°F

Answer 1

Using Newton's law of cooling,

T(t) = Troom - Troom + (To - Troome-kt

T₀ = initial temperature = 450^oF

T_room = room/environment temperature = 71^oF

k = cooling rate

After 5 minutes, T(t) = 300 ^oF

$\small 300=71+(450-71)e^{-k\times 5}$

$\small \frac{300-71}{450-71}=e^{-k\times 5}$

$\small \frac{229}{379}=e^{-k\times 5}$

1.6 = e-kx5

$\small ln(0.6)=ln(e^{-k\times 5})$

$\small -0.5108=-k\times 5$

$\small k=0.10216$ min^-1

a)

Temperature of pan = 130^oF

$\small 130=71+(450-71)e^{-0.10216t}$

$\small \frac{130-71}{450-71}=e^{-0.10216t}$

$\small \frac{59}{379}=e^{-0.10216t}$

$\small 0.15567=e^{-0.10216t}$

$\small ln(0.15567)=ln(e^{-0.10216t})$

$\small -1.86=-0.10216t$

$\small t=18$

At 7:18 PM, temperature of pan is 130^oF.

b)

T(t) = 230^oF

$\small 230=71+(450-71)e^{-0.10216t}$

$\small \frac{230-71}{450-71}=e^{-0.10216t}$

$\small \frac{159}{379}=e^{-0.10216t}$

$\small 0.42=e^{-0.10216t}$

$\small ln(0.42)=ln(e^{-0.10216t})$

$\small -0.8675=-0.10216t$

$\small t=8$

The time that needs to elapse before the pan is 230^oF is 8 minutes.

c)

As time tends to $\small \infty$ , $\small e^{-kt}$ tends to 1.

$\small T(t)=T_{room}+(T_{0}-T_{room})=T_{0}$

As time passes, temperature of pan is approaching 71^oF.

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