a person knows that to get well - done beef, it should be brought to a temperature of 170°f. he placed a…

a person knows that to get well - done beef, it should be brought to a temperature of 170°f. he placed a sirloin tip roast with a temperature of 40°f in an oven with a temperature of 329°f and after 4 hours the temperature of the roast was 139°f. how much longer must the roast be in the oven to get it well done? if the oven temperature is set at 170°f, how long will it take to get the roast well done? hint: the difference between the roast temperature and the oven temperature decreases exponentially. how much longer will it take to get the roast well done with an oven temperature of 329°? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. a. the roast must be in the oven hour minutes longer. (round to the nearest whole number as needed.) b. the roast will never get well done.

a person knows that to get well - done beef, it should be brought to a temperature of 170°f. he placed a sirloin tip roast with a temperature of 40°f in an oven with a temperature of 329°f and after 4 hours the temperature of the roast was 139°f. how much longer must the roast be in the oven to get it well done? if the oven temperature is set at 170°f, how long will it take to get the roast well done? hint: the difference between the roast temperature and the oven temperature decreases exponentially. how much longer will it take to get the roast well done with an oven temperature of 329°? select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. a. the roast must be in the oven hour minutes longer. (round to the nearest whole number as needed.) b. the roast will never get well done.

Answer

Explanation:

Step1: Set up the exponential decay model

Let (T(t)) be the temperature of the roast at time (t) (in hours). The general form of the exponential - decay model for the temperature difference is (T(t)=T_{oven}-(T_{oven}-T_{0})e^{-kt}), where (T_{0}) is the initial temperature of the roast, (T_{oven}) is the oven temperature, and (k) is the decay constant.

Given (T_{0} = 40^{\circ}F), (T_{oven}=329^{\circ}F), and when (t = 4), (T(4)=139^{\circ}F).

Substitute these values into the formula: (139 = 329-(329 - 40)e^{-4k}).

First, simplify the equation: ((329 - 40)e^{-4k}=329 - 139).

(289e^{-4k}=190).

(e^{-4k}=\frac{190}{289}).

Take the natural logarithm of both sides: (-4k=\ln(\frac{190}{289})).

(k=-\frac{1}{4}\ln(\frac{190}{289})\approx-\frac{1}{4}\times(-0.439)\approx0.10975).

Step2: Find the time (t) when (T(t) = 170^{\circ}F)

Substitute (T(t)=170), (T_{0} = 40), and (T_{oven}=329) into (T(t)=T_{oven}-(T_{oven}-T_{0})e^{-kt}).

(170 = 329-(329 - 40)e^{-0.10975t}).

((329 - 40)e^{-0.10975t}=329 - 170).

(289e^{-0.10975t}=159).

(e^{-0.10975t}=\frac{159}{289}).

Take the natural logarithm of both sides: (-0.10975t=\ln(\frac{159}{289})).

(t=\frac{\ln(\frac{159}{289})}{- 0.10975}).

(\ln(\frac{159}{289})\approx-0.597), so (t=\frac{-0.597}{-0.10975}\approx5.44) hours.

Since it has already been in the oven for (4) hours, the additional time is (t - 4\approx5.44-4 = 1.44) hours.

(0.44) hours (\times60) minutes/hour (=26.4) minutes.

Answer:

A. The roast must be in the oven (1) hour (26) minutes longer.