I have been able to figure out question a, but I have no idea how to even start b and c.
A fast-food restaurant wants a special container to hold coffee. The restaurant wishes the container to quickly cool the coffee from 200° to 130° F and keep the liquid between 110° and 130° F as long as possible . The restaurant has 3 containers to select from.
1. The CentiKeeper Company has a container that reduces the temperature of a liquid from 200° to 100° F in 30 minutes by maintaining a constant temperature of 70° F.
2. The TempControl Company has a container that reduces the temperature of a liquid from 200° to 110° F in 25 minutes by maintaining a constant temperature of 60° F.
3. The Hot’n’Cold Company has a container that reduces the temperature of a liquid from 200° to 120° in 20 minutes by maintaining a constant temperature of 65° F.
a. Use Newton’s Law of Cooling to find a function relating the temperature of the liquid over time for each container.
1. y-70=Cekt y=70+130ekt
200-70=Cek*0 100=70+130ek30
C=130 30=130ek30
30/130=ek30
Ln (30/130)=30k
Ln (30/130)/30=k
K=1/30 ln (30/130)=-0.048878
Y=70+130e^-0.048878t
2. y-60=Cekt y=60+140ekt
200-60=Cek*0 110=60+140ek25
C=140 50=140ek25
50/140=ek25
Ln (50/140)=25k
Ln (50/140) / 25=k
K= 1/25 ln (50/140)=-0.041185
Y=60+140e^-0.041185
3. y-65= Cekt y=65+135ekt
200-65= Cek*0 120=65+135ek20
C=135 55=135ek20
50/135=ek20
Ln (50/135)=20k
Ln (50/135) / 20= k
K=1/20 Ln (50/135)= -0.049663
Y=50+135e^-0.049663
b. How long does it take each container to lower the coffee temperature from 200° to 130° F?
c. How long will the coffee temperature remain between 110° and 130° F? This temperature is considered the optimal drinking temperature.
A fast-food restaurant wants a special container to hold coffee. The restaurant wishes the container to quickly cool the coffee from 200° to 130° F and keep the liquid between 110° and 130° F as long as possible . The restaurant has 3 containers to select from.
1. The CentiKeeper Company has a container that reduces the temperature of a liquid from 200° to 100° F in 30 minutes by maintaining a constant temperature of 70° F.
2. The TempControl Company has a container that reduces the temperature of a liquid from 200° to 110° F in 25 minutes by maintaining a constant temperature of 60° F.
3. The Hot’n’Cold Company has a container that reduces the temperature of a liquid from 200° to 120° in 20 minutes by maintaining a constant temperature of 65° F.
a. Use Newton’s Law of Cooling to find a function relating the temperature of the liquid over time for each container.
1. y-70=Cekt y=70+130ekt
200-70=Cek*0 100=70+130ek30
C=130 30=130ek30
30/130=ek30
Ln (30/130)=30k
Ln (30/130)/30=k
K=1/30 ln (30/130)=-0.048878
Y=70+130e^-0.048878t
2. y-60=Cekt y=60+140ekt
200-60=Cek*0 110=60+140ek25
C=140 50=140ek25
50/140=ek25
Ln (50/140)=25k
Ln (50/140) / 25=k
K= 1/25 ln (50/140)=-0.041185
Y=60+140e^-0.041185
3. y-65= Cekt y=65+135ekt
200-65= Cek*0 120=65+135ek20
C=135 55=135ek20
50/135=ek20
Ln (50/135)=20k
Ln (50/135) / 20= k
K=1/20 Ln (50/135)= -0.049663
Y=50+135e^-0.049663
b. How long does it take each container to lower the coffee temperature from 200° to 130° F?
c. How long will the coffee temperature remain between 110° and 130° F? This temperature is considered the optimal drinking temperature.
