Answer:
a. The exponential model is
b. It takes the cake 9.32 minutes to cool to the desired temperature
Step-by-step explanation:
Let us solve it by using the Newton's Law of cooling
, where
- T(t) is the temperature at any given time
- C is the surrounding temperature
is the initial temperature of the heated object- k is a negative constant
- t is the time
∵ A cake recipe says to bake the cake until the center is 180 °F
∴
= 180 ⇒ initial temperature
∵ The room temperature is 69 °F
∴ C = 69 ⇒ surrounding temperature
- Use the table to substitute t and T to find the constant k
∵ At t = 5 minutes, T = 125 °F
∴ ![125=69+(180-69)e^{5k}](https://tex.z-dn.net/?f=125%3D69%2B%28180-69%29e%5E%7B5k%7D)
- Subtract 69 from both sides
∴ ![56=(111)e^{5t}](https://tex.z-dn.net/?f=56%3D%28111%29e%5E%7B5t%7D)
- Divide both sides by 111
∴ ![\frac{56}{111}=e^{5k}](https://tex.z-dn.net/?f=%5Cfrac%7B56%7D%7B111%7D%3De%5E%7B5k%7D)
- Insert ㏑ for both sides
∴ ![ln(\frac{56}{111})=5k](https://tex.z-dn.net/?f=ln%28%5Cfrac%7B56%7D%7B111%7D%29%3D5k)
- Divide both sides by 5
∴ - 0.1368357021 = k
∴
a. The exponential model is
∵ The cake cool to 100 °F
∴ The desired temperature is 100°
∴ T(t) = 100 ⇒ cake's temperature at t minute
- Use the model above to find t
∵
∴ ![100=69+(111)e^{-0.1368357021t}](https://tex.z-dn.net/?f=100%3D69%2B%28111%29e%5E%7B-0.1368357021t%7D)
- Subtract 69 from both sides
∴ ![31=(111)e^{-0.1368357021t}](https://tex.z-dn.net/?f=31%3D%28111%29e%5E%7B-0.1368357021t%7D)
- Divide both sides by 111
∴ ![\frac{31}{111} =e^{-0.1368357021t}](https://tex.z-dn.net/?f=%5Cfrac%7B31%7D%7B111%7D%20%3De%5E%7B-0.1368357021t%7D)
- Insert ㏑ for both sides
∴ ![ln(\frac{31}{111})=-0.1368357021t](https://tex.z-dn.net/?f=ln%28%5Cfrac%7B31%7D%7B111%7D%29%3D-0.1368357021t)
- Divide both sides by - 0.136835702
∴ 9.321711938 = t
∴ t ≅ 9.32 minutes
b. It takes the cake 9.32 minutes to cool to the desired temperature