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
∴
- Subtract 69 from both sides
∴
- Divide both sides by 111
∴
- Insert ㏑ for both sides
∴
- 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
∵
∴
- Subtract 69 from both sides
∴
- Divide both sides by 111
∴
- Insert ㏑ for both sides
∴
- 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