Answer :
To solve this problem, we'll apply Newton's Law of Heating. According to this law, the rate of temperature change of an object is proportional to the difference between its current temperature and the surrounding temperature.
Here’s a step-by-step guide on how to find the value of [tex]\( k \)[/tex] and the temperature of the turkey at 3 p.m.:
### Step 1: Understand the Formula
The formula for Newton's Law of Heating is given by:
[tex]\[ T(t) = T_{\text{env}} + (T_{\text{initial}} - T_{\text{env}}) \times e^{-kt} \][/tex]
Where:
- [tex]\( T(t) \)[/tex] = Temperature of the turkey at time [tex]\( t \)[/tex]
- [tex]\( T_{\text{env}} \)[/tex] = Oven temperature, which is [tex]\( 325^\circ F \)[/tex]
- [tex]\( T_{\text{initial}} \)[/tex] = Initial turkey temperature, which is [tex]\( 68^\circ F \)[/tex]
- [tex]\( t \)[/tex] = Time in hours
- [tex]\( k \)[/tex] = Constant of proportionality
### Step 2: Use Given Data to Find [tex]\( k \)[/tex]
You know that after 2 hours, the turkey's temperature is [tex]\( 100^\circ F \)[/tex]. Plug these values into the formula:
[tex]\[ 100 = 325 + (68 - 325) \times e^{-k \times 2} \][/tex]
### Step 3: Calculate [tex]\( k \)[/tex]
Solve for [tex]\( k \)[/tex] in the equation above to determine the rate of heating.
The calculated value of [tex]\( k \)[/tex] is approximately [tex]\( 0.0665 \)[/tex].
### Step 4: Find the Turkey's Temperature at 3 p.m.
3 p.m. is 7 hours after 8 a.m., so [tex]\( t = 7 \)[/tex].
Using the formula:
[tex]\[ T(7) = 325 + (68 - 325) \times e^{-0.0665 \times 7} \][/tex]
### Step 5: Calculate the Temperature
The temperature of the turkey at 3 p.m. is approximately [tex]\( 163.64^\circ F \)[/tex].
So, the turkey reaches a temperature of about [tex]\( 163.64^\circ F \)[/tex] by 3 p.m.
Here’s a step-by-step guide on how to find the value of [tex]\( k \)[/tex] and the temperature of the turkey at 3 p.m.:
### Step 1: Understand the Formula
The formula for Newton's Law of Heating is given by:
[tex]\[ T(t) = T_{\text{env}} + (T_{\text{initial}} - T_{\text{env}}) \times e^{-kt} \][/tex]
Where:
- [tex]\( T(t) \)[/tex] = Temperature of the turkey at time [tex]\( t \)[/tex]
- [tex]\( T_{\text{env}} \)[/tex] = Oven temperature, which is [tex]\( 325^\circ F \)[/tex]
- [tex]\( T_{\text{initial}} \)[/tex] = Initial turkey temperature, which is [tex]\( 68^\circ F \)[/tex]
- [tex]\( t \)[/tex] = Time in hours
- [tex]\( k \)[/tex] = Constant of proportionality
### Step 2: Use Given Data to Find [tex]\( k \)[/tex]
You know that after 2 hours, the turkey's temperature is [tex]\( 100^\circ F \)[/tex]. Plug these values into the formula:
[tex]\[ 100 = 325 + (68 - 325) \times e^{-k \times 2} \][/tex]
### Step 3: Calculate [tex]\( k \)[/tex]
Solve for [tex]\( k \)[/tex] in the equation above to determine the rate of heating.
The calculated value of [tex]\( k \)[/tex] is approximately [tex]\( 0.0665 \)[/tex].
### Step 4: Find the Turkey's Temperature at 3 p.m.
3 p.m. is 7 hours after 8 a.m., so [tex]\( t = 7 \)[/tex].
Using the formula:
[tex]\[ T(7) = 325 + (68 - 325) \times e^{-0.0665 \times 7} \][/tex]
### Step 5: Calculate the Temperature
The temperature of the turkey at 3 p.m. is approximately [tex]\( 163.64^\circ F \)[/tex].
So, the turkey reaches a temperature of about [tex]\( 163.64^\circ F \)[/tex] by 3 p.m.
