Here’s the cleaned-up and formatted version:

Newton’s Law of Cooling

The formula:
$T(t) = T_r + (T_i - T_r) e^{kt}$

Where:

• $T(t)$: Temperature at time $t$
• $T_r$: Room temperature
• $T_i$: Initial temperature
• $k$: Cooling constant
• $t$: Time

Given:

• Room temperature ($T_r$) = $70^\circ$
• Initial temperature ($T_i$) = $180^\circ$
• At $t = 5$ minutes, $T(5) = 140^\circ$

Step 1: Determine the cooling constant $k$:

Using the formula:
$T(t) = T_r + (T_i - T_r) e^{kt}$

Substitute the known values:
$140 = 70 + (180 - 70) e^{5k}$

Simplify:
$140 = 70 + 110 e^{5k}$
$70 = 110 e^{5k}$
$\frac{70}{110} = e^{5k}$
$\frac{7}{11} = e^{5k}$

Take the natural logarithm ($\ln$) on both sides:
$\ln\left(\frac{7}{11}\right) = 5k$

Solve for $k$:
$k = \frac{\ln\left(\frac{7}{11}\right)}{5}$

Using a calculator:
$k \approx \frac{\ln(0.6364)}{5} \approx \frac{-0.452}{5} \approx -0.0904$

Thus:
$k \approx -0.0904$

Step 2a: Find the temperature after 10 minutes ($t = 10$):

Using the formula:
$T(t) = T_r + (T_i - T_r) e^{kt}$

Substitute the known values:
$T(10) = 70 + (180 - 70) e^{-0.0904 \cdot 10}$
$T(10) = 70 + 110 e^{-0.904}$

Using a calculator:
$e^{-0.904} \approx 0.405$
$T(10) = 70 + 110(0.405)$
$T(10) \approx 70 + 44.55 \approx 114.55^\circ$

Thus:
$T(10) \approx 114.6^\circ$

Step 2b: Find the time ($t$) for the temperature to cool to $95^\circ$:

Using the formula:
$T(t) = T_r + (T_i - T_r) e^{kt}$

Substitute the known values:
$95 = 70 + (180 - 70) e^{-0.0904 \cdot t}$
$95 = 70 + 110 e^{-0.0904 \cdot t}$
$25 = 110 e^{-0.0904 \cdot t}$
$\frac{25}{110} = e^{-0.0904 \cdot t}$
$\frac{5}{22} = e^{-0.0904 \cdot t}$

Take the natural logarithm ($\ln$) on both sides:
$\ln\left(\frac{5}{22}\right) = -0.0904 \cdot t$

Solve for $t$:
$t = \frac{\ln\left(\frac{5}{22}\right)}{-0.0904}$

Using a calculator:
$\ln\left(\frac{5}{22}\right) \approx \ln(0.2273) \approx -1.482$
$t \approx \frac{-1.482}{-0.0904} \approx 16.4 \, \text{minutes}$

Final Answers:

1. Temperature after 10 minutes: $114.6^\circ$
2. Time to cool to $95^\circ$: $16.4$ minutes

Let me know if you have any questions or need further clarification!