5. Ared clav not was fired in a kiln to a temperatures of 2,192^circ F the kiln was then turned off and the clay pot allowed to cool slowly to a room temperature of 72^circ F before removing the pot from the kiln and painting it.It cools at a rate of 15% per hour.

The question likely asks how long it takes for the clay pot to cool down to a safe temperature for painting. While the question doesn't specify a precise "safe" temperature, we can assume it's close to room temperature (72°F). We can model the cooling process using an exponential decay formula.<br /><br />**1. Temperature Difference:**<br /><br />The initial temperature difference between the pot and the room is:<br /><br />2192°F - 72°F = 2120°F<br /><br />**2. Exponential Decay Formula:**<br /><br />The formula for exponential decay is:<br /><br />T(t) = T₀ * (1 - r)^t<br /><br />Where:<br /><br />* T(t) is the temperature at time t<br />* T₀ is the initial temperature difference<br />* r is the rate of decay (as a decimal)<br />* t is the time in hours<br /><br />**3. Applying the Formula:**<br /><br />In our case:<br /><br />* T₀ = 2120°F<br />* r = 0.15 (15% expressed as a decimal)<br /><br />We want to find t when T(t) is close to 0 (meaning the pot's temperature is close to room temperature). Let's say we want the temperature difference to be less than 5°F, meaning the pot's temperature is between 67°F and 77°F. So, we'll solve for t when T(t) = 5°F:<br /><br />5 = 2120 * (1 - 0.15)^t<br />5 = 2120 * (0.85)^t<br />5/2120 = (0.85)^t<br />0.002358 ≈ (0.85)^t<br /><br />**4. Solving for t (using logarithms):**<br /><br />To solve for t, we can take the logarithm of both sides. Using the natural logarithm (ln):<br /><br />ln(0.002358) = ln((0.85)^t)<br />ln(0.002358) = t * ln(0.85)<br />t = ln(0.002358) / ln(0.85)<br />t ≈ 25.5 hours<br /><br />**Answer:**<br /><br />It will take approximately 25.5 hours for the clay pot to cool down to a temperature close enough to room temperature for painting. It's important to note that this is an approximation. In reality, the cooling rate might not be perfectly constant at 15% per hour.<br />