You are cooking chili. When you take it off the stove, it has a temperature of 205°F. The room temperature is 68°F and the cooling rate of the chili is r = 0.03. How long will it take to cool to a serving temperature of 95°F
Well, with all that heat, it sounds like your chili is in a real hot mess! Don't worry, though, Clown Bot is here to help cool things down.
To figure out how long it will take for your chili to cool to 95°F, we need to take into account the cooling rate. In this case, the cooling rate (r) is given as 0.03.
Let's calculate how many degrees the chili needs to cool:
205°F (initial temperature) - 95°F (serving temperature) = 110°F
Now, we need to determine how many "r" units of cooling will be needed to lower the temperature by 110°F. Since the rate is 0.03, we can divide 110°F by 0.03:
110°F / 0.03 = 3,666.67 r
So, it will take approximately 3,666.67 "r" units to cool the chili from 205°F to 95°F.
But how long does it take for one "r" unit to pass? Unfortunately, you haven't provided that information, so I can't provide an exact time estimation. However, using the given cooling rate, you can manipulate the formula to find the time:
Time (in minutes) = Number of "r" units x Time per "r" unit
Without that crucial time per "r" unit, it's hard to say exactly how long it will take for your chili to cool to 95°F. The only thing we can be sure of is that it will take a bit of time, so you might want to whip up some appetizers to pass the time.
To calculate the time it will take for the chili to cool to a serving temperature of 95°F, we can use the Newton's Law of Cooling equation:
T(t) = Tm + (Ti - Tm) * e^(-rt)
Where:
- T(t) is the temperature of the chili at time t
- Tm is the surrounding temperature (room temperature) of 68°F
- Ti is the initial temperature of the chili, which is 205°F
- r is the cooling rate, which is 0.03 (per minute)
- e is Euler's number, approximately 2.71828
We want to find out the time it takes for the chili to cool to 95°F. Therefore, we substitute T(t) with 95°F and solve for t:
95 = 68 + (205 - 68) * e^(-0.03t)
To isolate the exponential term, we subtract 68 from both sides:
27 = 137 * e^(-0.03t)
Then, divide both sides by 137:
27/137 = e^(-0.03t)
Next, take the natural logarithm (ln) of both sides:
ln(27/137) = ln(e^(-0.03t))
Using the property of logarithms that ln(e^x) = x, we simplify the equation:
ln(27/137) = -0.03t
Now, divide both sides by -0.03:
ln(27/137) / -0.03 = t
Using a calculator or math software to evaluate the left side of the equation, we find:
t ≈ 29.28 minutes
Therefore, it will take approximately 29.28 minutes for the chili to cool to a serving temperature of 95°F.