I know that I have to use log to solve, but I am having a hard time with the word problems. Please help
Coffee is best enjoyed at a temperature of 121° F. A restaurant owner wants to discover the temperature T at which he should serve his coffee so that it will have cooled to this ideal temperature in 6 minutes. He discovers that a cup of coffee served at 198° F cools to 185° F in one minute when his restaurant is at 70° F. If he maintains the restaurant temperature at 70° F, at what temperature should he serve the coffee to meet his goal
To solve this word problem, we can use the concept of exponential decay. Let's break down the problem and figure out the steps to find the answer.
Step 1: Define your variables:
- T: The temperature at which the coffee should be served so that it cools to the ideal temperature of 121° F.
- t: The time it takes for the coffee to cool to the ideal temperature of 121° F.
- C: The starting temperature of the coffee.
- r: The rate at which the coffee cools per minute.
Step 2: Identify the given information:
- C (starting temperature of the coffee) = 198° F.
- The coffee cools to 185° F in one minute.
- The desired time for the coffee to cool to the ideal temperature is 6 minutes.
- The restaurant temperature is maintained at 70° F.
Step 3: Determine the rate of cooling, r:
To find the rate of cooling, we need to calculate the difference between the current temperature and the temperature after one minute. From the given information, the coffee cools from 198° F to 185° F in one minute. Therefore, the rate of cooling, r, is 198° F - 185° F = 13° F per minute.
Step 4: Set up the exponential decay equation:
We can use the formula for exponential decay: C(t) = C * e^(-rt), where C(t) is the temperature of the coffee at time t, C is the starting temperature, r is the rate of decay, and e is Euler's number (approximately 2.71828).
Since we want to find the temperature at which the coffee should be served (T), we can rewrite the equation as: T = C * e^(-rt).
Step 5: Solve for T:
Using the given information:
- C = 198° F
- r = 13° F per minute
- t = 6 minutes
Plugging these values into the equation, we get:
T = 198 * e^(-13 * 6)
Step 6: Use a calculator to solve for T:
Using a calculator, multiply -13 and 6 to get -78. Then calculate e^(-78) to find the value of e raised to the power of -78. Finally, multiply the result by 198 to find the final value of T.
Please note that the exact value of T will depend on the accuracy of the calculator used.