I don't know where to start on this one.
A thermometer is taken from an inside room to the outside, where the air temperature is ten degrees Fahrenheit. After one minute the thermometer reads 40 degrees Fahrenheit, and after two minutes it reads twenty-five degrees Fahrenheit. What is the initial temperature of the inside room?
Start with Newton's Law of cooling. Using your numbers, with To the initial temperature,
T(t) = 10 + (To-10)e^(-kt)
Now you have
T(1) = 10 + (To-10)e^-k = 40
T(2) = 10 + (To-10)e^-2k = 25
(To-10)e^-k = 30
(To-10)e^-2k = 15
(To-10)^2 e^-2k = 900
(To-10) e^-2k = 25
Now divide and the e^-2k stuff goes away, leaving
(To-10) = 900/25 = 36
To = 46
To solve this problem, we need to analyze the information given and determine the relationship between the time and temperature.
Step 1: Analyze the temperature readings over time.
According to the information provided, the temperature recorded by the thermometer increased from 10 degrees Fahrenheit to 40 degrees Fahrenheit in one minute. Then, it decreased from 40 degrees Fahrenheit to 25 degrees Fahrenheit in the following minute.
Step 2: Determine the slope of the temperature change.
To determine the relationship between time and temperature, we can calculate the change in temperature divided by the change in time. For the first minute, the change in temperature is 40 - 10 = 30 degrees Fahrenheit. The change in time is 1 minute. Therefore, the slope for the first minute is 30/1 = 30 degrees Fahrenheit per minute.
Step 3: Determine the temperature change per minute.
Next, we need to determine the temperature change per minute for the second reading. From the second minute to the third minute, the temperature decreased from 40 to 25 degrees Fahrenheit. The change in temperature is 25 - 40 = -15 degrees Fahrenheit. The change in time is 1 minute. Therefore, the slope for the second minute is -15/1 = -15 degrees Fahrenheit per minute.
Step 4: Determine the initial temperature.
Now that we have the slopes for each minute, we can determine the initial temperature of the inside room. Since the temperature increased in the first minute and then decreased in the second minute, we can infer that the initial temperature was between 10 and 40 degrees Fahrenheit.
Step 5: Calculate the initial temperature range.
To calculate the initial temperature range, you can take the average of the two slopes. Adding the two slopes (30 + (-15)) gives us 15 degrees Fahrenheit. Dividing this sum by 2 gives us 15/2 = 7.5 degrees Fahrenheit.
Step 6: Determine the initial temperature.
To determine the initial temperature of the inside room, subtract the average temperature change per minute (7.5) from the final temperature of the second minute (25). This gives us 25 - 7.5 = 17.5 degrees Fahrenheit.
Therefore, the initial temperature of the inside room is 17.5 degrees Fahrenheit.
To find the initial temperature of the inside room, we need to analyze the given information and use a mathematical approach.
Let's assign variables to the temperatures involved.
Let T be the initial temperature of the inside room.
Let A be the air temperature outside at ten degrees Fahrenheit.
Let T1 be the temperature of the thermometer after one minute (40 degrees Fahrenheit).
Let T2 be the temperature of the thermometer after two minutes (25 degrees Fahrenheit).
The key observation here is that the temperature change of the thermometer is proportional to the difference between the current temperature and the air temperature.
We can represent this relationship with a simple linear equation:
Temperature change ∝ (Current temperature - Air temperature)
Using this equation, we can calculate the temperature change after one minute:
T1 - A = k * (T - A) (k is the proportionality constant)
Similarly, we can calculate the temperature change after two minutes:
T2 - A = k * (T1 - A)
To solve for T, we can rewrite the second equation in terms of T:
T2 - A = k * (T1 - A)
T2 - A = k * (T - A)
T2 - A = k * T - k * A
T2 = k * T - k * A + A
T2 = k * T - A * (k - 1)
Now, we have two equations with two unknowns (T1 and T2) that we can solve simultaneously:
T1 - A = k * (T - A) ---> Equation 1
T2 = k * T - A * (k - 1) ---> Equation 2
By rearranging Equation 1, we can solve it for k:
k = (T1 - A) / (T - A)
Now, substitute this value of k in Equation 2:
T2 = [(T1 - A) / (T - A)] * T - A * {[(T1 - A) / (T - A)] - 1}
Simplifying further:
T2 = (T1 - A) + T - A - T1 + A
T2 = T - T1
Finally, rearrange the equation to solve for T, the initial temperature of the inside room:
T = T2 + T1
Substituting the given values:
T = 25 + 40
T = 65 degrees Fahrenheit
Therefore, the initial temperature of the inside room is 65 degrees Fahrenheit.