A thermometer reading 7 degrees C is brought into a room with a constant temperature of 29 degrees C.

If the thermometer reads 15 degrees C after 4 minutes, what will it read in 6 minutes? 11 minutes?

So I guess I need to put into newton's law of cooling:
u(t) = T + (u sub zero - T ) e^kt

How do I solve for k so can plug in my T's. and what is u sub zero? 7 degrees?

To solve for the constant k in Newton's Law of Cooling and find the value of u sub zero, we will need to use the given information.

Newton's Law of Cooling states that the rate of change of temperature of an object is proportional to the difference between the object's temperature and the ambient temperature. Mathematically, it can be written as:

u(t) = T + (u sub zero - T) * e^(k * t)

Where:
- u(t) is the temperature of the object at time t.
- T is the ambient temperature of the surrounding environment.
- u sub zero is the initial temperature of the object at t = 0.
- e is the base of the natural logarithm.
- k is the cooling constant we need to determine.

Given information:
- The initial temperature of the object (u sub zero) is 7 degrees Celsius.
- The ambient temperature (T) is 29 degrees Celsius.
- After 4 minutes, the thermometer reads 15 degrees Celsius.

Now, let's solve for k and find u(t) for 6 minutes and 11 minutes.

Step 1: Substitute the known values into the equation using the initial temperature measurement:
15 = 29 + (7 - 29) * e^(4k)

Step 2: Simplify the equation:
15 = 29 - 22 * e^(4k)

Step 3: Rearrange the equation to isolate the exponential term:
22 * e^(4k) = 29 - 15
22 * e^(4k) = 14

Step 4: Divide both sides by 22:
e^(4k) = 14 / 22
e^(4k) = 0.6364

Step 5: Take the natural logarithm of both sides:
ln(e^(4k)) = ln(0.6364)
4k = ln(0.6364)

Step 6: Solve for k:
k = ln(0.6364) / 4

Now that we have the value of k, we can substitute it back into the equation to find the temperature at 6 minutes and 11 minutes.

For t = 6 minutes:
u(6) = 29 + (7 - 29) * e^(k * 6)

For t = 11 minutes:
u(11) = 29 + (7 - 29) * e^(k * 11)

Plug in the value of k obtained in step 6 and evaluate u(6) and u(11) to find their respective temperatures.

well I put the law like this:

T(t) = Ts - (Ts-To)e^(-kt)
Ts = T surroundings = 29
To = initial Temp = 7

now put in at 4 min
T(4) = 15 = 29 - (29-7)e^-4k
15 = 29 - 22 e^(-4k)
-14 = 22 e^-4k
- .636363... = e^-4k
ln (-.636363 ... ) = -4k
-4k = -.452
k = .113
so you go on and put 6 minutes in for t