Please help me on this calculus question. I am getting the wrong diff. equation and so my answers to the remaining parts are wrong.
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A thermometer reading 85 degress outside is brought into a room with a constant temperature of 72 degrees. The rate of change, dT/dt is directly proportional to the difference between the temperatures.
A. Write the differential equation
B. Solve the diff. equation that you wrote subject to the fact that when t=0, the thermometer was 85 degrees
C. In 2 minutes the temperature is 83 degrees. Find K (proportionality constant)
D. How long till the temperature drops to 75 degrees
Sure! Let's break down each part of the problem step by step.
A. To write the differential equation, we need to express the rate of change of temperature, dT/dt, in terms of the temperature difference between the thermometer and the room. Since the rate of change is directly proportional to this temperature difference, we can write:
dT/dt = k(T - R),
where T is the thermometer temperature, R is the room temperature, and k is the proportionality constant.
B. To solve the differential equation, we need to separate the variables and integrate. Rearranging the equation from part A:
dT / (T - R) = k dt.
Integrating both sides:
ln|T - R| = kt + C,
where C is the constant of integration.
Exponentiating both sides:
T - R = e^(kt+C).
Using the fact that when t = 0, the thermometer was at 85 degrees (T = 85):
85 - R = e^C.
Combining these terms:
T - R = e^(kt) * e^C.
Now, let's define A as the constant e^C:
T - R = Ae^(kt).
C. In 2 minutes, the temperature is 83 degrees (T = 83). We can plug these values into the equation:
83 - 72 = Ae^(2k).
Simplifying:
11 = Ae^(2k).
To find the value of k, we need more information or data.
D. To find out how long it takes for the temperature to drop to 75 degrees, we set T = 75 in the general solution:
75 - 72 = Ae^(kt).
Simplifying:
3 = Ae^(kt).
To find the time, we need to know the value of the constant A or k.
To solve this calculus question, let's break it down step by step:
A. Write the differential equation:
The rate of change of temperature, dT/dt, is directly proportional to the difference between the temperatures. Let's denote the room temperature as T_room and the thermometer reading as T. We can write the differential equation as follows:
dT/dt = -k(T - T_room) (Note the negative sign since the temperature is decreasing)
B. Solve the differential equation subject to the fact that when t=0, the thermometer was 85 degrees:
To solve this first-order linear ordinary differential equation, we can separate the variables and then integrate both sides. Rearranging the equation, we have:
dT/(T - T_room) = -k * dt
Now, let's integrate both sides with respect to their respective variables:
∫ dT / (T - T_room) = -k ∫ dt
Integrating the left side using the natural logarithm:
ln(T - T_room) = -kt + C
Exponentiating both sides:
T - T_room = e^(-kt + C)
Since the thermometer was 85 degrees when t = 0, we can substitute these values into the equation and solve for C:
85 - 72 = e^(0 + C)
13 = e^C
Therefore, the equation becomes:
T - T_room = 13 * e^(-kt)
C. In 2 minutes, the temperature is 83 degrees. Find K (proportionality constant):
Given that in 2 minutes the temperature is 83 degrees and T_room is 72 degrees, we can substitute these values into the equation and solve for K:
83 - 72 = 13 * e^(-2k)
11 = 13 * e^(-2k)
e^(-2k) = 11/13
Taking the natural logarithm of both sides:
-2k = ln(11/13)
Dividing by -2:
k = ln(11/13) / -2
Therefore, K is equal to ln(11/13) divided by -2.
D. How long until the temperature drops to 75 degrees:
To find the time it takes for the temperature to drop to 75 degrees, we can substitute this value into the equation and solve for t:
75 - 72 = 13 * e^(-kt)
3 = 13 * e^(-kt)
e^(-kt) = 3/13
Taking the natural logarithm of both sides:
-kt = ln(3/13)
Solving for t:
t = ln(3/13) / -k
Substituting the value of k, which we derived from part C, we can calculate the value for t.
Note: Make sure to evaluate the natural logarithms using an appropriate calculator or software.