Please help :)

A potato is put into an oven that has been heated to 350F. Its temperature as a function of time is given by T(t) = a(1 - e-kt) + b. The potato was 50F when it was first put into the oven.

What is the value of b in this context? Explain.

What is the value of a in this context? Explain.

Thanks :D

e^-kt -> 0 as x->∞

1-e^-kt -> 1

So, T(t) -> a+b

at t=0, T(t) = b

So, b is the initial temperature of the potato, 50°F

a is the initial temperature difference between the potato and the oven: 300 in this case. You know this because it is clear that as time passes, the temperature of the spud approaches the temp of the oven.

To determine the value of b in this context, let's consider the given information: the potato was 50°F when it was first put into the oven.

In the equation T(t) = a(1 - e^(-kt)) + b, T(t) represents the temperature of the potato at time t. Since the potato's initial temperature is 50°F, we can substitute these values into the equation:

T(0) = a(1 - e^(-k*0)) + b
50 = a(1 - e^0) + b
50 = a(1 - 1) + b
50 = a + b

Therefore, the value of b is 50°F.

Now, let's determine the value of a in this context. The value of a represents the difference between the maximum temperature the potato will reach and the initial temperature.

Since the equation T(t) represents the temperature of the potato as a function of time, we can rewrite it as T(t) = a(1 - e^(-kt)) + 50. The maximum temperature the potato will reach is represented by a.

So, in this context, the value of a represents the difference between the maximum temperature and the initial temperature (50°F).

Hence, the value of a is the maximum temperature the potato will reach in the oven above its initial temperature.

To find the value of b in this context, we need to use the information given and the formula for T(t). We know that the potato was 50F when it was first put into the oven. This initial temperature corresponds to the time t = 0.

Substituting t = 0 into the formula T(t), we get:

T(0) = a(1 - e^(0*k)) + b

Since e^0 is equal to 1, the equation simplifies to:

T(0) = a(1 - 1) + b

T(0) = 0 + b

Therefore, T(0) = b.

This means that the value of b in this context is equal to the initial temperature of the potato, which is 50F.

Now, let's move on to finding the value of a.

The formula for T(t) gives the temperature of the potato as a function of time. It's in the form of T(t) = a(1 - e^(-kt)) + b.

To find the value of a, we need to use the given information and the equation. Unfortunately, the information provided does not directly give us the value of a.

However, if we have additional information such as the temperature of the potato at a certain time, we can use that to solve for a.

Without any additional information, we can make an assumption based on the formula. The term (1 - e^(-kt)) represents how the temperature changes over time. The constant "a" can be considered as a scaling factor, determining the overall shape and magnitude of the temperature curve.

Typically, "a" would be determined by experimental data or additional information specific to the situation. However, if no additional information is given, we can assume a value of 1 for "a" as a starting point.

So, in this context, the value of "a" can be assumed as 1, but its actual value will depend on the specifics of the situation or experimental data if available.

I hope this explanation helps! Let me know if you have any further questions.