Key Concepts and Formulas

• Newton's Law of Cooling: $\frac{dT}{dt} = -K(T - T_a)$, where $T(t)$ is the temperature at time $t$, $T_a$ is the ambient temperature, and $K$ is a positive constant.
• Solution to a Separable Differential Equation: If $\frac{dy}{dx} = f(x)g(y)$, then $\int \frac{dy}{g(y)} = \int f(x) dx$.
• Exponential Decay: Solutions to equations of the form $\frac{dy}{dt} = -ky$ are of the form $y(t) = y_0e^{-kt}$, where $y_0$ is the initial value.

Step-by-Step Solution

Step 1: Separate variables and integrate the differential equation.

We are given the differential equation $\frac{dT}{dt} = -K(T - 80)$. We separate the variables to get $\frac{dT}{T - 80} = -K dt$.

Integrating both sides, we have $\int \frac{dT}{T - 80} = \int -K dt$. This gives us $\ln|T - 80| = -Kt + C$, where $C$ is the constant of integration.

Step 2: Solve for T(t).

Exponentiating both sides, we get $|T - 80| = e^{-Kt + C} = e^C e^{-Kt}$. Since $e^C$ is also a constant, we can write $T - 80 = Ae^{-Kt}$, where $A = \pm e^C$ is a constant. Therefore, $T(t) = 80 + Ae^{-Kt}$.

Step 3: Use the initial condition T(0) = 160 to find A.

We are given that $T(0) = 160$. Substituting $t = 0$ into the equation $T(t) = 80 + Ae^{-Kt}$, we get $160 = 80 + Ae^{-K(0)} = 80 + A$. Thus, $A = 160 - 80 = 80$. So, $T(t) = 80 + 80e^{-Kt}$.

Step 4: Use the condition T(15) = 120 to find K.

We are given that $T(15) = 120$. Substituting $t = 15$ into the equation $T(t) = 80 + 80e^{-Kt}$, we get $120 = 80 + 80e^{-15K}$. Subtracting 80 from both sides gives $40 = 80e^{-15K}$. Dividing by 80 gives $\frac{1}{2} = e^{-15K}$. Taking the natural logarithm of both sides gives $\ln(\frac{1}{2}) = -15K$, so $-\ln 2 = -15K$, which means $K = \frac{\ln 2}{15}$.

Step 5: Find T(45).

We want to find $T(45)$. We have $T(t) = 80 + 80e^{-Kt}$, and we found $K = \frac{\ln 2}{15}$. So, $T(45) = 80 + 80e^{-(\frac{\ln 2}{15})(45)} = 80 + 80e^{-3\ln 2} = 80 + 80e^{\ln(2^{-3})} = 80 + 80(2^{-3}) = 80 + 80(\frac{1}{8}) = 80 + 10 = 90$.

Common Mistakes & Tips

• Sign Errors: Be careful with the negative sign in Newton's Law of Cooling.
• Constant of Integration: Don't forget the constant of integration when integrating.
• Simplifying Exponentials and Logarithms: Remember the properties of exponents and logarithms to simplify expressions.

Summary

We solved the differential equation $\frac{dT}{dt} = -K(T - 80)$ using separation of variables and integration. We then used the initial condition $T(0) = 160$ to find the constant $A$ and the condition $T(15) = 120$ to find the constant $K$. Finally, we used the equation $T(t) = 80 + 80e^{-Kt}$ with the calculated value of $K$ to find $T(45) = 90$.

Final Answer

The final answer is \boxed{90}, which corresponds to option (A).