91Ó°ÊÓ

The temperature function is crucial when examining how temperature changes over time, particularly when considering exponential growth and decay. In this exercise, the roast's temperature function is defined by the equation \[ R(t) = 325 - 280 e^{-0.005t} \]This equation lets us predict the roast's temperature, \( R \), at any given time \( t \), measured in minutes. In essence, it accounts for the oven's maximum temperature (325°F) and the initial rapid increase in temperature as the roast sits in the oven. The expression \[ 280 e^{-0.005t} \] represents how the initial temperature difference decays over time. By plugging different values of \( t \) into this function, one can determine how the roast's temperature changes as it bakes.

Functional notation is a mathematical way to represent functions and their evaluations. In this exercise, we use the functional notation \( R(t) \) to denote the roast's temperature after being placed in the oven for \( t \) minutes. This format helps to clearly express and compute how the roast's temperature changes over time. When a specific time is given, you can replace \( t \) with that value to calculate the roast's temperature at that exact moment. For instance, to express the temperature after 30 minutes, the functional notation is \( R(30) \). This indicates that the function should be evaluated at \( t = 30 \). Functional notation is therefore useful for describing the time-dependent relationship between variables in a concise manner.

The initial temperature of an object is the temperature before any external changes occur. In this case, it's the temperature of the roast when first removed from the refrigerator. To find this in our exercise, we set \( t = 0 \) in the temperature function:\[ R(0) = 325 - 280 e^{0} \]Since \( e^{0} = 1 \), the equation simplifies to: \[ R(0) = 325 - 280 = 45 \]As such, the initial temperature of the roast is 45 degrees Fahrenheit. This value provides a baseline, marking the starting point for any subsequent temperature changes as the roast heats up in the oven.

Evaluating an exponential function involves substituting specific values into the function and performing calculations to obtain a numerical result. In the given equation, \[ R(t) = 325 - 280 e^{-0.005t} \]we replace \( t \) with numbers that correspond to specific instances, such as 10, 30, or 60 minutes. After substituting, it's necessary to calculate the value of the exponential expression, \[ e^{-0.005t} \]. This requires understanding of properties of exponential functions and might involve using a calculator to find accurate approximations, like \( e^{-0.15} \approx 0.8607 \). Once evaluated, these numbers reflect how the roast's temperature evolves. By computing specific values, one can observe the decay behavior from high initial rates of change, slowing down as the roast approaches the oven's temperature.