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An exponential function is a mathematical expression where a constant base is raised to a power denoted by a variable. In our example, the function is defined as \( s(t) = 12 e^{0.3 t} t^2 \). This equation includes an exponential component \( e^{0.3t} \), where \( e \) is Euler's number, approximately equal to \( 2.71828 \).
Exponential functions are essential in modeling growth or decay processes since they can rapidly increase or decrease. The base \( e \), known as the natural exponential base, is particularly common in continuous growth models. Here, the rate of growth or decay is defined by the coefficient \( 0.3 \) in the exponent.
Exponential functions can take various shapes on a graph, but in essence, they all showcase that the function's rate of change is proportional to the function's current value. With the function \( s(t) \), as the value of \( t \) increases, \( s(t) \) will grow significantly due to the exponential factor.

To evaluate a function effectively, replace the variable in the function with a specific value. In our problem, the function \( s(t) = 12 e^{0.3 t} t^2 \) needs to be evaluated for specific values of \( t \).
Let's take the function for two values, \( t = 10 \) and \( t = 1 \). For \( t = 10 \), substitute \( 10 \) into the function:

• Compute the exponent: \( 0.3 \times 10 = 3 \).
• Calculate the value of \( e^3 \), resulting in approximately \( 20.0855 \).
• Multiply the results: \( 12 \times 20.0855 \times 100 \), which equals approximately \( 24,102.6 \).

Function evaluation involves understanding how to substitute and perform these calculations correctly.
For \( t = 1 \), simply follow a similar approach:

• Calculate the exponent: \( 0.3 \times 1 = 0.3 \).
• Find the value of \( e^{0.3} \), which is approximately \( 1.3499 \).
• Final multiplication: \( 12 \times 1.3499 \times 1 = 16.199 \).

Evaluating functions is a key skill in understanding not just exponential functions but many other types as well.

Rounding is the process of adjusting a number to make it simpler and more understandable, usually by keeping only significant figures up to a prescribed number of decimal places. In our problem, the results need to be rounded to three decimal places.
Rounding a number involves following a straightforward process:

• Identify the position of the digit to round to, which in this case, is the third decimal place.
• Look at the digit immediately to the right of the target decimal place.
• If this digit is 5 or greater, increase the target digit by one. If it is less than 5, keep the target digit as is.

For the value \( 24,102.6 \), the digit right of the target decimal place is already handled, resulting in \( 24,102.6 \) when rounded to one decimal point already.
Similarly, for \( 16.199 \), it remains \( 16.199 \) after rounding to three decimal places as the digits are already aligned.
Rounding ensures results are concise, practical for interpretation, and consistent in presentation.