Given the surge function $C(t)=10 t \cdot e^{-0.5 t}$, at $t=1, C(t)$ is:

Select one:
decreasing
increasing
at a maximum
at an inflection point

Step 1 ：We are given the surge function \(C(t)=10 t \cdot e^{-0.5 t}\) and we are asked to determine the behavior of the function at \(t=1\).

Step 2 ：The function is a product of two functions, \(10t\) and \(e^{-0.5t}\). We need to use the product rule to find the derivative. The product rule states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

Step 3 ：Let's calculate the derivative. The derivative of \(10t\) is \(10\) and the derivative of \(e^{-0.5t}\) is \(-0.5e^{-0.5t}\).

Step 4 ：Applying the product rule, we get \(C'(t) = 10 \cdot e^{-0.5t} + 10t \cdot -0.5e^{-0.5t} = -5.0t \cdot e^{-0.5t} + 10 \cdot e^{-0.5t}\).

Step 5 ：We evaluate the derivative at \(t=1\), \(C'(1) = -5.0 \cdot e^{-0.5} + 10 \cdot e^{-0.5} = 3.03265329856317\).

Step 6 ：The derivative of the function at \(t=1\) is positive, which means the function is increasing at \(t=1\).

Step 7 ：Final Answer: The function \(C(t)=10 t \cdot e^{-0.5 t}\) is \(\boxed{increasing}\) at \(t=1\).