A company determines that its marginal revenue per day is given by $R^{\prime}(t)$, where $R(t)$ is the total accumulated revenue, in dollars, on the th day. The company's marginal cost per day is given by $C^{\prime}(t)$, where $C(t)$ is the total accumulated cost, in dollars, on the th day. \[ R^{\prime}(t)=80 e^{t}, R(0)=0 ; C^{\prime}(t)=80-0.3 t, C(0)=0 \]

Step 1 ：Given the marginal revenue per day as \(R^{\prime}(t)=80 e^{t}\) and the marginal cost per day as \(C^{\prime}(t)=80-0.3 t\), with initial conditions \(R(0)=0\) and \(C(0)=0\).

Step 2 ：To find the total accumulated revenue and cost on the th day, we need to integrate the marginal revenue and cost functions, respectively.

Step 3 ：Integrating the marginal revenue function \(R^{\prime}(t)=80 e^{t}\) with respect to t, we get the total accumulated revenue on the th day as \(R(t) = 80e^{t}\).

Step 4 ：Integrating the marginal cost function \(C^{\prime}(t)=80-0.3 t\) with respect to t, we get the total accumulated cost on the th day as \(C(t) = -0.15t^{2} + 80t\).

Step 5 ：\(\boxed{\text{Final Answer: The total accumulated revenue, } R(t) \text{, on the th day is given by } 80e^{t} \text{ and the total accumulated cost, } C(t) \text{, on the th day is given by } -0.15t^{2} + 80t}\)