**Given Information:**

- eg(t) = 2.5t V
- Circuit shown in the figure

**To Find:**

- Values of i(t) and vL(t) at t = 4 seconds

**Circuit Description:**

The circuit shown in the figure consists of a resistor (R), an inductor (L), and a voltage source (eg(t)).

To find the values of i(t) and vL(t) at t = 4 seconds, we need to analyze the circuit using Kirchhoff's laws and the properties of inductors.

**Analysis of the Circuit:**

To analyze the circuit, we will use the following steps:

1. **Determine the Current (i(t)):**
- The current flowing through the inductor (i(t)) can be found using the equation V = L(di/dt), where V is the voltage across the inductor and L is the inductance.
- In this circuit, the voltage across the inductor (vL(t)) is the same as the voltage source (eg(t)) because there are no other elements connected in parallel with the inductor.
- So, vL(t) = eg(t) = 2.5t V
- Now, differentiate vL(t) to find di/dt:
- di/dt = d(2.5t)/dt = 2.5 A/s
- Therefore, the current i(t) flowing through the inductor is 2.5 A/s.

2. **Determine the Voltage across the Inductor (vL(t)):**
- As mentioned earlier, the voltage across the inductor (vL(t)) is equal to the voltage source (eg(t)).
- At t = 4 seconds, we substitute t = 4 in the equation eg(t) = 2.5t to find vL(t):
- vL(t) = eg(t) = 2.5 * 4 = 10 V
- Therefore, the voltage across the inductor vL(t) at t = 4 seconds is 10 V.

**Summary:**

- The value of the current i(t) at t = 4 seconds is 2.5 A/s.
- The value of the voltage across the inductor vL(t) at t = 4 seconds is 10 V.

Note: The circuit diagram is not provided in the question, so it is not possible to provide a detailed analysis based on the specific circuit configuration. The answer provided assumes a basic circuit configuration with a resistor and an inductor connected in series with a voltage source.