Given Information:
The charge through a cross section of a conductor is given by q = 5t^2 - 2t coulomb, where t represents time in seconds.

Objective:
To find the average current through the conductor in the interval t1 = 2 sec to t2 = 4 sec.

Solution:

1. Definition of Average Current:
The average current (I_avg) through a conductor is defined as the total charge (Q) passing through a cross section of the conductor divided by the time interval (Δt) during which the charge flows.

2. Calculation of Total Charge:
To find the total charge passing through the conductor, we need to integrate the given charge function with respect to time (t) over the interval t1 to t2:

Q = ∫[t1 to t2] q dt

3. Integration:
The given charge function is q = 5t^2 - 2t. Integrating this function with respect to time, we get:

Q = ∫[t1 to t2] (5t^2 - 2t) dt

4. Integration of 5t^2:
Using the power rule of integration, the integral of 5t^2 with respect to t is:

∫(5t^2) dt = (5/3) * t^3 + C

5. Integration of -2t:
Similarly, the integral of -2t with respect to t is:

∫(-2t) dt = -t^2 + C

6. Evaluating the Definite Integral:
Substituting the limits of integration (t1 = 2 sec and t2 = 4 sec) into the integrated functions, we can evaluate the definite integral:

Q = [(5/3) * t^3 - t^2] from 2 to 4

Q = [(5/3) * 4^3 - 4^2] - [(5/3) * 2^3 - 2^2]

Q = [(5/3) * 64 - 16] - [(5/3) * 8 - 4]

Q = (320/3 - 16) - (40/3 - 4)

Q = (320/3 - 16) - (40/3 - 12/3)

Q = (320/3 - 16) - (40/3 - 12/3)

Q = (320 - 48) / 3

Q = 272 / 3 coulomb

7. Calculation of Time Interval:
The time interval (Δt) is given as t2 - t1:

Δt = t2 - t1
= 4 - 2
= 2 sec

8. Calculation of Average Current:
Using the formula for average current, we can calculate the average current (I_avg) through the conductor:

I_avg = Q / Δt
= (272 / 3) / 2
= (272 / 3) * (1/2)
= 136 / 3
≈ 45.33 A