To determine the E. coli population 4 hours after the culture was started, we first need to convert hours into 20-minute increments.
4 hours is equivalent to:
\[
4 \text{ hours} \times 60 \text{ minutes/hour} = 240 \text{ minutes}
\]
\[
\frac{240 \text{ minutes}}{20 \text{ minutes/increment}} = 12 \text{ increments}
\]
Next, we can use the formula given for the population of E. coli:
\[
f(t) = 10 \cdot 2^t
\]
where \( t = 12 \) (the number of 20-minute increments).
Now we can calculate:
\[
f(12) = 10 \cdot 2^{12}
\]
Calculating \( 2^{12} \):
\[
2^{12} = 4096
\]
Then:
\[
f(12) = 10 \cdot 4096 = 40960
\]
Thus, the E. coli population 4 hours after the culture was started is:
\[
\boxed{40,960}
\]