What dictates the aspect ratio of an aircraft's wing?
Now I am going to commit a heresy, but keep reading to get an explanation:

Increasing the aspect ratio of a wing will not change its induced drag. Increasing the span will.

The induced drag coefficient of a wing is $c_{Di} = \frac{c_L^2}{\pi\cdot AR\cdot\epsilon}$, and this seems to indicate that a bigger aspect ratio AR would lower the induced drag coefficient $c_{Di}$. But only at the same lift coefficient $c_L$!

Now let's look at the real numbers and compare two wings of the same span, but different aspect ratios. For simplicity, wing 1 has an AR of 5 and wing 2 has an AR of 10. Let's further assume that both wings have the same mass. Since both wings have the same span, wing 1 has twice the wing area of wing 2. To create the same lift, wing 1 needs only half the lift per area than wing 2! This means its $c_L$ is only half as big as that of wing 2, and now lets look at the induced drag again: $D_i = q_\infty\cdot S\cdot c_{Di}$

Wing 1: $D_{i_1} = q_\infty\cdot S_1\cdot\frac{c_{L_1}^2}{\pi\cdot AR_1\cdot\epsilon}$

Wing 2: $D_{i_2} = q_\infty\cdot S_2\cdot\frac{c_{L_2}^2}{\pi\cdot AR_2\cdot\epsilon} = q_\infty\cdot 0.5\cdot S_1\cdot\frac{4\cdot c_{L_1}^2}{\pi\cdot 2\cdot AR_1\cdot\epsilon} = D_{i_1}$

If both have the same span efficiency $\epsilon$, both have the same induced drag at the same lift. To reduce induced drag requires a span increase, regardless of aspect ratio.

However, a higher aspect ratio wing does have advantages:


Lower surface area means less friction drag
Lower surface area also means less mass, at least at moderate aspect ratios.
Smaller pitching moments, requiring a smaller tailplane


but also disadvantages:


Less internal volume for fuel or the landing gear
Needs more complex high lift devices for the same landing speed


In the end, the wing chord is chosen to minimize wing mass and to yield the minimum required fuel volume, and the aspect ratio is just a consequence of the selected wing span. Driving wing mass down is also reducing induced drag, and the $\epsilon$ of modern airliner wings is only 0.75 to 0.8, which shows how little importance the induced drag coefficient has for finding an overall optimum.