If the wind speed increases from 4.0 m/s to 5.0 m/s, the power output from an ideal windmill will increase by a factor of about:

The correct option is ~2.

To understand why this is the correct answer, let us look at the relationship between wind speed and the power output of an ideal windmill.

The power output $P$ of an ideal windmill is directly proportional to the cube of the wind speed $v$:
$P\propto v^3$

Let $v_1=4.0\text{m/s}$ be the initial wind speed, and $v_2=5.0\text{m/s}$ be the final wind speed. The factor $f$ by which the power output increases is given by the ratio of the final power $P_2$ to the initial power $P_1$:
$f=\frac{P_2}{P_1}={\left(\frac{v_2}{v_1}\right)}^3$

Substituting the given values into the equation:
$f={\left(\frac{5.0}{4.0}\right)}^3={\left(1.25\right)}^3$

Calculating the cube of 1.25:
$f=1.25\times 1.25\times 1.25=1.953125$

Since 1.953125 is approximately equal to 2, the power output from the ideal windmill will increase by a factor of about 2.