Answer
$l=57.4in$
Work Step by Step
The length of the belt still in contact with the pullet is $\frac{2}{3}$ of the circumference of the circle (because we have $360° - 120° = 240°$). The rest of the length of the belt is calculated using the triangle formed by the radius of the circle, the distance from the spindle to the center of the pulley, and the tangent of the circle that intercepts the spindle.
Let us call the center of the circle ($O$), the tangent of the circle ($T$), and the spindle of the wheel ($S$).
We have the variables:
$d=15in$
$OS=15in$
$OT=7.5in$
$\lt OTS = 90°$ because the tangent is always perpendicular to the radius.
So we use the Pythagorean Theorem to find the length of the rest of the belt ($x$):
$x=TS=\sqrt {(OS)^{2}-(OT)^{2}}=\sqrt {(15in)^{2}-(7.5in)^{2}}=12.99in$
The total length of the belt is calculated with the values found above:
$l=\frac{2}{3}C+2x=\frac{2}{3}\pi d+2x=2(\frac{1}{3}\pi d+x)=2(\frac{1}{3}\pi (15in)+12.99in)=57.4in$
