#### Answer

86$^{\circ}$F

#### Work Step by Step

Since June is represented by $x=5$, we substitute $x=5$ in the equation to find the value of $t$:
$t=60-30\cos\frac{x\pi}{6}$
$t=60-30\cos\frac{5\pi}{6}$
$t=60-30(-\frac{\sqrt 3}{2})$
$t=60-15(-\sqrt 3)$
$t=60+15\sqrt 3$
$t=60+25.981$
$t=85.981\approx86$
Therefore, the maximum afternoon temperature for the month of June is 86$^{\circ}$F.