Answer
After $5$ seconds, the first ball passes the second ball when both balls are at a height of 272 feet.
Work Step by Step
We can write an equation for the vertical position of the first ball:
$s_1(t) = -16t^2+48t+432$
We can write an equation for the vertical position of the second ball:
$s_2(t) = -16(t-1)^2+24(t-1)+432,~~~~~t \geq 1$
Note that the second ball is thrown 1 second later, so we need to adjust the second equation by 1 second.
We can find $t$ when $s_1 = s_2$:
$s_1 = s_2$
$-16t^2+48t+432 = -16(t-1)^2+24(t-1)+432$
$-16t^2+48t = -16(t^2-2t+1)+24t-24$
$-16t^2+48t = -16t^2+32t-16+24t-24$
$48t = 32t-16+24t-24$
$8t = 40$
$t = 5$
After $5$ seconds, the first ball passes the second ball.
We can verify this by finding the height of each ball after 5 seconds:
$s_1(5) = -16(5)^2+48(5)+432 = 272$
$s_2(5) = -16(5-1)^2+24(5-1)+432 = 272$
After $5$ seconds, the first ball passes the second ball when both balls are at a height of 272 feet.
