Answer
$=.559$ inches
Work Step by Step
$V=\pi*r^2*h$
$400=V_{1}-V_{2}$
$V_{1}=\pi*(2+x)^2*50$
$V_{2}=\pi*2^2*50$
$V_{2}=200*\pi$
$V_{2}=628.32$
$V_{1}=\pi*(x^2+4x+4)*50$
$V_{1}=\pi*(50x^2+200x+200)$
$400=V_{1}-628.32$
$400+628.32=V_{1}-628.32+628.32$
$1028.32 = V_{1}$
$1028.32 = \pi*(50x^2+200x+200)$
$1028.32/\pi = \pi*(50x^2+200x+200)/\pi$
$327.49 = (50x^2+200x+200)$
$327.49 = 50x^2+200x+200$
$127.49 = 50x^2+200x$
$127.49-127.49= 50x^2+200x-127.49$
$0=50x^2+200x-127.49$
$x=(-b±\sqrt {b^2-4ac})/2a$
$x=(-(200)±\sqrt {(200)^2-4*50*(-127.49)})/2*50$
$x=(-(200)±\sqrt {(40000-200*(-127.49)})/100$
$x=(-200)±\sqrt {(40000+25498)})/100$
$x=(-200)±\sqrt {(65498)})/100$
$x=(-200±255.9)/100$
$x=(-200+255.9)/100$
$x=55.9/100 = .559$
$x=(-200-255.9)/100$
$x=-455.9/100 = -4.559$ (we can't have a negative length, so this answer is invalid)
