Question 104901
You know the height is 7 inches and the volume is 600 cubic inches.
You also know that a formula for volume, V(x),is given by the function,
{{{V(x)=63x + 42x^2}}} where x is the width. 
Solve for x when V(x)=600.
{{{63x+42x^2=600}}}
{{{42x^2+63x-600=0}}} Move all terms to one side. 
{{{14x^2+21x-200=0}}} Divide both sides by 6.
Use the quadratic formulat to solve for x. 
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
where a=14, b=21, and c=-200. 
{{{x = (-21 +- sqrt( 21^2-4*14*(-200) ))/(2*14) }}} 
{{{x = (-21 +- sqrt( 441+11200 ))/(28) }}} 
{{{x = (-21 +- sqrt( 11641 ))/(28) }}} 
{{{sqrt(11641)= (107.89)}}}
{{{x = (-21 +- (107.89))/(28) }}} 
{{{x=-4.6}}} and {{{x=3.1}}}
Since a negative width does not make sense, there is one solution. 
x=3.1 inches.
The volume of a rectangular prism is also given by, V=LxWxH. 
Since H=7 inches and W=3.1 inches, then
{{{LWH=600}}}
{{{L(7)(3.1)=600}}}
{{{L=600/(7*3.1)}}}
L=27.6 inches