Question 259834
Let 

L = Length, W = Width, H = Height


Recall that the formula for the volume of the box is: {{{V=LWH}}} (ie multiply the length, width, and height to find the volume)



In addition, the surface area is simply the sum of the individual areas of the 6 faces on the box. Since there are essentially 2 of each face, we have the basic surface area formula 


Surface Area = 2*Length*Width+2*Width*Height+2*Length*Height


which can be rewritten as {{{S=2LW+2WH+2LH}}}



Now because "A box with rectangular sides has width twice the length of the base", we can say that {{{W=2L}}} (ie double the length to get the width)



So for each equation above, we can plug in {{{W=2L}}} to eliminate W altogether. What will result will be two equations with two unknowns (which is now possible to solve, if there is a solution).



So plug in {{{W=2L}}} into {{{V=LWH}}} to get {{{V=2L^2H}}}. Also, plug in {{{V=24}}} (the given volume) to get {{{24=2L^2H}}}. This will be equation 1.



Now plug in {{{W=2L}}} into {{{S=2LW+2WH+2LH}}} to get {{{S=4L^2+4LH+2LH}}}. Also, plug in {{{S=52}}} (the given surface area) to get {{{52=4L^2+4LH+2LH}}}. We'll refer to this as equation 2.



So you should now have the system of equations


{{{system(24=2L^2H,52=4L^2+4LH+2LH)}}}



To solve this system, simply isolate one variable and use substitution. I'll let you finish the problem. Let me know if you still need help.