Sabrina is building a rectangular raised flower bed. The boards on the two sides are 6 inches thick, and the boards on the longer side are 4 inches thick.

Question

Sabrina wants the outer length of her bed to be 4 times its height and the outer width to be 2 times its height. She also wants the boards to rise 4 inches above the level of the soil in the bed. What should the outer dimensions of the bed be if she wants it to hold 3136 cubic inches of soil?

Answer 1

Let 'h' = height of bed.

 

Then, Outer Length of bed = 4h" ( " =inches)

 

and,   Outer Width of bed = 2h"  

 

Volume of space inside the bed (space where soil goes) = (4h-12)(2h-8)(h) inch-cubed

 

Volume occupied by soil = (4h-12)(2h-8)(h-4) inch-cubed  (height of soil = h-4 in. because soil is 4 in. below top of box)

 

Now,  (4h-12)(2h-8)(h-4) = 3136 inch-cubed  (given)

 

Or, simplifying, (4)(h-3)(2)(h-4)(h-4) = 3136  

 

Or, (h-3)(h-4)^2 = 392

 

Or,  (h-4)^2 = 392/(h-3)

 

Or,  h-4 = Sq.Rt.[392/(h-3)]

 

Finally, we have, h = Sq.Rt.[392/(h-3)] + 4

 

It gets a bit tricky here. You have to factorize 392.  392 = 2X2X2X49

 

By trial & error, for h=11, the equation will balance. That is, when h=11 (left side of equation), the right side of equation will also yield 11, because Right Side = Sq.Rt.[392/(11-3)] + 4 = (Sq.Rt. 49) + 4 = 7 + 4 = 11

 

So, the Answer is:   Outer length of bed = 4h = 44 inches

                          & Outer width of bed = 2h = 22 inches