Answer:
The outer dimensions of the flower bed would be 32 in × 14 in × 7 in
Step-by-step explanation:
The outer dimensions of the flower bed is given by:
L is the length of the longer boards
W is the length of the shorter boards
H is the height of the flower bed
As the two shorter sides are 6 inches thick and the two longer sides are 4 inches thick, the inner dimensions are:
L-12: is the inner dimension because there are two shorter boards
W-8: is the inner dimension because there are two longer boards
The boards to rise 4 inches above the level of the soil in the bed, then:
H-4 is the height of soil
The volume of soil is would be:
(L-12)×(W-8)×(H-4)= 3136
We also know that the outer length must be 4 times its height and the outer width must be 2 times its height, so:
L=4H
W=2H
We solve the equation:
(L-12)×(W-8)×(H-4)= 3136
(4H-12)×(2H-8)×(H-4)= 3136
(8H²-32H-24H+96)×(H-4)= 3136
(8H²-56H+96)×(H-4)= 3136
8H³-56H²+96H-32H²+224H-384= 3136
8H³-88H²+320H-384= 3136
8H³-88H²+320H-384-3136=0
8H³-88H²+320H-3520=0
We divide the equation by 8 in both members:
H³-11H²+40H-440=0
This equation has 1 real solution (H=11) and two complex solutions
The adecuate solution of the problem is H=11
Hence, the outer dimensions of the flower bed are:
L=4H=44 in
W=2H=22 in
Outer dimensions 44 in × 22 in × 11 in
