Answer:
Width: 20 in
Length: 25 in
Step-by-step explanation:
We can represent the situation with the following figure
Where x is the width of the rectangular piece of metal, (x + 5) is the length of the rectangular because it is 5 in longer than its wide, and the corners have squares of side 1 in.
Therefore, the volume of the box will be equal to
Volume = Length · Width · Height
Volume = (x + 5 - 1 - 1) · (x - 1 - 1) · (1)
Volume = (x + 3)(x - 2)(1)
Volume = (x + 3)(x - 2)
Because the length of the box will be the length of the rectangle less the length of the squares and the width of the box will be the length of the rectangle less the width of the squares.
The volume is 414 in³, so we need to solve the following equation:
414 = (x + 3)(x - 2)
414 = x² + 3x - 2x + 3(-2)
414 = x² + x - 6
414 - 414 = x² + x - 6 - 414
0 = x² + x - 420
Factorizing x² + x - 420, we get:
(x + 21)(x - 20) = 0
Then
x + 21 = 0
x + 21 - 21 = 0 - 21
x = -21
or
x - 20 = 0
x - 20 + 20 = 0 + 20
x = 20
Since x = -21 doesn't have sense, the width is x = 20 and the length is:
x + 5 = 20 + 5 = 25 in.
So, the original width is 20 in and the original length is 25 in