A rectangular piece of metal is 15in longer than it is wide. Squares with sides 3 in long are cut from four corners and the flaps are folded up to form an open box. If the volume is 1218in^3, what we’re the original deminsions of the piece of metal?
original piece is x by x+15
base of box = x-6 by x+9
height of box = 3
volume = 3(x-6)(x+9) = 1218
x^2 + 3x - 54 = 406
x^2 + 3x - 460 = 0
(x - 20)(x + 23) = 0
x = 20 or a negative
the original piece was 20 by 35 inches
Interesting. Are you able to visualize this? Any idea how to get started?
X^2+3x-1272=0
Needs to be factored
To find the original dimensions of the piece of metal, we need to follow these steps:
Step 1: Determine the width of the rectangular piece of metal.
Let's assume the width of the metal is x inches. According to the problem, the metal is 15 inches longer than its width, so the length would be x + 15 inches.
Step 2: Calculate the dimensions of the box after the corners are cut and folded up.
When each corner is cut, a square with sides of 3 inches is removed. This reduces the width and length of the metal by 6 inches each, as 3 inches are removed from both ends. After folding up the flaps, the height of the box would also be 3 inches.
So, the dimensions of the box would be (x - 6) inches for the width, (x + 15 - 6) inches for the length, and 3 inches for the height.
Step 3: Calculate the volume of the box.
To calculate the volume of the box, multiply the width by the length and the height:
Volume = (x - 6) * (x + 15 - 6) * 3
Step 4: Solve for the width.
Set the volume equal to 1218in^3 and solve for x:
(x - 6) * (x + 15 - 6) * 3 = 1218
Step 5: Solve the equation.
Simplify the equation:
(x - 6) * (x + 9) * 3 = 1218
(x^2 + 9x - 6x - 54) * 3 = 1218
(x^2 + 3x - 54) * 3 = 1218
3x^2 + 9x - 162 = 1218
3x^2 + 9x - 1380 = 0
Solve the quadratic equation to find the value of x.
After applying the quadratic formula:
x = (-9 ± √(9^2 - 4 * 3 * -1380)) / (2 * 3)
Simplifying further, we get:
x = (-9 ± √(81 + 16560)) / 6
x = (-9 ± √16641) / 6
x = (-9 ± 129) / 6
Solving for both possibilities, we get:
x₁ = (-9 + 129) / 6
x₁ = 120 / 6
x₁ = 20
x₂ = (-9 - 129) / 6
x₂ = -138 / 6
x₂ = -23
Since the width cannot be negative, the original width of the rectangular piece of metal is 20 inches.
Step 6: Calculate the length.
Using the width we found, the length would be:
Length = x + 15 = 20 + 15 = 35 inches
Therefore, the original dimensions of the piece of metal are 20 inches in width and 35 inches in length.