To find the width of the box, we need to use the formula for the volume of a rectangular prism, which is:
Volume = Length × Width × Height
Given:
- Volume = 2720 in³
- Height = 17 in
- Length = 2w + 4 (where w is the width)
We can set up the equation:
2720 = (2w + 4) × w × 17
First, we can simplify this equation:
2720 = 17(2w + 4)w
Next, divide both sides by 17:
160 = (2w + 4)w
Expanding this gives:
160 = 2w² + 4w
Rearranging this into a standard quadratic form:
2w² + 4w – 160 = 0
Dividing the entire equation by 2:
w² + 2w – 80 = 0
Now we can use the quadratic formula to find the width:
w = (-b ± √(b² – 4ac)) / (2a)
Here, a = 1, b = 2, and c = -80. Plugging in these values:
w = (-2 ± √(2² – 4 × 1 × -80)) / (2 × 1)
w = (-2 ± √(4 + 320)) / 2
w = (-2 ± √324) / 2
w = (-2 ± 18) / 2
This gives us two potential solutions:
w = (16) / 2 = 8 or w = (-20) / 2 = -10
Since width cannot be negative, the width of the box is:
w = 8 inches
Thus, the width of the box is 8 inches.