The length of a rectangular piece of cardboard is three more than twice the width. A square 2 cm on a side is cut out of each corner. The sides are folded up to form an open box. if the volume of the box is 120cm^3, what were the original dimensions of the cardboard.
L=3+2W
Now, for the box,
L=originalL-4=3+2originalW-4
W=originalW-4
volume=LW*2
120=(3+2originalW-4)(originalW-4)*2
let me now use for originalW, w'
60=(2w'-1)(w'-4)
60=2w'^2-8w'-w'+4
56=2w'^2-9w'
2w'^2-9w'-56=0
(2w+7)(w'-8)=0
w'=8
l'=2(8)+3=19
check...volume of box then is
V=2*(8-4)(19-4)=4*15*2=120
To solve this problem, we can follow these steps:
1. Let's assume the width of the rectangular piece of cardboard is "x" cm. According to the problem, the length is three more than twice the width, so the length would be (2x + 3) cm.
2. After cutting out the squares from each corner, the width of the box will be reduced by 4 cm (2 cm from each side). Therefore, the width of the box would be (x - 4) cm.
3. Similarly, the length of the box would be (2x + 3 - 4) cm, which simplifies to (2x - 1) cm.
4. The height of the box will be 2 cm since the squares cut from the corners are used to form the sides of the box.
5. Now, we can calculate the volume of the box using the formula: Volume = Length * Width * Height. From the problem, we know that the volume is 120 cm³, so we have the equation: 120 = (2x - 1) * (x - 4) * 2.
6. Simplifying the equation further, we get: 120 = 2(2x - 1)(x - 4).
7. Expanding the equation, we have: 120 = 4x² - 12x - 2x + 8.
8. Combining like terms, we get: 120 = 4x² - 14x + 8.
9. Rearranging the equation to form a quadratic equation equal to zero, we have: 4x² - 14x + 8 - 120 = 0.
10. Simplifying further, we get: 4x² - 14x - 112 = 0.
11. Now, we can solve this quadratic equation. We can either factorize it or use the quadratic formula. Factoring the equation, we have: (2x - 16)(2x + 7) = 0.
12. Setting each factor equal to zero, we have: 2x - 16 = 0 or 2x + 7 = 0.
13. Solving these equations, we get: x = 8 or x = -7/2. Since the width cannot be negative, we discard the second solution.
14. Therefore, the width of the cardboard is 8 cm.
15. Using the width, we can find the length using the equation: Length = 2x + 3 = 2(8) + 3 = 16 + 3 = 19 cm.
16. So the original dimensions of the cardboard were a width of 8 cm and a length of 19 cm.