Arectangular piece of cardboard, whose area is 216 square centimeters, is made into an open box by cutting a 2-centimeter square from each corner and turning up the sides. if the box is to have a volume of 224 cubic centimeters, what should the original dimensions of the cardboard start with

Length: 18 cm

Width: 12 cm

Step-by-step explanation:

A better explanation for this is the following:

First we know that the area of the card is 216 cm². This can be expressed as:

A = L * W   so

216 = L * W  (1)

Now, let's change the variables here, we will denote the Length as "x" and the width as "y". Then (1) can be rewritten as:

216 = xy   (2)

Now, we know that in order to make the open box, it was cut from each corner of the card, 2 cm², and the volume of the box is 224 cm³. According to this, we know that the volume of the box is:

V = L' * W' * H (3)

The Height of the box would be the 2 cm that were cut and L' and W' would be x and y. However, as the Length and Width has been cut, then the expression for both of them is the following

For the Length:

L' = x - 4

For the Width:

W'= y - 4

Replacing in expression (3):

224 = 2 * (x-4) * (y-4)

112 = (x-4)(y-4)  (4)

Now, in (2) we can solve either x or y to make a new expression and then, do the same in (4), and thus, we can actually solve for one of the dimensions. In this case, we will solve for y first, so let's solve for y in (2) and (4):

216 = xy

y = 216/x (5)

Solving now for y, from (4):

112/x-4 = y - 4

y = (112/x-4) + 4 (6)

So now, all we have to do is equal (5) and (6), and in that way we can find the value of x:

216/x = (112/x-4) + 4

216/x = 112 + 4(x-4) / (x-4)

216(x-4) = x(112 + 4x - 16)

216x - 864 = 112x + 4x² - 16x

4x² - 120x + 864 = 0 (7)

From here, we can either do the general expression and solve for x, or we can just factorize (7) and get the 2 values of x at once. In this case let's use the general expression. Although is longer, but we will get the correct result using this method so:

4x² - 120x + 864 = 0    (Divide by 4 all terms)

x² - 30x + 216 = 0  (8)

the general equation:

x = -b ±√b² - 4ac / 2a

From (8), we know that a = 1, b = -30, c = 216. Replacing:

x = 30 ± √(-30)² - 4*1*216 / 2

x = 30 ±√900 - 864 / 2

x = 30 ± 6 / 2

x1 = 30 + 6 / 2 = 18

x2 = 30 - 6 / 2 = 12

So the values for the Length and width are:

L = 18 cm

W = 12 cm

If you put this numbers into equations (2) and (4):

216/18 = 12

(18-4)(12-4) = 112

207/8 is the right