Kling1982
22.07.2019 •
Mathematics
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
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Ответ:
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
Ответ:
1359
may be my answer is helpful for u