Hakika
23.11.2019 •
Mathematics
An open box is to be made out of a 8-inch by 16-inch piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. find the dimensions of the resulting box that has the largest volume.
dimensions of the bottom of the box: x
height of the box:
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Ответ:
Dimension of bottom =Length x Breadth
= 12.619 inch x 4.619 inch
≈ 13 inch x 5 inch (To nearest whole number)
Height of box = 1.6905 inch
≈ 2 inch
Step-by-step explanation:
Length of the cardboard = 16 inch
Breadth of the cardboard = 8 inch
Since the cardboard has four corners, we would cut out four (4 ) squares.
Assuming the length of the square =y
length of open box = 16-2y (1)
breadth of open box = 8-2y(2)
height of open box = y
Volume of box = Length x breadth x height
= (16-2y) x ( 8-2y) x y
= ( 128-48y + 4y²)y
= 128y-48y² + 4y³
To maximize the volume of the box, V(y); dV/dy =0
dV/dy = 128- 96y + 12y²
12y²-96y+128 = 0
a = 12 , b= -96 , c= 128
y= [-b± √(b²-4ac)]/2a
= [96 ±√(96²- (4 x 12 x128)]/2 (12)
y= 6.3094 or y =1.6905
Substituting both values of y into (1) and (2)
When y = 6.3094
Length of box = 16- 2(6.3094)
= 3.3812 inch
Breadth of box =8- 2 (6.3094)
= -4.6188 inch
Since the breadth cannot be negative, this value of y is therefore an extraneous root.
So the correct value of y is the smaller of the two roots of equation,
y = 1.6905 inch
Substituting both values of y into (1) and (2)
When y = 1.6905
Length of box = 16- 2(1.6905)
= 12.619 inch
≈ 13 inch
Breadth of box =8- 2 (1.6905)
= 4.619 inch
≈ 5 inch
So the Dimension of the bottom (to nearest whole number):
Length x breadth = 13 inch x 5 inch
Height of the box = 1.6905 inch
≈ 2 inch
Ответ:
4
Step-by-step explanation:
plug 4 into x
parentheses first, 4x4 is 16, 16 - 10 is 6
6 times 1.25 is 7.5