A candy box is to be made out of a piece of cardboard that measures 8 by 12 inches. Squares of equal size will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box. What size square should be cut from each corner to obtain a maximum volume

Length of square cut = 1.569 inches

Step-by-step explanation:

given data

Length of cardboard = 12 inches

Breadth of cardboard = 8 inches

solution

we will consider here side of the square = x

when we cutting out the square then there Length and breadth of candy box will be

Length of box = (12 - x - x)

Length of box = (12 - 2x) inches.

and

Breadth of box = (8 - x - x)

Breadth of box = (8 - 2x) inches.

and Height of candy box  wil be = x inches

so

Volume of a cuboid = L × b × h    1

Volume of a cuboid = (12 - 2x) × (8 - 2x) × x

Volume of a cuboid = 96x - 40x² + 4x³

now we Differentiate with respect to x

V' = 96 - 80x + 12x²

and for maximum volume we put V' = 0

0 = 96 - 80x + 12x²

solve it we get

x = 5.097

x = 1.569

when x = 5.097 inches

Breadth of candy box = 8 - 10.194 = -2.194 inches

but we know breadth never be negative,

so we take

Length of square cut = 1.569 inches

1.4

2.3

step-by-step explanation:

the formula for the constant of variation is

y= kx

divide each side by x

y/x =kx/x

y/x =k

we know a and y

1.   y/x = k

4/1 =4

check the other values to make sure they are the same

8/2 = 4   12/3 =4   and 16/4 =4  

the constant of variation is 4

2.   y/x = k

3/1 =3

check the other values to make sure they are the same

6/2 = 3   9/3 =3   and 12/4 =3  

the constant of variation is 3