A sheet of metal 12 inches by 10 inches is to be used to make a open box. Squares of equal sides x are cut out of each corner then the sides are folded to make the box. Find the value of x that makes the volume maximum. *Round to 2 decimal places and write out "inches"

1.81 inches

Step-by-step explanation:

We are told that the sheet of metal to be used measures 12 inches by 10 inches.

Squares of equal sides x are cut out of each corner. Thus, it means total of 2x is cut of the 12 inches side and 10 inches side.

Thus;

Length; L = 12 - 2x

Width; W = 10 - 2x

Height; H = x

Volume of the box is;

V = LWH

V = (12 - 2x)(10 - 2x)x

Factorizing out, we have;

V = 4x[(6 - x)(5 - x)]

Equating V to zero, we can find the domain of the function;

At V = 0,

4x = 0; 6 - x = 0 ; 5 - x = 0

x = 0, x = 6, x = 5

The dimensions have to be positive. Thus;

0 ≤ x ≤ 5

Expanding the volume equation;

V = 4x[30 - 5x - 6x + x²]

V = 4x[30 - 11x + x²]

V = 4x³ - 44x² + 120x

dV/dx = 12x² - 88x + 120

Equating to zero to find the value of x that makes volume maximum;

12x² - 88x + 120 = 0

Using quadratic formula gives;

x = 1.81 or 5.52

From our domain, we have 0 ≤ x ≤ 5.

1.81 falls within the domain and therefore we will pick it as the height that makes the volume maximum.

The first one is wrong

Step-by-step explanation:

The student it wrong because the x is going up twice so it would be wrong.

the x going up by one makes y go up by 1.25 so they're wrong.

you could do 6 and 7.5

and then with 8 and 10.0

with those two numbers you can subtract it and then you would get 2 and 2.5 which if you divide by two you would get 1.25.

So the answer is they're wrong and that the constant is 1.25