Answer:
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.
