Respuesta :
Answer:
96.256 in³
Step-by-step explanation:
Given,
The dimension of the sheet = 9-inch by 13.5-inch,
Suppose x be the side of the square which is cut from each corner,
So, the dimension of the resultant open container,
(9-2x) inch × (13.5 - 2x) inch × x inch
Thus, the volume of the box,
V(x) = (9-2x) × (13.5 - 2x) × x
[tex]V(x) = (9x-2x^2)(13.5 - 2x)[/tex]
Differentiating with respect to x,
[tex]V'(x) = (9x - 2x^2)(-2) + (9 - 4x)(13.5 - 2x)[/tex]
[tex]=-18x + 4x^2 + 121.5 - 18x - 54x + 8x^2[/tex]
[tex]=12x^2 - 90x + 121.5[/tex]
Again differentiating,
[tex]V''(x) = 24x - 90[/tex]
For maxima or minima,
[tex]V'(X) = 0[/tex]
[tex]12x^2 - 90x + 121.5=0[/tex]
Using the quadratic formula,
[tex]x =\frac{90\pm \sqrt{90^2 - 4\times 12\times 121.5}}{24}[/tex]
[tex]\implies x\approx 1.766\text{ or }x=5.734[/tex]
For x = 1.766, V''(x) = negative,
Thus, V(x) is maximum at x = 1.766,
For x =5.734, V''(x) = positive,
Thus, V(x) is minimum at x = 5.734,
Hence, the maximum possible volume, V(1.766) = (9-2(1.766)) × (13.5 - 2(1.766)) × 1.766
≈ 96.256 in³
