Respuesta :
Answer:
480
Step-by-step explanation:
Since, for making a box from a cardboard,
We need to cut four congruent pieces from each corner of the cardboard,
Let x be the side of a piece ( in inches ),
Given,
The dimensions of the cardboard are 16 in by 22,
So, the dimension of the box would be (16-2x) in by (22-2x) in by x in,
Thus, the volume of the box,
[tex]V(x)=(16-2x)(22-2x)x=4x^3-76x^2+352x[/tex]
Differentiating with respect to x,
[tex]V'(x) = 12x^2-152x+352[/tex]
Again differentiating with respect to x,
[tex]V''(x) = 24x-152[/tex]
For maxima or minima,
V'(x) = 0
[tex]\implies 12x^2-152x+352=0[/tex]
By the quadratic formula,
[tex]x=\frac{-(-152)\pm \sqrt{-152^2-4\times 12\times 352}}{24}[/tex]
[tex]x=\frac{152\pm \sqrt{6208}}{24}[/tex]
[tex]\implies x\approx 9.62 \text{ or }x\approx 3.05[/tex]
Since, for x = 9.62, V''(x) = positive,
While for x = 3.05, v''(x) = negative,
Hence, volume is maximum for x = 3.05,
And, maximum volume,
[tex]V(3.05) = 4(3.05)^3-76(3.05)^2+352(3.05)=480.1005\approx 480\text{ cube in}[/tex]
