A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the volume V of the box as a function of x.
Well the volume of a box is V = width (w) * height(h) * length (l)
If you draw your box you will notice that the height will be x and that the height and length depend on x. In fact l = 20-2x and w = 12 – 2x
So the volume is:
V(x) = whl = (12-2x)(x)(20-2x)
expand to get
V(x) = 4x^3 – 64x^2 + 240x
[We can know find what value of x gives the largest volume by differentiating and finding where the gradient of the curve is zero.
V’ = 12x^2 – 128x + 240
0 = 12x^2 – 128x + 240
x = 8.4 or 2.4
Sub these back into our equation for V(x) and see which gives the bigger volume…V(8.4) = -129.024 which is obviously incorrect as it’s negative so the answer must be x = 2.4 which gives a volume of 262.656.
Hope that helps!
considering the fact which you will cut back x out of each and every corner, the dimensions x would be subtracted two times (2 corners in step with area). try this for each area and multiply. undergo in strategies quantity equals length cases width cases height. this could effect in V=(11-2x)(22-2x). Then multiply with the aid of x to element interior the peak. the only perfect equation is V=(11-2x)(22-2x)(x), or, simplified, (4x^3)-(66x^2)+242.