Question

An open box is to be made from a square piece of material 24 cm on a side by cutting equal squares from the corners and turning up the sides as shown express the volume v of the box as a function of x

Answer 1

Answer:

V = 4(x³ - 24x² + 144x)

Step-by-step explanation:

An open box is to be made from a square piece of material, 24 cm on a side, by cutting equal squares from the corners and turning up the sides.

Size of square piece = 24 * 24  cm²

Let say Size of csquare  cut from corner = x * x   cm²

Then sides of Open box would be be

24 - 2x  , 24 - 2x  & x

Volume of the open box = (24-2x)(24-2x)x

= x(24 - 2x)²

= x * 2²(12 - x)²

= 4x * (x² + 144 - 24x)

= 4(x³ - 24x² + 144x)

V = 4(x³ - 24x² + 144x)

dV/dx = 4(3x² - 48x + 144)

dV/dx  = 0

4(3x² - 48x + 144)  = 0

=> x² - 16x + 48 = 0

x² - 12x - 4x + 48 = 0

x(x-12) -4(x-12) = 0

(x - 4)(x-12) = 0

x = 12 is not possible as then no box will be left

x = 4 will give max volume