A square of side x cm is cut from each of the corners of a rectangular piece of cardboard 15 cm by 24 cm. The cardboard is then folded to form an open box of depth x cm. Show that the volume of the box is (4x³ - 78x² + 360x) cm³. Find the value of x for which the volume is a maximum.
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Answer:
Visualise the box:
The dotted red box is the base of the open box.
So what we have in the end is a box with base area (15 - 2x)(24 - 2x) with depth x cm
Hence, volume of box is given by
V = x(15 - 2x)(24 - 2x)
= x (360 - 78x + 4x²)
= 4x³ - 78x² + 360x (shown)
To find the value of x for which volume is maximum,
dV/dx = 12x² - 156x + 360
when dV/dx = 0,
12x² - 156x + 360 = 0
x² - 13x + 30 = 0
(x - 3)(x - 10) = 0
x = 3 or x = 10
d²V/dx² = 24x - 156
When x = 3, d²V/dx² = -84
When x = 10, d²V/dx² = 84
Hence, V is a maximum when x = 3.
