A square of side x=cm is cut from each corner of a square metal sheet of side 17cm the remaining sheet is folded into a cuboid if x is a natural number then find the maximum possible volume of the cuboid so formed
Answers
363 cm³ would be maximum volume and x = 3 x being natural number and square of x cm is cut from each corner of a square metal sheet of side 17 cm
Step-by-step explanation:
Size of square piece = 17 * 17 cm²
Let say Size of square cut from corner = x * x cm²
Then sides of Open box would be be
17 - 2x , 17 - 2x & x
Volume of the open box = (17-2x)(17-2x)x
= x(17 - 2x)²
= x(289 + 4x² - 68x)
= 4x³ - 68x² + 289x
V = 4x³ - 68x² + 289x
dV/dx = 12x² - 136x + 289
dV/dx = 0
12x² - 136x + 289 =
12x² - 34x - 102x + 289 =0
2x(6x - 17) - 17(6x - 17) = 0
=> x = 17/2 or x = 17/6
d²V/dx² = 24x - 136
d²V/dx² = -ve at x = 17/6 hence maximum volume
d²V/dx² = +ve at x = 17/2 hence minimum volume
x = 17/2 not possible as then nothing will remain
x = 17/6 = 2.833
x is a natural number => x = 3
x = 3 would give maximum Volume
Maximum volume = 4x³ - 68x² + 289x
= 4 * 3³ - 68*3² + 289*3
= 108 - 612 + 867
= 363 cm³
or
(17-2x)(17-2x)x
= (17 - 2*3)(17 - 2*3)3
= 11 * 11 * 3 = 363 cm³
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