Megha wants to prepare a handmade gift box for her friend's birthday at home. For making lower part of box, she takes a square piece of cardboard of side 20 cm.
Answers
Solution :-
i)
→ Side of square = 20 cm .
So,
→ x ∈ (0 , 10) (b) .
ii)
→ Let height of open box = x cm
→ Length of open box = (20 - 2x) cm
→ Breadth of open box = (20 - 2x) cm .
So,
→ Volume of open box = L * B * H = x(20 - 2x)(20 - 2x) (a)
iii)
→ V = x(20 - 2x)²
→ dV/dx = x•2•(20 - 2x)•(-2) + (20 - 2x)²
→ dV/dx = (-4x)(20 - 2x) + (20 - 2x)²
→ dV/dx = (20 - 2x)[(-4x) + (20 - 2x)]
→ dV/dx = (20 - 2x)(20 - 6x)
when dV/dx = 0,
→ 20 - 2x = 0
→ 20 = 2x
→ x = 10
and,
→ 20 - 6x = 0
→ 20 = 6x
→ x = (10/3)
therefore, values of x are (d) (10 , 10/3) .
iv)
→ dV/dx = (20 - 2x)(20 - 6x)
→ d²V/dx = (20 - 2x)•(-6) + (20 - 2x)•(-2)
→ d²V/dx = (-120 + 12x) + (-40 + 12x)
→ d²V/dx = (24x - 160)
now at x = 10,
→ d²V/dx = (24 * 10 - 160) = 80 > 0
and, at x = (10/3)
→ d²V/dx = (24 * 10/3 - 160) = 80 - 160 = (-80) < 0 .
therefore, for maximum volume of box, side of square to be cut off should be (c) x = (10/3) .
v)
→ The maximum value of the volume = x(20 - 2x)²
putting x = (10/3)
→ The maximum value of the volume = (10/3)[20 - (20/3)]²
→ The maximum value of the volume = (10/3)[40/3)²
→ The maximum value of the volume = (10/3) * (1600/9)
→ The maximum value of the volume = (16000/27) cm³ (d)
Learn more :-
19. Embroidery on cloth
Meera embroidered a leaf by
knitting on her table cloth. Her
son traced the design on a
coordina...
brainly.in/question/34722753
u par LARKI TWAR
17. Mapping of Village: A patwari decides to map his native village by assigning coordinates to each ho...
brainly.in/question/33329177
Three girls Rani, Mansi and Sneha are talking to each other and maintaining a social distance due to COVID-19. They are ...
brainly.in/question/33571589
