Respuesta :
Answer:
The dimensions of the box is 20 cm by 20 cm by 2 cm.
The minimum cost of manufacturing the box is 120 cents.
Step-by-step explanation:
Given that, the volume of of rectangular is 800 cm³ with a square base, an open top.
Consider the side of the square base is x cm.
Then length = width = x cm.
and the height be h cm.
The volume of the box is = length×width×height
=(x.x.h) cm³
=x²h cm³
Therefore
x²h=800
[tex]\Rightarrow h=\frac{800}{x^2}[/tex]
The area of the base is = x² cm² [ since the base is square in shape]
The area of the sides = 2 (length×width) height
=2(x+x)h cm²
=2(2x)h cm²
= 4xh cm²
The cost of the material for the base is 0.1 cents per square and the cost of the material for the the sides is 0.5 cents.
Total cost of material is C(x)=[(x²h×0.1)+ (4xh×0.5)] cents
∴ C(x)=[(x²×0.1)+ (4xh×0.5)]
⇒C(x)=(0.1x²+2xh)
Putting [tex]h=\frac{800}{x^2}[/tex]
[tex]C(x)= 0.1 x^2+2x.\frac{800}{x^2}[/tex]
[tex]\Rightarrow C(x)= 0.1 x^2+\frac{1600}{x}[/tex] .......(1)
Differentiating with respect to x
[tex]C'(x)=0.2x-\frac{1600}{x^2}[/tex]
Again differentiating with respect to x
[tex]C''(x)=0.2+\frac{3200}{x^3}[/tex]
To find the maximum or minimum cost, we set C'(x)=0
[tex]\therefore 0.2x-\frac{1600}{x^2}=0[/tex]
[tex]\Rightarrow 0.2x=\frac{1600}{x^2}[/tex]
[tex]\Rightarrow 0.2x^3=1600[/tex]
[tex]\Rightarrow x^3=\frac{1600}{0.2}[/tex]
[tex]\Rightarrow x^3=8000[/tex]
[tex]\Rightarrow x=20[/tex]
At x=20 ,the value of C(x) >0. Then at x=20 the value of cost will be maximum.
At x=20 ,the value of C(x) <0. Then at x=20 the value of cost will be minimum.
[tex]\therefore C''(x)|_{x=20}= 0.2+\frac{3200}{20^3}>0[/tex]
The cost of manufacturing the box is minimum when x=20 cm.
Then the height h [tex]=\frac{800}{20^2}[/tex]
=2 cm.
The dimensions of the box is 20 cm by 20 cm by 2 cm.
Now putting the value of x in (1)
[tex]\therefore C(20)= 0.1\times 20^2+\frac{1600}{20}[/tex]
=120 cent.
The minimum cost of manufacturing the box is 120 cents.
