Answer:
Minimum surface area =[tex]70.77 cm^2[/tex]
Step-by-step explanation:
We are given that
Width of container=x cm
Length of container=2x cm
Volume of container=[tex]54 cm^3[/tex]
We have to find the minimum surface areas that this container will have.
Volume of container=[tex]l\times b\times h[/tex]
[tex]x\times 2x\times h=54[/tex]
[tex]2x^2h=54[/tex]
[tex]h=\frac{54}{2x^2}=\frac{27}{x^2}[/tex]
Surface area of container=[tex]2(b+l)h+lb[/tex]
Because the container does not have lid
Surface area of container=[tex]S=2(2x+x)\times \frac{27}{x^2}+2x\times x[/tex]
[tex]S=\frac{162}{x}+2x^2[/tex]
Differentiate w.r.t x
[tex]\frac{dS}{dx}=-\frac{162}{x^2}+4x[/tex]
[tex]\frac{dx^n}{dx}=nx^{n-1}[/tex]
Substitute [tex]\frac{dS}{dx}=0[/tex]
[tex]-\frac{162}{x^2}+4x=0[/tex]
[tex]4x=\frac{162}{x^2}[/tex]
[tex]x^3=\frac{162}{4}=40.5[/tex]
[tex]x^3=40.5[/tex]
[tex]x=(40.5)^{\frac{1}{3}}[/tex]
[tex]x=3.4[/tex]
Again differentiate w.r.t x
[tex]\frac{d^2S}{dx^2}=\frac{324}{x^3}+4[/tex]
Substitute x=3.4
[tex]\frac{d^2S}{dx^2}=\frac{324}{(3.4)^3}+4=12.24>0[/tex]
Hence, function is minimum at x=3.4
Substitute x=3.4
Then, we get
Minimum surface area =[tex]\frac{162}{(3.4)}+2(3.4)^2=70.77 cm^2[/tex]
