Answer:
106.71 cm²
Step-by-step explanation:
Let 'h' be the height of the container. If the width is x cm and the length is 2x cm, then the volume and surface area are given by:
[tex]V=x*2x*h=2hx^2\\100=2hx^2\\A=2*(xh)+2*(2x*h)+(2x*x)\\A=6xh+2x^2[/tex]
Rewriting the area function as a function of 'x':
[tex]100=2hx^2\\h=\frac{50}{x^2} \\A=6xh+2x^2\\A=\frac{300}{x}+2x^2[/tex]
The value of 'x' for which the derivate of the area function is zero, is the one that yields the minimum surface area:
[tex]A=\frac{300}{x}+2x^2\\\frac{dA}{dx}=0=\frac{-300}{x^2}+4x\\4x^3=300\\x=4.217 cm[/tex]
Therefore, the minimum area is:
[tex]A_{min}=\frac{300}{4.217}+2*(4.217^2) \\A_{min}= 106.71\ cm^2[/tex]
The container will have a minimum surface area of 106.71 cm²
