Maths-
General
Easy

Question

A Company is evaluating two packaging options for its product line. The more efficient design will have the lesser ratio of surface area to volume. Should the company use packages that are cylinders or rectangular prisms ?

hintHint:

Surface area of a cylinder: 2 pi r squared h
Surface area of a rectangular prism:2 left parenthesis w l plus w h plus l h right parenthesis
Volume of a cylinder: 2 pi r left parenthesis h plus r right parenthesis
Volume of a rectangular prism: V equals w h l
We are asked to find which packaging is more efficient, cylinder or prism.

The correct answer is: evident that the cylinder has lesser ratio of surface area to volume. Hence, it is the most efficient packaging


    Step 1 of 3:
    Let x be the side length of the prism and h be the height of the prism.
    The volume of a rectangular prism of given by: V equals w h l, where w ,h and l are the width, height and length respectively.
    Here, the length and width are same since it’s a square. So, the volume is:
    V equals w h l
    equals h left parenthesis x right parenthesis left parenthesis x right parenthesis
    equals h x squared
    The surface area is:
    A equals 2 left parenthesis w l plus h l plus h w right parenthesis
    equals 2 left parenthesis x x plus h x plus h x right parenthesis
    equals 2 open parentheses x squared plus 2 h x close parentheses
    equals 2 x squared plus 4 h x
    Their ratio is:
    fraction numerator 2 x squared plus 4 h x over denominator h x squared end fraction equals fraction numerator 2 left parenthesis x plus 2 h right parenthesis over denominator h x end fraction
    Step 2 of 3:
    The volume of a cylinder is given by:V equals pi r squared h , where r and h are the radius and height of the cylinder respectively.
    Here, the radius is same as the side length of the prism and the height is same as the height of the prism. So, we have;
    V equals pi r squared h
    equals pi x squared h
    The surface area is:
    A equals 2 pi r h plus 2 pi r squared
    equals 2 pi x h plus 2 pi x squared
    equals 2 pi x left parenthesis h plus x right parenthesis
    Their ratio is:
    fraction numerator 2 pi x left parenthesis h plus x right parenthesis over denominator pi x squared h end fraction equals fraction numerator 2 left parenthesis h plus x right parenthesis over denominator h x end fraction
    Step 3 of 3:
    Comparing the ratios,fraction numerator 2 pi x left parenthesis h plus x right parenthesis over denominator pi x squared h end fraction equals fraction numerator 2 left parenthesis h plus x right parenthesis over denominator h x end fraction (cylinder) and fraction numerator 2 x squared plus 4 h x over denominator h x squared end fraction equals fraction numerator 2 left parenthesis x plus 2 h right parenthesis over denominator h x end fraction(rectangle). It is evident that the cylinder has lesser ratio of surface area to volume. Hence, it is the most efficient packaging.

    Ratio between two values can be written as x colon y text  or  end text x over y.

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