Question

A closed box with a square base of length x and height of length y is to be constructed so that it's surface area is S=12(pi)ft^2. Express the volume of the box V, as a function of the width of the box, x

Answer 1

A closed box with a square or rectangular base will be a cuboid. To find the volume of this box let's know the things given in the question,

• Closed box with a square base with each side of 'x' units.
• Height of the box = y units
• Surface area of the box = 12π square feet

Volume of a cuboid will be given by,

     V = Length × Width × Height

Substitute the measures in the expression,

     V = $x\times x\times y$ [Since, base is a square]

     V = $x^2y$ ---------(1)

Since, surface area of a cuboid is given by,

      S = 2(lb + bh + hl)

Here, l = Length, b = width and h = height

Therefore, S = $2(x\times x)+2(x\times y)+2(y\times x)$

                  S = $2(x^2+2xy)$

Now substitute the value of Surface area,

      12π = $2(x^2+2xy)$

      $y=\frac{6\pi-x^2}{2x}$

By substituting the value of y in expression (1),

      $V=\frac{x^2(6\pi-x^2)}{2x}$

         Therefore, expression for the volume of the box will be $V=\frac{x^2(12\pi-x^2)}{2x}$

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