Respuesta :
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 = [tex]x\times x\times y[/tex] [Since, base is a square]
V = [tex]x^2y[/tex] ---------(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 = [tex]2(x\times x)+2(x\times y)+2(y\times x)[/tex]
S = [tex]2(x^2+2xy)[/tex]
Now substitute the value of Surface area,
12π = [tex]2(x^2+2xy)[/tex]
[tex]y=\frac{6\pi-x^2}{2x}[/tex]
By substituting the value of y in expression (1),
[tex]V=\frac{x^2(6\pi-x^2)}{2x}[/tex]
Therefore, expression for the volume of the box will be [tex]V=\frac{x^2(12\pi-x^2)}{2x}[/tex]
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