Given: A box with a square base and volume of 1200 cubic inches
To Determine: The dimension of the box
Solution
The box has a shape of a rectangular prism.
The volume of a box can calculated by the formula below
[tex]\begin{gathered} Volume(box)=base-area\times height \\ Area(square-base)=l^2 \\ Where:l=side-length \\ So, \\ Volume(box)=l^2h \end{gathered}[/tex][tex]\begin{gathered} Volume(box)=1200in^3 \\ Therefore \\ l^2h=1200 \\ h=\frac{1200}{l^2} \end{gathered}[/tex]The area of the box is the addition of the area of the top and the bottom and the 4 sides
The top and the bottom are square in shape and the sides have the shape of a rectangle
Therefore, the area of the top and and bottom is
[tex]\begin{gathered} Area(bottom)=l^2 \\ Area(top)=l^2 \\ Area-of-sides=4\times h\times l=4hl \end{gathered}[/tex][tex]\begin{gathered} Area(top,and,bottom)=2\times area(sides)===given \\ l^2+l^2=2\times4hl \\ 2l^2=8hl \\ h=\frac{2l^2}{8l} \\ h=\frac{l}{4} \end{gathered}[/tex]Since h is same, therefore
[tex]\begin{gathered} \frac{1200}{l^2}=\frac{l}{4} \\ l^3=4800 \\ l=\sqrt[3]{4800} \\ l=16.87in \end{gathered}[/tex][tex]\begin{gathered} h=\frac{l}{4} \\ h=\frac{16.87}{4} \\ h=4.22in \end{gathered}[/tex]Hence, the dimension of the box are 16.87 inches by 16.87 inches by 4.22 inches
