The largest possible volume of the box is [tex]\boxed{108000{\text{ c}}{{\text{m}}^3}}.[/tex]
Further explanation:
Given:
The area of the material is [tex]10800{\text{ c}}{{\text{m}}^2}.[/tex]
Explanation:
Consider the base length of the square box as “x”.
Consider the height of the box as “y”.
The surface area of the open box can be expressed as follows,
[tex]\boxed{{\text{Surface Area}} = 4xy + {x^2}}[/tex]
The surface area of the box is [tex]10800{\text{ c}}{{\text{m}}^2}.[/tex]
[tex]\begin{aligned}4xy + {x^2}&= 10800\\4xy&= 10800 - {x^2}\\y&= \frac{{10800 - {x^2}}}{{4x}}\\y&=\frac{{2700 - 0.25{x^2}}}{x}\\\end{aligned}[/tex]
The volume of the box can be expressed as follows,
[tex]\begin{aligned}V&= {x^2}y\\&= {x^2}\times \left({\frac{{2700 - 0.25{x^2}}}{x}} \right)\\&= \left( x \right)\times \left({2700 - 0.25{x^2}} \right)\\&= 2700x - 0.25{x^3}\\\end{aligned}[/tex]
Differentiate the volume with respect to “x”.
[tex]\begin{aligned}\frac{{dV}}{{dx}}&= \frac{d}{{dx}}\left({2700x - 0.25{x^3}}\right)\\&= 2700 - 0.75{x^2}\\\end{aligned}[/tex]
Substitute 0 for [tex]\dfrac{{dV}}{{dx}}[/tex] in above equation to obtain the value of x.
[tex]\begin{aligned}0&= 2700 - 0.75{x^2}\\0.75{x^2} &= 2700\\{x^2}&= \frac{{2700}}{{0.75}}\\{x^2}&= 3600\\x&= 60\\\end{aligned}[/tex]
The side of the base is [tex]60{\text{ cm}}.[/tex]
The height of the box can be obtained as follows,
[tex]\begin{aligned}y&= \frac{{2700 - 0.25{{\left( {60} \right)}^2}}}{{60}}\\&=\frac{{2700 - 900}}{{60}}\\&=\frac{{1800}}{{60}}\\&=30\\\end{aligned}[/tex]
The height of the box is [tex]y = 30{\text{ cm}}.[/tex]
The volume of the box can be calculated as follows,
[tex]\begin{aligned}V&={\left(60}\right)^2}\times\left({30}\right)\\&=3600\times30\\&=108000 \\\end{aligned}[/tex]
The largest possible volume of the boxis [tex]\boxed{108000{\text{ c}}{{\text{m}}^3}}[/tex].
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of Derivatives
Keywords: square, box, material, square base, volume of the box, largest, open from the top derivative, surface area.
