Question

if 10 800 cm2 of material is available to make a box with a square base and an open top find the largest possible volume of the box.

Answer 1

Let

x--------> the length side of the square base of the box

y-------> the height of the box

we know that

The surface area of the box is equal to

$SA=4xy+x^{2}$

$SA=10,800\ cm^{2}$

so

$10,800=4xy+x^{2}$

$y=(10,800-x^{2})/(4x)$

$y=(2,700-0.25x^{2})/(x)$ --------> equation A

the volume of the box is equal to

$V=x^{2}y$ --------> equation B

Substitute the equation A in the equation B

$V=x^{2}*(2,700-0.25x^{2})/(x)$

$V=x*(2,700-0.25x^{2})$

$V=(2,700x-0.25x^{3})$

using a graphing tool

see the attached figure

For $x=60\ cm$

$Volume=108,000\ cm^{3}$

the point $(60,108,000)$ is a maximum of the function

Find the dimensions of the box

$x=60\ cm$

Find the value of y

$V=x^{2}y$

$y=V/x^{2}$

$y=108,000/60^{2}$

$y=30\ cm$

The dimensions of the box are

$60\ cm*60\ cm*30\ cm$

The largest possible volume of the box is

$108,000\ cm^{3}$

Answer 2

The largest possible volume of the box is $\boxed{108000{\text{ c}}{{\text{m}}^3}}.$

Further explanation:

Given:

The area of the material is $10800{\text{ c}}{{\text{m}}^2}.$

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,

$\boxed{{\text{Surface Area}} = 4xy + {x^2}}$

The surface area of the box is $10800{\text{ c}}{{\text{m}}^2}.$

$\begin{aligned}4xy + {x^2}&= 10800\\4xy&= 10800 - {x^2}\\y&= \frac{{10800 - {x^2}}}{{4x}}\\y&=\frac{{2700 - 0.25{x^2}}}{x}\\\end{aligned}$

The volume of the box can be expressed as follows,

$\begin{aligned}V&= {x^2}y\\&= {x^2}\times \left({\frac{{2700 - 0.25{x^2}}}{x}} \right)\\&= \left( x \right)\times \left({2700 - 0.25{x^2}} \right)\\&= 2700x - 0.25{x^3}\\\end{aligned}$

Differentiate the volume with respect to “x”.

$\begin{aligned}\frac{{dV}}{{dx}}&= \frac{d}{{dx}}\left({2700x - 0.25{x^3}}\right)\\&= 2700 - 0.75{x^2}\\\end{aligned}$

Substitute 0 for $\dfrac{{dV}}{{dx}}$ in above equation to obtain the value of x.

$\begin{aligned}0&= 2700 - 0.75{x^2}\\0.75{x^2} &= 2700\\{x^2}&= \frac{{2700}}{{0.75}}\\{x^2}&= 3600\\x&= 60\\\end{aligned}$

The side of the base is $60{\text{ cm}}.$

The height of the box can be obtained as follows,

$\begin{aligned}y&= \frac{{2700 - 0.25{{\left( {60} \right)}^2}}}{{60}}\\&=\frac{{2700 - 900}}{{60}}\\&=\frac{{1800}}{{60}}\\&=30\\\end{aligned}$

The height of the box is $y = 30{\text{ cm}}.$

The volume of the box can be calculated as follows,

$\begin{aligned}V&={\left(60}\right)^2}\times\left({30}\right)\\&=3600\times30\\&=108000 \\\end{aligned}$

The largest possible volume of the boxis  $\boxed{108000{\text{ c}}{{\text{m}}^3}}$.

Learn more:

1. Learn more about inverse of the functionhttps://brainly.com/question/1632445.

2. Learn more about equation of circle brainly.com/question/1506955.

3. Learn more about range and domain of the function brainly.com/question/3412497

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.