Question

If 300 cm2 of material is available to make a box with a square base and an open top, find the maximum volume of the box in cubic centimeters. Answer to the nearest cubic centimeter without commas.

Answer 1

Area of the box:

A = 2 a^2 + 4 a h = 300 cm^2

h = a ( the box is a cube )

A = 3 a^2 = 300 cm^2

a^2 = 100

A = 10 cm

V = a^3 = 10^3 = 1000 cm^3

Answer 2

Answer:

Maximum volume of box is 500 cubic centimeters.

Step-by-step explaination:

Given 300 $cm^{2}$ of material available to make a box with a square base and an open top that means given area of box is 300 $cm^{2}$. Now, we have to find the maximum volume of box.

$Area=300cm^{2}$

Now, area of box with square box and open top = area square base + 4ab

                                                     = $a^{2}+4ab$

⇒ $a^{2}+4ab$ = 300

⇒ $b=\frac{300-a^{2} }{4a}$    →  (1)

Now, Volume of box  i.e $V=a^{2}b$

Using eq (1),                     $V=a^{2}\frac{300-a^{2} }{4a}$

                                                  = $\frac{1}{4}(300a-a^{2})$

To find maximum volume differentiate above eq w.r.t aand then and equate to 0, we get

 $\frac{dV}{da}=\frac{1}{4}(300-3a^{2})=0$

⇒         $3a^{2}=300$

⇒           $a^{2}=100$ ⇒ a=10

⇒                 $b=\frac{300-100}{40}=5$

hence, Volume=100(5)=500 $cm^{2}$