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

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=(300-a^(2) )/(4a)` → (1)

Now, Volume of box i.e
`V=a^(2)b`

Using eq (1),
`V=a^(2)(300-a^(2) )/(4a)`

=
`(1)/(4)(300a-a^(2))`

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

`(dV)/(da)=(1)/(4)(300-3a^(2))=0`

⇒
`3a^(2)=300`

⇒
`a^(2)=100` ⇒ a=10

⇒
`b=(300-100)/(40)=5`

hence, Volume=100(5)=500
`cm^(2)`