Respuesta :
Answer:
Maximum volume of box is 500 cubic centimeters.
Step-by-step explaination:
Given 300 [tex]cm^{2}[/tex] of material available to make a box with a square base and an open top that means given area of box is 300 [tex]cm^{2}[/tex]. Now, we have to find the maximum volume of box.
[tex]Area=300cm^{2}[/tex]
Now, area of box with square box and open top = area square base + 4ab
= [tex]a^{2}+4ab[/tex]
⇒ [tex]a^{2}+4ab[/tex] = 300
⇒ [tex]b=\frac{300-a^{2} }{4a}[/tex] → (1)
Now, Volume of box i.e [tex]V=a^{2}b[/tex]
Using eq (1), [tex]V=a^{2}\frac{300-a^{2} }{4a}[/tex]
= [tex]\frac{1}{4}(300a-a^{2})[/tex]
To find maximum volume differentiate above eq w.r.t aand then and equate to 0, we get
[tex]\frac{dV}{da}=\frac{1}{4}(300-3a^{2})=0[/tex]
⇒ [tex]3a^{2}=300[/tex]
⇒ [tex]a^{2}=100[/tex] ⇒ a=10
⇒ [tex]b=\frac{300-100}{40}=5[/tex]
hence, Volume=100(5)=500 [tex]cm^{2}[/tex]
