Q:

If a total of 1700 square centimeters of material is to be used to make a box with a square base and open top, find the largest possible volume of such a box?

Accepted Solution

A:
Answer: [tex]V_{(max)}=6744.60 cm^{3}[/tex]Step-by-step explanation:Here we have to find the value for [tex]x[/tex] in order to have the maximum volume ([tex]V_{max}[/tex]) of the box, where the area of the box is given by:[tex]A_{(x)}=1700=x^{2} + 4hx[/tex] (1)And the volume:[tex]V_{(x)}=(x)(x)(h)=x^{2}h[/tex] (2)Isolating [tex]h[/tex] from (1):[tex]h=\frac{1700-x^{2}}{4x}[/tex] (3)Substituting (3) in (2):[tex]V_{(x)}=(x)(x)(h)=x^{2}(\frac{1700-x^{2}}{4x})=-\frac{1}{4}x^{3}+425 x[/tex] (4)So, we have to derive this expression, then make it equal to zero and solve the equation to find the critical point:[tex]V_{x}^{'}=-\frac{3}{4}x^{2} +425[/tex] (5)Making [tex]V_{h}^{'}=0[/tex]:[tex]-\frac{3}{4}x^{2} +425=0[/tex] (6)The critical point is:[tex]x=23.80 cm[/tex] (7)This means the maximum Volume is when [tex]x=23.80 cm[/tex]. Now with this value we can find [tex]h[/tex]:[tex]h=\frac{1700 cm^{2}-(23.80 cm)^{2}}{4(23.80 cm)}=11.907 cm[/tex] (8)Hence:[tex]V_{(max)}=(23.8 cm)^{2}(11.90 cm)=6744.60 cm^{3}[/tex] (9) This is the maximum volume of the box