Answer:
[tex]V = 6746.30cm^3[/tex]
Step-by-step explanation:
Given
[tex]Surface\ Area = 1700cm^2[/tex]
Required
Determine the largest possible volume
Represent the dimension with:
[tex]Height = h[/tex]
[tex]Sides = x[/tex]
Surface Area with an open top is:
[tex]Surface\ Area = x^2+ 4hx[/tex]
Substitute 1700 for Surface Area
[tex]1700 = x^2+ 4hx[/tex]
Make h the subject
[tex]4hx = 1700 - x^2[/tex]
[tex]h = \frac{1700 - x^2}{4x}[/tex]
The volume (V) is calculated as:
[tex]V = x^2h[/tex]
Substitute value for h in [tex]V = x^2h[/tex]
[tex]V = x^2*\frac{1700 - x^2}{4x}[/tex]
[tex]V = x*\frac{1700 - x^2}{4}[/tex]
[tex]V = \frac{1700x - x^3}{4}[/tex]
[tex]V = \frac{1700x}{4} - \frac{x^3}{4}[/tex]
[tex]V = 425x - \frac{x^3}{4}[/tex]
[tex]V = 425x - \frac{1}{4}x^3[/tex]
Differentiate and equate to 0, afterwards
[tex]V' =425 - \frac{3}{4}x^2[/tex]
[tex]425 - \frac{3}{4}x^2 = 0[/tex]
[tex]\frac{3}{4}x^2 = 425[/tex]
Solve for [tex]x^2[/tex]
[tex]x^2 = 425 * \frac{4}{3}[/tex]
[tex]x^2 = 566.67[/tex]
Take the square root of both sides
[tex]x = \sqrt{566.67[/tex]
[tex]x = 23.80[/tex]
Substitute 23.80 for x in [tex]h = \frac{1700 - x^2}{4x}[/tex]
[tex]h = \frac{1700 - 23.80^2}{4*23.80}[/tex]
[tex]h = \frac{1133.56}{95.20}[/tex]
[tex]h = 11.91[/tex]
So, the largest possible volume is:
[tex]V = x^2h[/tex]
[tex]V = 23.80^2 * 11.91[/tex]
[tex]V = 6746.30cm^3[/tex]
