Respuesta :
Answer:
The largest possible volume of the box is 500,000 cm³
Step-by-step explanation:
Let the a side of the square base = x
let the height of the box, = h
volume of the box, v = hx²
Surface area of the box, A = 4hx + x²
30000 = 4hx + x²
make h the subject of the formula;
[tex]4hx + x^2 = 30000\\\\4hx = 30000 -x^2\\\\h = \frac{30000 -x^2}{4x}[/tex]
Volume of the box, V = hx²
[tex]V = h*x^2 = (\frac{30000-x^2}{4x} )x^2\\\\V = \frac{30000x^2 -x^4}{4x} \\\\V = \frac{30000x^2}{4x} - \frac{x^4}{4x} \\\\V = 7500x- \frac{x^3}{4}[/tex]
At maximum value of V, dv/dx = 0
[tex]\frac{dv}{dx} = 7500 - \frac{3}{4} x^2\\\\0 = 7500 - \frac{3}{4} x^2\\\\\frac{3}{4} x^2 = 7500\\\\x^2 = 10000\\\\x = \sqrt{10000} \\\\x = 100[/tex]
Substitute in the value of x to determine the maximum volume, V;
[tex]V = 7500x - \frac{x^3}{4} \\\\V = 7500(100) - \frac{1}{4} (100)^3\\\\V = 750000- 250000\\\\V = 500,000 \ cm^3[/tex]
Therefore, the largest possible volume of the box is 500,000 cm³
The largest possible volume of the open box given with a surface area of 30000 cm² is;
V = 500000 cm³
Since the base of the open box is a square, then;
Length = x
width = x
Let height be h
Thus, Surface area of the open box is;
A = x² + 4hx
We are told that 30000 cm² of material is available to make the box.
Thus;
x² + 4hx = 30000
Making h the subject gives;
h = (30000 - x²)/4x
Now, formula for volume of a box is;
Volume = length × width × height
Thus;
V = x²h
put (30000 - x²)/4x for h to get;
V = x²((30000 - x²)/4x
V = 7500x - x³/4
Largest possible volume will have dimensions at dV/dx = 0
Thus;
dV/dx = 7500 - 3x²/4
at dV/dx = 0
7500 - 3x²/4 = 0
Thus, x = 100 cm
Put 100 for x in V = 7500x - x³/4 to get;
V = 7500(100) - (100)³/4
V = 500000 cm³
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