Question

If 30,000 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box. g

Answer 1

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;

     $4hx + x^2 = 30000\\\\4hx = 30000 -x^2\\\\h = \frac{30000 -x^2}{4x}$

Volume of the box, V = hx²

$V = h*x^2 = (\frac{30000-x^2}{4x} )x^2\\\\V = \frac{30000x^2 -x^4}{4x} \\\\V = \frac{30000x^2}{4x} - \frac{x^4}{4x} \\\\V = 7500x- \frac{x^3}{4}$

At maximum value of V, dv/dx = 0

$\frac{dv}{dx} = 7500 - \frac{3}{4} x^2\\\\0 = 7500 - \frac{3}{4} x^2\\\\\frac{3}{4} x^2 = 7500\\\\x^2 = 10000\\\\x = \sqrt{10000} \\\\x = 100$

Substitute in the value of x to determine the maximum volume, V;

$V = 7500x - \frac{x^3}{4} \\\\V = 7500(100) - \frac{1}{4} (100)^3\\\\V = 750000- 250000\\\\V = 500,000 \ cm^3$

Therefore, the largest possible volume of the box is 500,000 cm³