If 2000 square centimeters of material is available to make a box with a square base and an open top, find the largest possible volume of the box. volume=cm^3

8586.7cm^3

Step-by-step explanation:

We can denote the side of the base square as "x" in cm

We can also denote the height as "y" in cm

(x^2 * y) = V(x,y)

The surface area can be written as

(x^2 + 4xy = 2000)

We can make "y" as subject of the formula

4xy = 2000 - x^2

y=( 2000 - x^2)/ 4xeqn(*)

Substitute "y" into V(x,y)

(x*y)= x^2 * (2000-x^2)/4x = V(x)

We can divide the numerator by the denominator

y= x^2 (500/x - x/4)eqn#)

We can take the derivatives

V'(x)= 500 - 3/4x^2

If we set V'(x)=0 we have

500=3/4x^2

2000=3x^2

x^2=666.7

x=√666.7

x=25.8cm

From eqn(*)

y=( 2000 - x^2)/ 4x

y=(2000 - 25.8^2)/(4×25.8)

y=12.9cm

(x^2 * y) = V(x,y)

25.8^2 × 12.9

V(x,y)= 8586.7cm^3

Hence find the largest possible volume of the box is 8586.7cm^3

one that is bigger than 3 1/2