Answer:
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)/ 4x----------eqn(*)
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
