Answer:
The volume V of the box can be expressed as a function of the length of its base x as:
[tex]\[ V(x) = \frac{2000x - x^3}{2} \][/tex]
Step-by-step explanation:
Let's denote the length of the base of the box as \( x \) centimeters. Since the base is square, the width and height of the box are also \( x \) centimeters.
The surface area A of the box is given by the formula:
[tex]\[ A = 2lw + 2lh + 2wh \][/tex]
Given that the length of the base is x centimeters, we can substitute[tex]\( x \) for \( l \), \( w \), and \( h \):[/tex]
[tex]\[ 4000 = 2x^2 + 2xh + 2xh \][/tex]
[tex]\[ 4000 = 2x^2 + 4xh \][/tex]
To express the volume V of the box as a function of the length of its base x, we need to first solve for the height h in terms of x, and then use it to express the volume.
From the equation [tex]\( 4000 = 2x^2 + 4xh \), we can solve for \( h \)[/tex]:
[tex]\[ 4000 = 2x^2 + 4xh \][/tex]
[tex]\[ 2000 = x^2 + 2xh \][/tex]
[tex]\[ 2xh = 2000 - x^2 \][/tex]
[tex]\[ h = \frac{2000 - x^2}{2x} \][/tex]
Now, the volume V of the box is given by the formula:
[tex]\[ V = l.w.h \][/tex]
Substituting the expressions for[tex]\( l \), \( w \), and \( h \):[/tex]
[tex]\[ V(x) = x \times x \times \frac{2000 - x^2}{2x} \][/tex]
[tex]\[ V(x) = \frac{x^2(2000 - x^2)}{2x} \][/tex]
[tex]\[ V(x) = \frac{x(2000 - x^2)}{2} \][/tex]
[tex]\[ V(x) = \frac{2000x - x^3}{2} \][/tex]
So, the volume V of the box can be expressed as a function of the length of its base x as:
[tex]\[ V(x) = \frac{2000x - x^3}{2} \][/tex]
