Answer:
∛18 * ∛18 * 18/(∛18)²
Step-by-step explanation:
Let the surface area of the box be expressed as S = 2(LB+BH+LH) where
L is the length of the box = x
B is the breadth of the box = x
H is the height of the box = h
Substituting this variables into the formula, we will have;
S = 2(x(x)+xh+xh)
S = 2x²+2xh+2xh
S = 2x² + 4xh and the Volume V = x²h
If V = x²h; h = V/x²
Substituting h = V/x² into the surface area will give;
S = 2x² + 4x(V/x²)
Since the volume V = 18cm³
S = 2x² + 4x(18/x²)
S = 2x² + 72/x
Differentiating the function with respect to x to get the minimal point, we will have;
dS/dx = 4x - 72/x²
at dS/dx = 0
4x - 72/x² = 0
- 72/x² = -4x
72 = 4x³
x³ = 72/4
x³ = 18
[tex]x = \sqrt[3]{18}[/tex]
Critical point is at [tex]x = \sqrt[3]{18}[/tex]
If x²h = 18
(∛18)²h =18
h = 18/(∛18)²
Hence the dimension is ∛18 * ∛18 * 18/(∛18)²
