Answer:
6 in x 6 in x 3 in.
Step-by-step explanation:
Given
V = xyz = 108 ⇒ z = 108/(xy)
The amount of the material used is
S = xy + 2yz + 2xz
Put value of z from the volume
S = xy + 2y*108/(xy) + 2x*108/(xy) = xy + 216/x + 216/y
Now, we find the relative minimum of the function S(x,y)
First, we find the critical point. Set Sx = 0 and Sy = 0
and solve this system:
Sx(x,y) = y - (216/x²) = 0
Sy(x,y) = x - (216/y²) = 0
From the first equation we have
y = 216/x²
Put it in the second equation and find x
x - (216/(216/x²)²) = 0
⇒ x*(1 - (x³/216)) = 0
⇒ x₁ = 0 and x₂ = 6
Now, we can find y as follows
y₁ = 216/(0)² which is undefined
y₂ = 216/(6)² = 6
Hence, the only critical point of S is (6, 6). Next, we calculate the second ordered derivatives that we need for the second derivative test:
Sxx(x,y) = 432/x³
Sxy(x,y) = 1
Syy(x,y) = 432/y³
Applying the second derivative test
D(6, 6) = Sxx(6, 6)*Syy(6, 6) - S²xy(6, 6) = 2*2 - 1² = 4 -1 = 3 > 0
Sxx(6, 6) = 2 > 0
Since D(6, 6) > 0 and Sxx(6, 6) > 0 we can conclude that S has a relative minimum at (6, 6).
z coordinate is:
z = 108/(xy) = 108 / (6*6) = 3
Finally, the dimentions of a box are 6 in x 6 in x 3 in.
