Answer:
dimensions to minimize cost is 6 inches x 6 inches x 6 inches
Step-by-step explanation:
Since the box has a square bottom, then it means length and width are the same value. Let the length and width be x. Let the depth by y.
Thus;
Volume is; V = x²y
We are given volume as 216 in³
Thus, V = 216
x²y = 216
y = 216/x²
Surface area of box will be;
S = 2x² + 4xy
Since box is to be made of sheet of paper that coast 1 cent per square inch.
It's means per Sq.m is $0.01
Thus;
C(x) = 2 × 0.01(x²) + 4 × 0.01(xy)
C = 0.02x² + 0.04xy
Put 216/x² for y;
C = 0.02x² + 0.04x(216/x²)
C = 0.02x² + 8.64/x
dC/dx = 0.04x - 8.64/x²
At dC/dx = 0, cost is minimum
Thus;
0.04x - 8.64/x² = 0
0.04x = 8.64/x²
x³ = 8.64/0.04
x³ = 216
x = 6
From y = 216/x²
y = 216/6²
y = 6
Thus,dimensions to minimize cost is 6 inches x 6 inches x 6 inches
