Answer:
The dimensions of the box are 5.67 cm by 5.67 cm by 8.51 cm.
The total minimum cost = 28.97 cents.
Step-by-step explanation:
Let the base dimensions are a cm by a cm and the height is h cm.
So, a²h = 274 ............. (1)
And, total cost, C = 0.3a² + 0.1 × 4ah = 0.3a² + 0.4ah
C = 0.3a² + 0.4 × (274/a) ................. (2)
Now, for minimum total cost, the condition is [tex]\frac{dC}{da} = 0 = 0.6a - \frac{0.4 \times 274}{a^{2} }[/tex]
⇒ [tex]a^{3} = \frac{0.4 \times 274}{0.6} = 182.67[/tex]
⇒ a = 5.67 cm
So, [tex]h = \frac{274}{a^{2}} = 8.51[/tex] cm.
Therefore, the dimensions of the box are 5.67 cm by 5.67 cm by 8.51 cm.
And the total minimum cost = [tex]C_{min} = 0.3 (5.67)^{2} + 0.4 \times \frac{274}{5.67} = 28.97[/tex] cents. (Answer)
