Answer:
Cost of material is $245.31
Step-by-step explanation:
Let dimension of box,
The length of this base is twice the width.
∴ x = 2y
Volume of box = 10 m³
∴ xyz = 10
⇒ 2y²z = 10
⇒ y²z = 5
[tex]\Rightarrow z=\dfrac{5}{y^2}\ \ \ ...(i)[/tex]
Material for the base costs $15 per square meter.
Total cost of base = 15xy
Total cost of base = 30y² [∵ x = 2y ]
Material for the sides costs $9 per square meter.
Total cost of side = 9(2xz+2zy)
Total cost of side = 18(xz+yz)
Total cost of material for container = 30y² + 18(xz+yz)
[tex]C(y)=30y^2+18(2y\cdot \dfrac{5}{y^2}+y \dfrac{5}{y^2})[/tex] [From (i)]
[tex]C(y)=30y^2+\dfrac{270}{y}[/tex]
Differentiate w.r.t y
[tex]C'(y)=60y-\dfrac{270}{y^2}[/tex]
For critical point , C'(y)=0
[tex]60y-\dfrac{270}{y^2}=[/tex]
[tex]y=\sqrt[3]{\dfrac{9}{2}}[/tex]
[tex]x=2y=2\sqrt[3]{\dfrac{9}{2}}[/tex]
[tex]z=\dfrac{5}{y^2}=5\sqrt[3]{\dfrac{4}{81}}[/tex]
The minimum cost of container material at [tex]y=\sqrt[3]{\dfrac{9}{2}}[/tex]
[tex]C_{min}=30\sqrt[3]{\dfrac{81}{4}}+270\sqrt[3]{\dfrac{2}{9}}[/tex]
[tex]C_{min}=245.31[/tex]
Hence, the cheapest cost of material for container is $245.31
