Answer:
Cost of material is $245.31
Step-by-step explanation:
Let dimension of box,
<em>The length of this base is twice the width.</em>
<em>∴ </em>x = 2y
Volume of box = 10 m³
∴ xyz = 10
⇒ 2y²z = 10
⇒ y²z = 5
![\Rightarrow z=\dfrac{5}{y^2}\ \ \ ...(i)](https://tex.z-dn.net/?f=%5CRightarrow%20z%3D%5Cdfrac%7B5%7D%7By%5E2%7D%5C%20%5C%20%5C%20...%28i%29)
<em>Material for the base costs $15 per square meter.</em>
Total cost of base = 15xy
Total cost of base = 30y² [∵ x = 2y ]
<em>Material for the sides costs $9 per square meter.</em>
Total cost of side = 9(2xz+2zy)
Total cost of side = 18(xz+yz)
Total cost of material for container = 30y² + 18(xz+yz)
[From (i)]
![C(y)=30y^2+\dfrac{270}{y}](https://tex.z-dn.net/?f=C%28y%29%3D30y%5E2%2B%5Cdfrac%7B270%7D%7By%7D)
Differentiate w.r.t y
![C'(y)=60y-\dfrac{270}{y^2}](https://tex.z-dn.net/?f=C%27%28y%29%3D60y-%5Cdfrac%7B270%7D%7By%5E2%7D)
For critical point , C'(y)=0
![60y-\dfrac{270}{y^2}=](https://tex.z-dn.net/?f=60y-%5Cdfrac%7B270%7D%7By%5E2%7D%3D)
![y=\sqrt[3]{\dfrac{9}{2}}](https://tex.z-dn.net/?f=y%3D%5Csqrt%5B3%5D%7B%5Cdfrac%7B9%7D%7B2%7D%7D)
![x=2y=2\sqrt[3]{\dfrac{9}{2}}](https://tex.z-dn.net/?f=x%3D2y%3D2%5Csqrt%5B3%5D%7B%5Cdfrac%7B9%7D%7B2%7D%7D)
![z=\dfrac{5}{y^2}=5\sqrt[3]{\dfrac{4}{81}}](https://tex.z-dn.net/?f=z%3D%5Cdfrac%7B5%7D%7By%5E2%7D%3D5%5Csqrt%5B3%5D%7B%5Cdfrac%7B4%7D%7B81%7D%7D)
The minimum cost of container material at ![y=\sqrt[3]{\dfrac{9}{2}}](https://tex.z-dn.net/?f=y%3D%5Csqrt%5B3%5D%7B%5Cdfrac%7B9%7D%7B2%7D%7D)
![C_{min}=30\sqrt[3]{\dfrac{81}{4}}+270\sqrt[3]{\dfrac{2}{9}}](https://tex.z-dn.net/?f=C_%7Bmin%7D%3D30%5Csqrt%5B3%5D%7B%5Cdfrac%7B81%7D%7B4%7D%7D%2B270%5Csqrt%5B3%5D%7B%5Cdfrac%7B2%7D%7B9%7D%7D)
![C_{min}=245.31](https://tex.z-dn.net/?f=C_%7Bmin%7D%3D245.31)
Hence, the cheapest cost of material for container is $245.31