A trash company is designing an open-top, rectangular container that will have a volume of 3645 ft cubed. The cost of making the bottom of the container is $5 per square foot, and the cost of the sides is $4 per square foot. Find the dimensions of the container that will minimize total cost.

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nuuk nuuk · Answer 1 · 2019-08-06T12:23:14+02:00

Answer:L=9.3 ft

b=9.3 ft

h=42.14 ft

Step-by-step explanation:

Given

volume(V)=[tex]3645 ft^3[/tex]

let L,b,h be length ,breadth and height of cube

Bottom cost[tex](C_1)[/tex]=5Lb

Side Costs[tex](C_2)[/tex]=8Lh+8bh

Total cost(C)=5Lb+8Lh+8bh

C=[tex]5\times \frac{3645}{h}+8h\left ( L+b\right )[/tex]

considering to be fixed ,cost become the function of L+b

and if h is fixed then Lb is also fixed and for cost to be minimum L+b should be minimum therefore L=b is necessary

thus [tex]b^2=\frac{3645}{h}[/tex]

C=[tex]5b^2+\frac{16\times 3645}{b}[/tex]

For minimum cost differentiate w.r.t b

[tex]\frac{\mathrm{d}C}{\mathrm{d} b}=10b-\frac{16\times 3645}{b}[/tex]

[tex]\frac{\mathrm{d}C}{\mathrm{d} b}=0[/tex]

[tex]10b-\frac{16\times 3645}{b}=0[/tex]

[tex]b=9.29\approx 9.3 ft[/tex]

L=9.3 ft

h=42.14 ft