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.
Accepted Solution
A:
Answer:L=9.3 ftb=9.3 fth=42.14 ftStep-by-step explanation:Given volume(V)=[tex]3645 ft^3[/tex]let L,b,h be length ,breadth and height of cubeBottom cost[tex](C_1)[/tex]=5LbSide Costs[tex](C_2)[/tex]=8Lh+8bhTotal cost(C)=5Lb+8Lh+8bhC=[tex]5\times \frac{3645}{h}+8h\left ( L+b\right )[/tex]considering to be fixed ,cost become the function of L+band if h is fixed then Lb is also fixed and for cost to be minimum L+b should be minimum therefore L=b is necessarythus [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