Question

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.

Answer 1

Answer:L=9.3 ft

b=9.3 ft

h=42.14 ft

Step-by-step explanation:

Given

volume(V)=$3645 ft^3$

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

Bottom cost$(C_1)$=5Lb

Side Costs$(C_2)$=8Lh+8bh

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

C=$5\times \frac{3645}{h}+8h\left ( L+b\right )$

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 $b^2=\frac{3645}{h}$

C=$5b^2+\frac{16\times 3645}{b}$

For minimum cost differentiate w.r.t b

$\frac{\mathrm{d}C}{\mathrm{d} b}=10b-\frac{16\times 3645}{b}$

$\frac{\mathrm{d}C}{\mathrm{d} b}=0$

$10b-\frac{16\times 3645}{b}=0$

$b=9.29\approx 9.3 ft$

L=9.3 ft

h=42.14 ft