Respuesta :
Answer:
The dimensions that will minimize the cost of the cage are: base 5.85 x 5.85 ft and heght 29.2 ft.
Step-by-step explanation:
We have a cage with dimensions x,y,z, with a fixed volume of 1000 ft3.
The sides cost 5 times less per unit of area than the base and top.
The volume can be written as:
[tex]V=x\cdot y\cdot z=1000[/tex]
The cost function is
[tex]C=top+base+sides=5xy+5xy+(2xz+2yz)[/tex]
The base will be square, so we can simplify as:
[tex]V=x^2z=1000\\\\z=1000/x^2[/tex]
The cost become
[tex]C=10xy+2(xz+yz)=10x^2+4xz=10x^2+4x(\frac{1000}{x^2}) \\\\C=10x^2+4000/x[/tex]
To minimize the cost, we derive the cost function and equal to zero
[tex]dC/dx=20x-4000x^{-2}=0\\\\20x=4000x^{-2}\\\\x^{1+2}=4000/20\\\\x^3=200\\\\x=\sqrt[3]{200}=5.85[/tex]
The base sides are 5.85 ft.
The height of the box (z) is:
[tex]z=1000/x^2=1000/5.85^2=1000/34.2225=29.2[/tex]
