Respuesta :
Answer:
L = W = 6.810 cm
H = 10.22 cm
Explanation:
given data
volume L W H = 474 cm³
sides of the box cost = 2 cent/cm²
top and bottom cost 3 cent/cm²
to find out
dimensions of the box that minimize the total cost of material use
solution
we know here L W H = 474
here L is length and B is width and H is height
so when we minimize the cost function
C( L, W, H ) = (2) 2 H ( L + W ) + (3) 2 L W
so put here H
substitute H = [tex]\frac{474}{LW}[/tex]
we get
C( L, W ) = 1896 ( [tex]\frac{1}{W} + \frac{1}{L}[/tex] ) + 6 LW
so
minimum cost will be when the two partial derivatives is 0
so
[tex]\frac{dC}{dL}[/tex] = 6W - [tex]\frac{1896}{L^2}[/tex] = 0
so
[tex]\frac{dC}{dW}[/tex] = 6L - [tex]\frac{1896}{W^2}[/tex] = 0
L = [tex]\frac{316}{W^2}[/tex]
so
by solving above equation we get
L = W = [tex]({{316})^{1/3}[/tex] = 6.810 cm
and
H = [tex]\frac{474}{6.810^2}[/tex]
H = 10.22 cm
