Respuesta :
Answer:
base side = 9.037 inches
height = 6.024 inches
Minimum cost = 196 cents
Step-by-step explanation:
The volume of the bin is given by:
[tex]Volume = side^2 * height[/tex]
and the surface area of the bin is given by:
[tex]Surface\ area = side^2 + 4*side*height[/tex]
The cost of the bin will be:
[tex]Cost = 0.8*side^2 + 0.6*4*side*height[/tex]
[tex]Cost = 0.8*side^2 + 2.4*side*height[/tex]
From the volume equation, we have:
[tex]height = 492 / side^2[/tex]
Now the cost will be:
[tex]Cost = 0.8*side^2 + 2.4*side*492/side^2[/tex]
[tex]Cost = 0.8*side^2 + 1180.8/side[/tex]
To find the side that gives the minimum cost, we can find the derivative of Cost in relation to side and then make it equal zero:
Abbreviating Cost as C and side as s, we have:
[tex]dC/ds = 0.8*2*s - 1180.8/s^2[/tex]
[tex]1.6s - 1180.8/s^2 = 0[/tex]
[tex]1.6s = 1180.8/s^2[/tex]
[tex]1.6s^3 = 1180.8[/tex]
[tex]s^3 = 738[/tex]
[tex]s = 9.037\ in[/tex]
Finding the height of the bin, we have:
[tex]height = 492 / 9.037^2[/tex]
[tex]height = 6.024\ in[/tex]
The minimum cost is:
[tex]Cost = 0.8*9.037^2 + 1180.8/9.037 = 196\ cents[/tex]
