A cylindrical drum is to be made out of material that costs 5 dollars per square foot for the base and top, and 3 dollars per square foot for the curved sides. What is the least possible cost of a drum with capacity 28 cubic feet?
Can someone help me?
Can someone help me?
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Area=A= 2 pi r^2 + 2 pi r h ( the area of two circles and the curved rectangle).
Volume=V= pi r^2 h = 28 --> h= 28/ (pi r^2)
Sub into area function above
A= 2 pi r^2 + 2 pi r [ 28/ pi r^2] = 2 pi r^2 + 56 / r
Actually, there way no real need to write out the area function as a whole.
Now I will take it apart again.
From the above we can see that
Area of base and top = 2 pi r^2
Area of curved rectangle = 56/r
Now we can come up with a cost function
C(r) = (2 pi r^2)(5) + (56/r)(3) = 10 pi r^2 + 168/r
Differentiate to find min/max points
dC/dr = 20 pi r -168/r^2 =0
20 pi r^3 - 168 =0
r= cube root ( 168 / (20 pi) ) = 1.39 ( 2 dp).
Test that it is a minimum
d^2C/dr^2 = 20 pi + 336/r^3 , which is positive for all r>0 , so we do have a minimum
Thus, minimum cost is C(1.39) = $181.56
Volume=V= pi r^2 h = 28 --> h= 28/ (pi r^2)
Sub into area function above
A= 2 pi r^2 + 2 pi r [ 28/ pi r^2] = 2 pi r^2 + 56 / r
Actually, there way no real need to write out the area function as a whole.
Now I will take it apart again.
From the above we can see that
Area of base and top = 2 pi r^2
Area of curved rectangle = 56/r
Now we can come up with a cost function
C(r) = (2 pi r^2)(5) + (56/r)(3) = 10 pi r^2 + 168/r
Differentiate to find min/max points
dC/dr = 20 pi r -168/r^2 =0
20 pi r^3 - 168 =0
r= cube root ( 168 / (20 pi) ) = 1.39 ( 2 dp).
Test that it is a minimum
d^2C/dr^2 = 20 pi + 336/r^3 , which is positive for all r>0 , so we do have a minimum
Thus, minimum cost is C(1.39) = $181.56
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You must provide some dimensions to work out the areas. Capacity of 28 cubic feet alone is not enough. At least, one more dimension would be helpful