SOLUTION: A clyinderical can of radius xcm has volume 144cm^3. the cost of producing the can is determined by it's surface area. (A) show that the height of the can is h=144/piex^2. (B

Algebra ->  Surface-area -> SOLUTION: A clyinderical can of radius xcm has volume 144cm^3. the cost of producing the can is determined by it's surface area. (A) show that the height of the can is h=144/piex^2. (B      Log On


   

Question 484460: A clyinderical can of radius xcm has volume 144cm^3. the cost of producing the can is determined by it's surface area.
(A) show that the height of the can is h=144/piex^2.
(B) find an expression for the surface area of the can
(C) find the dimensions that will minimise the cost of production.
All questions will help a lot, thanks!
Answer by htmentor(1343) About Me  (Show Source):
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A clyinderical can of radius xcm has volume 144cm^3. the cost of producing the can is determined by it's surface area.
(A) show that the height of the can is h=144/piex^2.
(B) find an expression for the surface area of the can
(C) find the dimensions that will minimise the cost of production.
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The volume of the can is V = pi%2Ax%5E2%2Ah where h = the height of the can
A) Solve for h in the above equation and substitute the value given for V:
h+=+144%2F%28pi%2Ax%5E2%29
B) The surface area of the can is given by S+=+2%2Api%2Ax%2Ah+%2B+2%2Api%2Ax%5E2
Substitute the expression for h derived in A) above:
S+=+2%2Api%2Ax%2A144%2F%28pi%2Ax%5E2%29+%2B+2%2Api%2Ax%5E2
S+=+288%2Fx+%2B+2%2Api%2Ax%5E2
C) The surface area will be a minimum where dS/dx = 0
dS%2Fdx+=+0+=+-288%2Fx%5E2+%2B+4%2Api%2Ax
Solve for x:
Multiply through by x%5E2
4%2Api%2Ax%5E3+=+288
x+=+%2872%2Fpi%29%5E%281%2F3%29
The value for h can then be obtained from the expression derived in A).