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Question 1036306: Find the dimensions that minimize the surface area for a cone with a volume of 225 cm^3
Answer by addingup(3677)   (Show Source):
You can put this solution on YOUR website! Area of the cylinder assuming it has a top and of course it has a bottom:
A=(2Pi*r*)h + 2(Pi*r^2)
= 2Pi*r*h+2Pi*r^2
Volume of the cylinder:
V=(Pi*r^2)h
=Pi*r^2h
Optimization equation using the formula for the area, since the problem says to minimize the surface area:
Constraint: Must hold a volume of 225cm^3:
225=Pi*r^2*h
h=225/Pi*r^2
A=2Pi*r(225/Pi*r^2)+2Pi*r^2
IN the first equation, Pi in the numerator cancels Pi in the denominator:
A= 2r(225/r^2)And
Again in the first equation, r in the top cancels the square in r^2 in the bottom:
A= 2(225/r)+2Pi*r^2
A=450/r+2Pi*r^2
So, we've simplified the equation as much as possible. Let's take the derivative next:
A=450/r+2Pi*r^2 In the first equation, move the r to the top:
A=450*r^-1+2Pi*r^2
A(prime)=-450*r^-2+4Pi*r
=4Pi*r-450/r^2 Solve for r.
First, find a common denominator. Multiply both sides times r^2. You get:
0=(4Pi*r^3-450)/r^2
∂=4(Pi*r^3-112.5)Divide both sides by 4, then add 112.5 on both sides:
112.5=Pi*r^3
112.5/Pi=r^3
r=cuberoot500/Pi This is the radius that minimizes the surface of the can.
h=225/Pi(cuberoot112.5/Pi)^2
=225/Pi(112.5/Pi)^2/3 This is the height of the can that minimizes the surface.
Run the numbers in your calculator.
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