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Calculus/Two tough calculus max and min problems???

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Expert: - 4/16/2008


Question
I have been having trouble with this section. especially this two word problems with max and min.

1. The annual cost C (in dollars) of a natural gas pipeline of radius r (in feet) is given approximately by C= ar(squared)+br(to the -5) where a and b are constants.  Determine the radius in terms of a and b for the annual cost is a minimum?

2. The efficiency E of a screw in mechanics is given as a function of H by E=h(1 - 0.5h)/(h+.05) where h is the tangent of the angle of pitch of the screw.  For what value of h is E a maximum?

If anyone can help thanks a lot!!

Answer
1. It can be written as ar^2 + b/r^5.  Take a derivative in terms of r and set it equal to 0 to find any extreme points.  The second derivative should be negative if we are looking for a maximum.

2. Multiply the top together to get f(h)/g(h).  

   (that g(h)=h+.05 and f(h)=h-0.5h^2.)

Differentiate using the quotient rule and set the top of the fraction equal to 0.  Plug back into the equation to find E.

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