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Calculus/Optimization Problems


An apple grower knows that the current yield per tree is 130 lbs, and the current price is $.73 per pound. Find the maximum revenue possible if the yield increases 5 pounds per week, and the price decreases 2 cents per week.

The changes in the amount of apples and the price is 130 + 5x and 73 - 2x,
where the money is kept in the number of pennies involved.

The amount of money is the amount produced times the sale price.
That makes it (130+5x)(73-2x).  Multiplying them out gives 9490 + 365x - 260x - 10x.
It is customary to put the highest power of x first and combine like terms.
This gives -10x + 105x + 9490.

If f(x) = -10x + 105x + 9490, then f'(x) = -20x + 105.
Setting this equal to 0 gives -20x + 105 = 0, so 20x = 105, so x = 105/20.

Now 105 and 20 are both divisible by 5, so that is x = 21/4.
To make this a proper fraction, that is x = 5 1/4.


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