The tennis club charges each of its 80 members $100 per month. The owner estimates that for each decrease of $1 in the monthly fee, two new people will join. What monthly fee will generate the most income for the owners?
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okay so you need to develop a formula
so both the price and number of members is dependent on the fee...
the income is the number of members times the fee [i=mf] where i is income m is members and f is fee
m is 80+2n where n is the number of dollars dropped
f is 100-n
we substitute these two equations into the original i=mf and we get i=[80+2n][100-n]
we expand this into the equation i=8000-80n+200n-2n^2
the amount the price changes is dependent on the variables so we can isolate this from the equation:
the final income is greatest when 120n-2n^2 is greatest. this is a negative quadratic so we can find the vertex and the n value for the point [x] will be our answer
the n value is unchanged by the 8000 so we can make the problem simpler by removing it
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