7. A national charity is planning a fund-raising campaign in a major United States city having a population of 2 million. The percentage of the population who will make a donation is estimated by the function R = 1 – e(-0.025x) Where R equals the percentage of the population and x equals the number of days the campaign is conducted. Past experience indicates that the average contribution in this city is $3.25 per donor. Costs of the campaign are estimated at $11250 per day. a. How many days should the campaign be conducted if the objective is to maximize net proceeds (total contributions minus total costs) from the campaign? b. What are maximum net proceeds expected to equal? What percentage of the population is expected to donate?
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Number of donors=Rp where p is the population size (P=2000000). From this number of donors amount A=$Rpc is raised where c (c=$3.25) is the contribution per donor on average. 

Net profit P=A-Cx where C is the costs per day (C=$11250/day).

(a) P(x)=(1-e⁻⁰˙⁰²⁵ˣ)(2000000)(3.25)-11250x=

6500000(1-e⁻⁰˙⁰²⁵ˣ)-11250x.

dP/dx=0.025(6500000)e⁻⁰˙⁰²⁵ˣ-11250=0 at maximum.

(d²P/dx²<0 indicating maximum.)

162500e⁻⁰˙⁰²⁵ˣ=11250,

-0.025x=ln(11250/162500)=-2.6703 approx.

Therefore, x=107 days approx. (P(107)=$4,848,355.62.) The campaign needs to run for 107 days for maximum profit.

(b) Maximum net proceeds are expected to come to about $4,848,356. R=0.9311 or 93.11% of the population.

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