Question 190669
Remember, the revenue is simply the money that a individual/company has made BEFORE any expenses or taxes are accounted for. So this amount of money is solely determined by the amount of products/services that have been sold.



So basically, the revenue is


Revenue = (Price Per Product) x (Number of Products Sold)



In this case, since "a vendor charges p dollars each for rugby shirts", this means that the price is simply "p" and because "he expects to sell 2000-100p", this means that the number of products sold is {{{2000-100p}}}



So just multiply the price by the quantity to get


{{{R(p)=p(2000-100p)}}}



{{{R(p)=2000p-100p^2}}} Distribute



{{{R(p)=-100p^2+2000p}}} Rearrange the terms.



So the polynomial that represents the revenue is {{{R(p)=-100p^2+2000p}}}


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Find R(5):



{{{R(p)=-100p^2+2000p}}} Start with the given equation.



{{{R(5)=-100(5)^2+2000(5)}}} Plug in {{{p=5}}}.



{{{R(5)=-100(25)+2000(5)}}} Square {{{5}}} to get {{{25}}}.



{{{R(5)=-2500+2000(5)}}} Multiply {{{-100}}} and {{{25}}} to get {{{-2500}}}.



{{{R(5)=-2500+10000}}} Multiply {{{2000}}} and {{{5}}} to get {{{10000}}}.



{{{R(5)=7500}}} Combine like terms.



So this means that if the price is $5 a shirt, then he'll expect to make $7,500



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Find R(10):



{{{R(p)=-100p^2+2000p}}} Start with the given equation.



{{{R(10)=-100(10)^2+2000(10)}}} Plug in {{{p=10}}}.



{{{R(10)=-100(100)+2000(10)}}} Square {{{10}}} to get {{{100}}}.



{{{R(10)=-10000+2000(10)}}} Multiply {{{-100}}} and {{{100}}} to get {{{-10000}}}.



{{{R(10)=-10000+20000}}} Multiply {{{2000}}} and {{{10}}} to get {{{20000}}}.



{{{R(10)=10000}}} Combine like terms.




So this means that if the price is $10 a shirt, then he'll expect to make $10,000


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Find R(20):



{{{R(p)=-100p^2+2000p}}} Start with the given equation.



{{{R(20)=-100(20)^2+2000(20)}}} Plug in {{{p=20}}}.



{{{R(20)=-100(400)+2000(20)}}} Square {{{20}}} to get {{{400}}}.



{{{R(20)=-40000+2000(20)}}} Multiply {{{-100}}} and {{{400}}} to get {{{-40000}}}.



{{{R(20)=-40000+40000}}} Multiply {{{2000}}} and {{{20}}} to get {{{40000}}}.



{{{R(20)=0}}} Combine like terms.



So if the price is $20 per shirt, then he'll break even (he will neither gain or lose money).