$AP Calculus problem$

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$AP Calculus problem$

Rosalie is organizing a circus performance to raise money for a charity. She is trying to decide how much to charge for tickets. From past experience, she knows that the number of people who will attend is a linear function of the price per ticket. If she charges 5 dollars, 1200 people will attend. If she charges 7 dollars, 970 people will attend. How much should she charge per ticket to make the most money?

AP Calculus Precalculus Word Problem

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Let $y$ be the number of people attending a performance and $x$ be the ticket price. The slope of the line is
\[m=\frac{1200-970}{5-7}=\frac{230}{-2}=-115.\]
Hence the equation of the is
\[y-1200=-115(x-5) \Rightarrow y=-115x+1775.\]
The money that Rosalie makes is
\[f=x \times y \Rightarrow f(x)=x (-115x+1775)=-115x^2+1775x.\]
We want to maximize $f$. So lets find the critical points:
\[f'(x)=-230x+1775=0 \Rightarrow x=\frac{1775}{230} \approx \$7.72.\]

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$AP Calculus problem$

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