Precalc question

Question

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?

Answer 1

The number of people who attend is dependent on the cost of the ticket. The cost is the independent variable and the number of people is dependent. The number of people f(x) is the function of the cost (x):

f(5)=1200

f(7)=970

The linear function has the form of f(x)=mx+b.

f(5)=m*5+b

1200=5m+b

Also,

f(7)=m*7+b

970=7m+b

Solve the system of two equations:

(1) 1200=5m+b

(2) 970=7m+b

Subtract (2) from (1):

230=-2m

m=-115

(1) 1200=-115*5+b

1200=-575+b

b=1775

Thus, the formula is f(x)=-115x+1775

Now, the revenue is the cost times how many people bought the ticket, right? So, the revenue is f(x)*x:

f(x)*x=-115x²+1775x

This is a downward concave quadratic function, and its maximum is its vertex. The x-coordinate of the vertex is -b/2a:

x=-1775/(-230)=7.72

That is how much she can charge per ticket to make the most money.