a football team plays in a large stadium. with a ticket price of $24, the average attendance at recent games…

a football team plays in a large stadium. with a ticket price of $24, the average attendance at recent games has been 30,000. a market survey indicates that for each $1 increase in the ticket price, attendance decreases by 400.\na. express the number of spectators at a football game, n, as a function of the ticket price, x.\nb. express the revenue from a football game, r, as a function of the ticket price, x.\na. write the function that expresses the number of spectators at a football game, n, as a function of the ticket price, x.\nn(x) = (simplify your answer.)

a football team plays in a large stadium. with a ticket price of $24, the average attendance at recent games has been 30,000. a market survey indicates that for each $1 increase in the ticket price, attendance decreases by 400.\na. express the number of spectators at a football game, n, as a function of the ticket price, x.\nb. express the revenue from a football game, r, as a function of the ticket price, x.\na. write the function that expresses the number of spectators at a football game, n, as a function of the ticket price, x.\nn(x) = (simplify your answer.)

Answer

Explanation:

Step1: Determine the price - increase amount

The initial price is $24 and the current price is $x$. So the price - increase amount is $x - 24$.

Step2: Calculate the decrease in attendance

For each $1 increase in price, attendance decreases by 400. So for an increase of $x - 24$, the decrease in attendance is $400(x - 24)$.

Step3: Find the number of spectators function

The initial attendance is 30000. So $N(x)=30000-400(x - 24)$. Simplify the function: [ \begin{align*} N(x)&=30000-400x+9600\ &=- 400x + 39600 \end{align*} ]

Step4: Find the revenue function

Revenue $R$ is the product of the ticket price $x$ and the number of spectators $N(x)$. So $R(x)=x\cdot N(x)=x(-400x + 39600)=-400x^{2}+39600x$.

Answer:

a. $N(x)=-400x + 39600$ b. $R(x)=-400x^{2}+39600x$