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The Last Laugh, a comedy club, hosts the No Funny Business Comedy Festival every year. The club owners have observed that when they increase the price for festival tickets, fewer people buy tickets overall. The club's festival-ticket revenue in dollars can be modeled by the expression p(1,400−20p)p(1,400 - 20p), where pp is the price per ticket in dollars. This expression can be written in factored form as −20p(p−70)-20p(p - 70).\newlineWhat does the number 7070 represent in the expression?\newlineChoices:\newline(A)the price per festival ticket in dollars that maximizes the club's total revenue\newline(B)the club's maximum total revenue in dollars\newline(C)the club's minimum total revenue in dollars\newline(D)the price per festival ticket in dollars so that the club's total revenue is zero\newline

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Q. The Last Laugh, a comedy club, hosts the No Funny Business Comedy Festival every year. The club owners have observed that when they increase the price for festival tickets, fewer people buy tickets overall. The club's festival-ticket revenue in dollars can be modeled by the expression p(1,400−20p)p(1,400 - 20p), where pp is the price per ticket in dollars. This expression can be written in factored form as −20p(p−70)-20p(p - 70).\newlineWhat does the number 7070 represent in the expression?\newlineChoices:\newline(A)the price per festival ticket in dollars that maximizes the club's total revenue\newline(B)the club's maximum total revenue in dollars\newline(C)the club's minimum total revenue in dollars\newline(D)the price per festival ticket in dollars so that the club's total revenue is zero\newline
  1. Factorize expression: The expression for the club's festival-ticket revenue is p(1,400−20p)p(1,400 - 20p). This can be factored to −20p(p−70)-20p(p - 70).
  2. Analyze factored form: To understand what 7070 represents, let's look at the factored form −20p(p−70)-20p(p - 70). This is a quadratic equation in standard form.
  3. Identify standard form: The standard form of a quadratic equation is ax2+bx+cax^2 + bx + c. In our case, a=−20a = -20, b=1400b = 1400, and the expression is set to zero for revenue.
  4. Find roots of equation: The roots of the quadratic equation are the values of pp that make the revenue zero. The factored form shows one root at p=70p = 70.
  5. Determine maximum revenue point: Since the quadratic represents revenue, p=70p = 70 is the price per ticket where the revenue starts to decrease again, meaning it's the maximum revenue point.
  6. Interpret price per ticket: Therefore, the number 7070 represents the price per festival ticket in dollars that maximizes the club's total revenue.

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