Maximizing Revenue Mona Kalini gives a walking tour of...

Maximizing Revenue Mona Kalini gives a walking tour of Honolulu to one person for $\$ 49 .$ To increase her business, she advertised at the National Orthodontist Convention that she would lower the price by $\$ 1$ per person for each additional person, up to 49 people. a. Write the price per person $p$ as a function of the number of people $n$ Minul See Exercise 101 of Section 1.4 b. Write her revenue as a function of the number of people on the tour. Whin Her revenue is the product of $n$ and $p$ c. What is the maximum revenue for her tour?

Maximizing Revenue Mona Kalini gives a walking tour of Honolulu to one person for $\$ 49 .$ To increase her business, she advertised at the National Orthodontist Convention that she would lower the price by $\$ 1$ per person for each additional person, up to 49 people.
a. Write the price per person $p$ as a function of the number of people $n$ Minul See Exercise 101 of Section 1.4
b. Write her revenue as a function of the number of people on the tour. Whin Her revenue is the product of $n$ and $p$
c. What is the maximum revenue for her tour?

Key Concepts

Optimization of Quadratic Functions

Optimizing a quadratic function requires locating the vertex of the parabola, as the vertex represents the maximum or minimum of the function depending on the direction of the parabola. In revenue maximization problems, finding the vertex is crucial because it determines the number of units (or people) that yield the maximum revenue.

Quadratic Functions

When the revenue is defined as the product of a linear function (price) and the number of persons, the resulting revenue function is quadratic. Quadratic functions, which are polynomial functions of degree two, have a parabolic graph whose maximum or minimum can be determined by finding its vertex.

Linear Price Function

This concept involves expressing the price per person as a linearly decreasing function of the number of people. In many business scenarios, the price is reduced by a fixed amount for each additional unit sold, which can be modeled with a linear equation where the price decreases as the number of persons increases.

Revenue Function

The revenue function is defined as the product of the number of units sold (in this case, the number of people on the tour) and the price per unit. Creating such a function allows one to understand how variations in price and quantity sold affect the overall revenue and provides a basis for further analysis in maximizing revenue.

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