The total revenue R earned per day (in dollars) from a...

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R(p)=-12 p^{2}+150 p,$ where $p$ is the price charged per pet (in dollars). (a) Find the revenues when the prices per pet are $\$ 4$ $\$ 6,$ and $\$ 8$ (b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

The total revenue $R$ earned per day (in dollars) from a pet-sitting service is given by $R(p)=-12 p^{2}+150 p,$ where $p$ is the price charged per pet (in dollars).
(a) Find the revenues when the prices per pet are $\$ 4$ $\$ 6,$ and $\$ 8$
(b) Find the unit price that will yield a maximum revenue. What is the maximum revenue? Explain your results.

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree two that can be expressed in the form ax² + bx + c. Their graphs are parabolas, and they are used in various real-world applications, including modeling revenue or cost functions. Key properties include the vertex, axis of symmetry, and the fact that the coefficient of the quadratic term determines whether the parabola opens upward or downward.

Revenue Modeling

Revenue modeling involves creating mathematical functions that represent the total revenue generated by a business or service based on certain variables, such as price or quantity sold. In this context, the revenue function is a quadratic function, which reflects how changes in price can affect the revenue collected from services.

Function Evaluation

Function evaluation is the process of computing the value of a function for given input values. In revenue modeling, this involves substituting specific prices into the revenue function to determine the corresponding revenue. This step is essential in assessing performance metrics at different levels of the independent variable.

Optimization in Quadratic Functions

Optimization in the context of quadratic functions typically involves finding the maximum or minimum value of the function, which corresponds to the vertex of the parabola. When the quadratic coefficient is negative, the parabola opens downward, meaning the vertex represents the maximum value. This method is widely used to determine optimal pricing or production levels that yield the highest revenue or profit.

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