On the basis of data from past years, a consultant informs...

On the basis of data from past years, a consultant informs Bob's Bicycles that its profit from selling $x$ bicycles is given by the function $$p(x)=250 x-x^{2} / 4-15,000$$ (a) How much profit do they make by selling 100 bicycles? By selling 400 bicycles? (b) How many bicycles should be sold to maximize profit? In that case, what will be the profit per bicycle?

On the basis of data from past years, a consultant informs Bob's Bicycles that its profit from selling $x$ bicycles is given by the function $$p(x)=250 x-x^{2} / 4-15,000$$
(a) How much profit do they make by selling 100 bicycles? By selling 400 bicycles?
(b) How many bicycles should be sold to maximize profit? In that case, what will be the profit per bicycle?

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Quadratic Functions

A quadratic function is a polynomial of degree two, typically written in the form ax² + bx + c. Its graph is a parabola, and these functions are widely used to model situations where there is a maximum or minimum value, making them ideal for problems involving optimization such as profit maximization.

Vertex of a Parabola

The vertex of a parabola, calculated using the formula -b/(2a) for a quadratic function in standard form, represents the point of maximum or minimum value. This concept is crucial in optimization problems to find the exact input value at which a function reaches its highest or lowest outcome.

Function Evaluation

Function evaluation involves substituting specific values into a function to calculate the output. This concept is essential for determining the performance of a modeled situation at particular input levels, such as calculating profit for a given number of items sold.

Unit Rate Analysis

Unit rate analysis is the process of determining the output per single unit by dividing a total quantity by the number of units. This concept is particularly useful when assessing efficiency or effectiveness, for example, calculating profit per item sold.

Transcript

Please give Ace some feedback