A school sells candy as a fund-raiser for their sports...

A school sells candy as a fund-raiser for their sports programs. They sell about 5500 candy bars at a profit of $\$ 0.75$ each. Each $\$ 0.50$ increase in the price will increase the profit of each candy bar by $\$ 0.50 .$ However, they will sell about 1000 fewer candy bars. a. What is the profit for this school if the profit is $\$ 0.75$ per candy bar? b. Find an equation for the model of the profit this school makes if they increase the price of each candy bar by $\$ 0.50 x$ times. c. Use your model to determine the profit if the price per candy bar is increased by $\$ 1 .$ d. Use your model to determine the profit if the price per candy bar is increased by $\$ 2 .$ e. What price increase will bring the school the highest profit? Explain.

A school sells candy as a fund-raiser for their sports programs. They sell about 5500 candy bars at a profit of $\$ 0.75$ each. Each $\$ 0.50$ increase in the price will increase the profit of each candy bar by $\$ 0.50 .$ However, they will sell about 1000 fewer candy bars.
a. What is the profit for this school if the profit is $\$ 0.75$ per candy bar?
b. Find an equation for the model of the profit this school makes if they increase the price of each candy bar by $\$ 0.50 x$ times.
c. Use your model to determine the profit if the price per candy bar is increased by $\$ 1 .$
d. Use your model to determine the profit if the price per candy bar is increased by $\$ 2 .$
e. What price increase will bring the school the highest profit? Explain.

Key Concepts

Profit Function Modeling

This concept involves creating a mathematical function that represents the profit of a business activity. Profit is typically calculated as the product of the profit per unit and the number of units sold, and the modeling process requires identifying how changes in one variable (such as unit price) affect both the profit per unit and the quantity sold.

Linear Relationships in Price and Demand

This concept examines the way in which incremental changes in price can lead to predictable, linear changes in both profit per unit and the number of units sold. It is the basis for establishing the relationship between price increases and decreases in demand, allowing for adjustments to be quantified and incorporated into a profit model.

Quadratic Functions

Quadratic functions arise when two linear expressions (for example, a linearly increasing profit per unit and a linearly decreasing unit sales volume) are multiplied. The resulting quadratic function typically has a parabolic shape, which is integral in understanding the overall behavior of the profit model and lends itself to analysis such as finding maximum or minimum values.

Optimization of Quadratic Functions

Optimization in this context refers to finding the maximum profit that can be achieved using the quadratic profit model. This is usually done by identifying the vertex of the parabola represented by the quadratic function. The vertex provides the point at which the function reaches its maximum (or minimum, for a parabola opening upward) value, which is critical for decision-making in scenarios like pricing strategies.

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