A community bird-watching society makes and sells simple...

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $$\$ 6$$, and the society sells an average of 20 per week at a price of $$\$ 10$$ each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week. (a) Find a function that models weekly profit in terms of price per feeder. (b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

A community bird-watching society makes and sells simple bird feeders to raise money for its conservation activities. The materials for each feeder cost $$\$ 6$$, and the society sells an average of 20 per week at a price of $$\$ 10$$ each. The society has been considering raising the price, so it conducts a survey and finds that for every dollar increase, it loses 2 sales per week.
(a) Find a function that models weekly profit in terms of price per feeder.
(b) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

Key Concepts

Linear Demand Relationship

This concept refers to a straight-line relationship between the price of a product and the quantity demanded by consumers. In many economic models, as the price increases, the quantity demanded decreases at a constant rate. This relationship is key when altering prices to predict their effect on sales volumes.

Profit Function

A profit function calculates profit as the difference between revenue and costs. Typically, revenue is determined by multiplying the selling price by the number of units sold, while cost is derived from the unit cost multiplied by the same number of units. Understanding how these elements interact allows one to model profit mathematically.

Quadratic Optimization

Many profit functions, especially those that involve linear changes in demand with price increases, take the form of a quadratic function. Quadratic optimization involves finding the maximum or minimum value of such functions, often using the vertex formula. This approach is critical in determining the optimal pricing strategy that maximizes profit.

Transcript

00:01 So part a of this problem asks us to find a function that models the weekly profit in terms of price for feeder.

00:08 So the way we do that is by letting x is equal to the number of one dollar increases.

00:30 And so our profit would be our revenue minus cost.

00:35 And our revenue is the price times quantity.

00:38 So the price would be the base price, which is $10 plus x, which is our $1 increases.

00:49 That will be our price per feeder.

00:52 And our quantity of feeder is we start at 20 per week.

00:59 And since for every dollar, since for every dollar increase, we lose two sales.

01:10 Per week, it'll be minus 2x.

01:14 So the 10 plus x represents our price and 20 minus 2x represents our quantity.

01:24 And so if we just expand this out, if we use foil, so our first term is 200.

01:35 Outer terms are negative 20x, inner terms are plus 20x, and last terms are minus 2x squared.

01:53 So this models are, yes, so this models are revenue.

02:03 So now we got to factor in the cost to find a total profit because profit is equal to revenue minus cost.

02:08 So the cost is equal to the cost per feeder times the number of feeders.

02:14 So the cost per feeder is $6.

02:19 Multiply by the number of feeders, which we found above.

02:22 That's the same thing as the quantity.

02:24 So that's 20 minus 2x.

02:29 And that is equal to, if you expand this out, 120 minus 12x.

02:38 So if we subtract these two equations, because this is our revenue, so capital is capital r, and our cost is capital c, and profit is equal to r minus c, which is equal to 200, minus 20x plus 20x is equal to 0, so 200 minus 2x squared, minus 120, plus 12x...

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