Acommunity 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 30 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 3 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?

The required function is:

The price for each feeder to maximize profit should be $13.

The maximum weekly profit is $147

Step-by-step explanation:

Consider the provided information.

Part (A)

The materials for each feeder cost $6, and the society sells an average of 30 per week at a price of $10 each.

Let x be the price per feeder.

Profit = Revenue - cost

So the profit per feeder is

If we increase then for every dollar increase, it loses 3 sales per week.

This can be written as:

Where x-10 represents the increase in price and 3  represents the decrease in sales per week.

Thus the profit will be:

Hence, the required function is:

Part (B) What price should the society charge for each feeder to maximize profits? What is the maximum weekly profit?

The above function is a downward parabola so the maximum will occur at vertex.

The x coordinate of the vertex of parabola is:

Substitute a=-3, b=78 and c=-360 in above.

Hence, the price for each feeder to maximize profit should be $13.

Now substitute the value of x in

Hence, the maximum weekly profit is $147

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.

step-by-step explanation:

ratios represent a pattern or relationship that can be scaled.

equivalent ratios are pairs of ratios that represent the same relationship between quantities, and can be generated by multiplying or dividing both quantities in the ratio by the same factor.

in this lesson, students will use double number lines and ratio reasoning to solve ratio problems. by partitioning double number lines into proportionate intervals, students will build an understanding of equivalent ratio relationships as those that are generated by multiplying or dividing two related quantities by the same number. this lesson will build upon students' prior understanding of multiplicative comparisons, number line reasoning and equivalent fractions (while highlighting the difference between equivalent fractions and equivalent ratios); and will build toward future work in this unit on building ratio tables, and work in future grades in proportionality, scaling, and expressing and graphing linear functions.