Question

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 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?

Answer 1

Answer:

The required function is: $P(x)=-3x^2+78x-360$

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 $x-6$

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

This can be written as:

$30-3(x-10)$

$30-3x+30$

$-3x+60$

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

Thus the profit will be:

$P(x)=(x-6)(-3x+60)$

$P(x)=-3x^2+78x-360$

Hence, the required function is: $P(x)=-3x^2+78x-360$

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: $x=\frac{-b}{2a}$

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

$x=\frac{-78}{2(-3)}=13$

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

Now substitute the value of x in $P(x)=-3x^2+78x-360$

$P(x)=-3(13)^2+78(13)-360= 147$

Hence, the maximum weekly profit is $147