Maximizing Profit:?

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 they sell an average of 20per week at a price of $10 each. They have been considering raising the price, so theyconduct a survey and find for every dollar increase they lose 2 sales per week.

*find a function that models weekly profit in terms of price per feeder.

*what price should the society charge for each feeder to maximize profits? What is maxium profit?

Answer 1

Profit = (Price-6)*(# sold)
Let Price = p
Let # sold = s

We know that for each $1 increase in p s decreases by 2. So s is a linear function of p as follows
s = -2p +b where b is how many would be sold if they were free.
We know that when p = $10, then s = 20
So, 20 = -2*10+b => b=40
So, s = -2p+40 (Sold = -2*Price+40)

Substituting in the profit equation now we have
Profit = (p-6)*(-2p+40) = -2p^2 +40p+12p-240 = -2p^2+52p-240

This is parabola which opens downwards so there is a maximum to this equation.
You can memorize the point of symmetry for a parabolla or you can re-write the equation. Since I don't like memorizing let's re-write this as follows:
Profit = -2(p^2 -26p +120)
= -2(p^2 -2*13p + 120)
= -2(p^2 -2*13p +13^2-13^2 + 120) (if we add and subtract the same amount whe have not changed the equation
= -2(p^2 -2*13p + 13^2 -169 +120)
= -2(p^2 -2*13p + 13^2 - 49)
= -2(p^2 -2*13p + 13^2) +2*49
=-2(p^2 -2*13p + 13^2) + 98
= -2(p-13)^2 +98

so we see that
Profit = -2(p-13)^2 + 98
and we see that when price (p) equals $13 then profit equals $98.
The society should charge $13 to maximize profits at $98 per week

Answer 2

Profit = (Price - 6) * (# sales).

Now you know that if the price is 10+x, the number of sales is 20-2x.
This is because you lose 2 sales for every dollar over 10.

Profit = (10+x-6) * (20-2x) = (4+x)*(20-2x) = 80+12x-2x^2

This is a parabola so to maximize it we take x=-b/2a = -12/(2*-2) = 3, which corresponds to 10+3=13 sales, and this gives a profit of 98.

Answer 3

Total Profit= Number sold x (Profit per item[PPI])
Currently,
80 = 20 (10-6)
Now number sold= function of (sales price-cost), so if No is a baseline: No-2 u= PPI+u, where u is the number of dollar change in PPI.
Total Profit = (20-2u)(4 +u) this is your function.
AT maximized profit, d(Total Profit)/du=0
so d(total profit) = d (80-12u-2u^2)/du= -12 -4u
Then at max profit 4u=12, u=3.
If we increase profit to 7 dollars (cost to $13), we sell 14 units for $98.