Respuesta :
Answer:
The required function is: [tex]P(x)=-3x^2+78x-360[/tex]
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 [tex]x-6[/tex]
If we increase then for every dollar increase, it loses 3 sales per week.
This can be written as:
[tex]30-3(x-10)[/tex]
[tex]30-3x+30[/tex]
[tex]-3x+60[/tex]
Where x-10 represents the increase in price and 3 represents the decrease in sales per week.
Thus the profit will be:
[tex]P(x)=(x-6)(-3x+60)[/tex]
[tex]P(x)=-3x^2+78x-360[/tex]
Hence, the required function is: [tex]P(x)=-3x^2+78x-360[/tex]
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: [tex]x=\frac{-b}{2a}[/tex]
Substitute a=-3, b=78 and c=-360 in above.
[tex]x=\frac{-78}{2(-3)}=13[/tex]
Hence, the price for each feeder to maximize profit should be $13.
Now substitute the value of x in [tex]P(x)=-3x^2+78x-360[/tex]
[tex]P(x)=-3(13)^2+78(13)-360=$147[/tex]
Hence, the maximum weekly profit is $147
