Respuesta :
Answer:
Thus, the coffee shop is willing to supply 6 pounds per week at a price of $4 per pound.
Step-by-step explanation:
We are given the following information in the question:
The marginal price per pound (in dollars) is given by:
[tex]p'(x) = \displaystyle\frac{208}{(x+7)^2}[/tex]
where x is the supply in pounds.
[tex]P(x) = \displaystyle\int p'(x)~dx =\displaystyle\int\displaystyle\frac{208}{(x+7)^2}~dx\\\\P(x) = \frac{-208}{(x+7)} + c\\\\\text{where c is the constant of integration.}[/tex]
The coffee shop is willing to supply 9 pounds per week at a price of $7 per pound.
Thus, we are given that
P(9) = 7
Putting the values, we get,
[tex]P(x) = \displaystyle\frac{-208}{(x+7)} + c\\\\P(9) = 7\\\\\displaystyle\frac{-208}{(9+7)} + c = 7\\\\c = 7 + \frac{208}{16} = 20[/tex]
[tex]P(x) = \displaystyle\frac{-208}{(x+7)} + 20[/tex]
Now, we have to find how many pounds it would be willing to supply at a price of $4 per pound.
P(x) = 4
[tex]P(x) = \displaystyle\frac{-208}{(x+7)} + 20 = 4\\\\\frac{-208}{x+7} = -16\\\\x + 7 = 13\\x = 6[/tex]
Thus, the coffee shop is willing to supply 6 pounds per week at a price of $4 per pound.
