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olasank [31]
3 years ago
12

The marginal price per pound (in dollars) at which a coffee store is willing to supply x pounds of Jamaican Blue Mountain coffee

per week is given by p ′ (x) = 208 (x + 7)2 . If the coffee shop is willing to supply 9 pounds per week at a price of $7 per pound, determine how many pounds it would be willing to supply at a price of $4 per pound?
Mathematics
1 answer:
il63 [147K]3 years ago
3 0

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:

p'(x) = \displaystyle\frac{208}{(x+7)^2}

where x is the supply in pounds.

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.}

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,

P(x) = \displaystyle\frac{-208}{(x+7)} + c\\\\P(9) = 7\\\\\displaystyle\frac{-208}{(9+7)} + c = 7\\\\c = 7 + \frac{208}{16} = 20

P(x) = \displaystyle\frac{-208}{(x+7)} + 20

Now, we have to find how many pounds it would be willing to supply at a price of $4 per pound.

P(x) = 4

P(x) = \displaystyle\frac{-208}{(x+7)} + 20 = 4\\\\\frac{-208}{x+7} = -16\\\\x + 7 = 13\\x = 6

Thus, the coffee shop is willing to supply 6 pounds per week at a price of $4 per pound.

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