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Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores in runs of up to 500. Its cost (in dollars) for a run of x pairs of cleats is C(x) = 2750 + 8x + 0.1x2 (0 ≤ x ≤ 500). Gymnast Clothing sells the cleats at $100 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?

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Answer:

Revenue function:
R(x)=100x

Profit function:
P(x)=-0.1 x^2 + 92 x - 2750

31 to 500 Gymnast Clothing are manufactured to make a profit.

Explanation:

The cost (in dollars) for a run of x pairs of cleats.


C(x)=2750 + 8x + 0.1x^2

Gymnast Clothing sells the cleats at $100 per pair.

The Revenue (in dollars) for a run of x pairs of cleats.


R(x)=100x

Profit = Revenue - Cost


P(x)=R(x)-C(x)


P(x)=100x-(2750 + 8x + 0.1x^2)


P(x)=-0.1 x^2 + 92 x - 2750

We need to find how many should Gymnast Clothing manufacture to make a profit.


P(x)\geq 0


-0.1 x^2 + 92 x - 2750\geq 0

From the given figure it is clear that P(x) is greater than or equal to 0 for 30.931 ≤ x ≤ 889.069.

The value of x can not be more than 500.

31 ≤ x ≤ 500

31 to 500 Gymnast Clothing are manufactured to make a profit.

Gymnast Clothing manufactures expensive soccer cleats for sale to college bookstores-example-1
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