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) = 3000 + 9x + 0.1x2 (0 ≤ x ≤ 500).
Gymnast Clothing sells the cleats at $130 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?
Revenue:
Profit:
At Least ____pairs
I don't know how to find the Revenue and Profit function with the formula given, how do I determine it?
Thanks
Revenue is price * units sold
Naturally, profit is revenue-cost
To find the revenue and profit functions, we need to understand the formulas and concepts involved.
Revenue is the total amount of money earned by selling a product. It is calculated by multiplying the selling price per unit by the number of units sold.
Profit, on the other hand, is the difference between revenue and cost. It represents the amount of money a company makes after deducting all expenses.
Now let's find the revenue function first. Given that Gymnast Clothing sells the cleats at $130 per pair and x represents the number of pairs produced, the revenue function (R) can be determined by multiplying the selling price by the number of pairs:
R(x) = 130x
Next, let's find the profit function. The cost function (C) is provided as:
C(x) = 3000 + 9x + 0.1x^2
We can define the profit function (P) as the difference between the revenue (R) and the cost (C):
P(x) = R(x) - C(x)
Substituting the expressions for R(x) and C(x) into the equation, we have:
P(x) = 130x - (3000 + 9x + 0.1x^2)
Simplifying further:
P(x) = 130x - 3000 - 9x - 0.1x^2
P(x) = -0.1x^2 + 121x - 3000
Now, to determine how many pairs Gymnast Clothing should manufacture to make a profit, we need to find the value of x where the profit function (P(x)) is positive. This means that the revenue generated is higher than the cost incurred.
To find the number of pairs, we can solve the inequality P(x) > 0.
-0.1x^2 + 121x - 3000 > 0
To solve this quadratic inequality, you can use either factoring, completing the square, or the quadratic formula. Once you find the solution(s) for x, you will know how many pairs Gymnast Clothing should manufacture to make a profit.
Note: The range of x given in the problem is 0 ≤ x ≤ 500, so make sure to consider this constraint when determining the number of pairs that Gymnast Clothing should manufacture.