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?
What is the break even cost?
(by using the formula
Revenue is price * units sold
Profit is revenue-cost
how do find the I break the equation, revenue and profit function?
I don't know how to break the equation given.
Thank you
To find the revenue and profit functions, we need to use the given information. The revenue is the total amount of money earned by selling a certain number of cleats, and the profit is the difference between the revenue and the cost.
1. Revenue Function:
The revenue function can be determined by multiplying the price per pair of cleats by the number of cleats sold. In this case, the price per pair is $130, and the number of cleats sold is represented by "x" pairs. Therefore, the revenue function, R(x), is given by:
R(x) = 130x
2. Cost Function:
The cost function, C(x), is already provided in the question. It is defined as:
C(x) = 3000 + 9x + 0.1x^2
This function includes the fixed cost of $3000, the variable cost of $9 per pair, and the quadratic cost component of 0.1x^2.
3. Profit Function:
The profit function, P(x), can be calculated by subtracting the cost function from the revenue function:
P(x) = R(x) - C(x) = 130x - (3000 + 9x + 0.1x^2)
Now, to find the number of cleats Gymnast Clothing needs to manufacture to make a profit, we need to determine where the profit function is positive (P(x) > 0). To do this, you can set the profit function equal to zero and solve for x:
130x - (3000 + 9x + 0.1x^2) > 0
Simplifying this inequality will give you the range of values for x, where the profit is positive. You can use techniques such as factoring, completing the square, or the quadratic formula to solve this inequality.
To find the break-even cost, you need to determine the point where revenue equals cost (R(x) = C(x)). This can be found by setting the revenue function equal to the cost function and solving for x:
130x = 3000 + 9x + 0.1x^2
Solving this equation will give you the break-even point.
Remember, these calculations involve manipulating equations and solving inequalities, so it may be helpful to use a graphing calculator or software to visualize and solve these functions efficiently.