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) = 2900 + 9x + 0.1x2 (0 ≤ x ≤ 500).
Gymnast Clothing sells the cleats at $150 per pair. Find the revenue and profit functions. How many should Gymnast Clothing manufacture to make a profit?
To find the revenue function, we need to multiply the number of cleats sold (x) by the selling price per pair ($150):
Revenue (R) = Selling price per pair (150) * Number of pairs (x)
Therefore, the revenue function is:
R(x) = 150x
To find the profit function, we subtract the cost function from the revenue function:
Profit (P) = Revenue (R) - Cost (C)
Substituting the revenue and cost functions:
P(x) = R(x) - C(x)
= 150x - (2900 + 9x + 0.1x^2)
Simplifying the equation:
P(x) = -0.1x^2 + 141x - 2900
Now, to find the point at which Gymnast Clothing makes a profit, we set the profit function greater than zero and solve for x:
P(x) > 0
-0.1x^2 + 141x - 2900 > 0
This quadratic inequality can be solved by factoring or using the quadratic formula. Factoring gives:
(-0.1x + 29)(x - 100) > 0
To determine when the inequality is true, we set each factor greater than zero:
-0.1x + 29 > 0 and x - 100 > 0
Solving each inequality separately:
-0.1x > -29
x < 290
x > 100
Since x represents the number of pairs of cleats, it cannot be negative or exceed 500. Therefore, Gymnast Clothing should manufacture between 100 and 290 pairs of cleats in order to make a profit.
To find the revenue function, we need to multiply the number of pairs of cleats (x) by the selling price per pair (150). Hence, the revenue function, R(x), is given by:
R(x) = 150x
To find the profit function, we subtract the cost function (C(x)) from the revenue function (R(x)). Therefore, the profit function, P(x), is given by:
P(x) = R(x) - C(x)
Substituting the values of R(x) and C(x) into the profit function, we have:
P(x) = 150x - (2900 + 9x + 0.1x^2)
P(x) = 150x - 2900 - 9x - 0.1x^2
P(x) = -0.1x^2 + 141x - 2900
To determine how many cleats Gymnast Clothing needs to manufacture to make a profit, we can set P(x) greater than zero (P(x) > 0). Solving this inequality will give us the range of x values that correspond to profit:
-0.1x^2 + 141x - 2900 > 0
To solve the inequality, we can use techniques such as factoring, graphing, or the quadratic formula. In this case, let's use the quadratic formula:
The solutions to the quadratic equation -0.1x^2 + 141x - 2900 = 0 are:
x = [ -141 ± √(141^2 - 4(-0.1)(-2900)) ] / (2(-0.1))
Simplifying this equation will give us the two values for x:
x = [ -141 ± √(19881 - 1160) ] / (-0.2)
x = [ -141 ± √(18721) ] / (-0.2)
x ≈ -17.11 or x ≈ 158.11
Since we cannot manufacture a negative quantity, we can discard the negative value. Therefore, the number of cleats Gymnast Clothing should manufacture to make a profit is approximately 158.
revenue = quantity * price
and of course,
profit = revenue - cost
just plug in the numbers given you