A company’s weekly revenue R is given by the formula R = -p2 + 30p, where p is the price of the company’s product. The company is considering hiring a distributor, which will cost the company 4p + 25 per week. Find the values of p for which the product will remain profitable if they hire the distributor. Which value of p will maximize the profit after including the cost?
To determine the values of p for which the product will remain profitable if they hire the distributor, we need to find the price range that ensures that the revenue is greater than the cost.
The weekly cost of hiring the distributor is given by 4p + 25.
The profit is calculated by subtracting the cost from the revenue: Profit = R - (4p + 25).
For the product to remain profitable, the profit needs to be greater than zero: Profit > 0.
Substituting the given formula for R, we can write this inequality as follows:
-p^2 + 30p - (4p + 25) > 0
Next, we can simplify the inequality:
-p^2 + 30p - 4p - 25 > 0
-p^2 + 26p - 25 > 0
To solve this inequality, we can factor the quadratic into two binomial expressions:
(p - 25)(p - 1) > 0
Next, we determine the critical points of this inequality by setting each factor equal to zero:
p - 25 = 0, p - 1 = 0
Solving these equations, we find that p = 25 and p = 1.
Plotting these points on a number line:
(-∞ , 1) 1 (1, 25) 25 (25, ∞)
Now, we can test a value from each interval to determine the sign of the inequality:
For p < 1, we can choose p = 0. Substituting this value into the inequality:
(-)(-) - 25 < 0
(positive) - 25 < 0
negative < 0
This interval satisfies the inequality.
For 1 < p < 25, we can choose p = 10. Substituting this value into the inequality:
(-)(positive) - 25 > 0
(negative) - 25 > 0
negative > 0
This interval does not satisfy the inequality.
For p > 25, we can choose p = 30. Substituting this value into the inequality:
(-)(positive) - 25 > 0
(negative) - 25 > 0
negative > 0
This interval does not satisfy the inequality.
Therefore, the values of p for which the product will remain profitable if they hire the distributor are p < 1.
To find the value of p that maximizes the profit, we need to determine the vertex of the parabola formed by the revenue function.
The formula for the vertex of a parabola in the form y = ax^2 + bx + c is given by x = -b / (2a).
In this case, a = -1 and b = 30.
Substituting these values into the formula, we find the x-coordinate of the vertex:
p = -30 / (2(-1))
p = 30 / 2
p = 15.
Therefore, the value of p that maximizes the profit after including the cost is p = 15.