The monthly profit P, in thousands of pesos of a company can be estimated by the formula P(x)= -3x^2+30x+2, where x is the number of units sold per month. Find the number of units that must be sold by the company to maximize its profit and then find the maximum profit.
since the parabola opens downward, its maximum is at the vertex.
As you recall, the vertex of ax^2+bx+c is at x = -b/2a
In this case, that is x = 30/6 = 5.
I expect you can evaluate P(5).
To find the number of units that must be sold by the company to maximize its profit, we need to determine the x-value at the maximum point of the quadratic function.
The function P(x) = -3x^2 + 30x + 2 is in the form of a quadratic equation in standard form: ax^2 + bx + c.
We can find the x-value of the maximum point using the formula: x = -b / (2a).
In this case, a = -3 and b = 30:
x = -30 / (2 * -3)
x = -30 / -6
x = 5
So, the company must sell 5 units to maximize its profit.
To find the maximum profit, substitute the x-value back into the equation P(x):
P(5) = -3(5)^2 + 30(5) + 2
P(5) = -3(25) + 150 + 2
P(5) = -75 + 150 + 2
P(5) = 77
Therefore, the maximum profit is 77,000 pesos.
To find the number of units that must be sold by the company to maximize its profit, we need to determine the x-value at which the profit function P(x) reaches its maximum value.
The profit function is given as P(x) = -3x^2 + 30x + 2. This is a quadratic function in standard form, where the coefficient of the x^2 term is negative.
To find the x-coordinate of the vertex, we can use the formula x = -b / (2a), where a is the coefficient of the x^2 term and b is the coefficient of the x term.
Using the values from the given profit function, we have a = -3 and b = 30. Plugging these values into the formula, we get x = -30 / (2 * -3) = 30 / 6 = 5.
The company must sell 5 units to maximize its profit.
To find the maximum profit, we substitute this value of x back into the profit function.
P(5) = -3(5)^2 + 30(5) + 2
= -3(25) + 150 + 2
= -75 + 150 + 2
= 77
The maximum profit is 77,000 pesos (since the profit function is in thousands of pesos).
Therefore, the company must sell 5 units to maximize its profit, and the maximum profit is 77,000 pesos.