The total cost of producting x radio sets per day is (1/4x^2 + 35x + 25) dollars. The price per set at which they may be sold is ( 50 -1/2x) dollars. What should be the daily output be to obtain the maximum profit?
Help!!!
To find the daily output that will yield the maximum profit, we need to determine the number of radio sets (x) that will maximize the profit.
The profit is given by the equation: profit = revenue - cost
We can express revenue as the price per set multiplied by the number of sets sold: revenue = (50 - 1/2x) * x
The cost is given by the equation: cost = 1/4x^2 + 35x + 25
Now, we can substitute the expressions for revenue and cost into the profit equation:
profit = (50 - 1/2x) * x - (1/4x^2 + 35x + 25)
Simplifying this equation gives us the following quadratic equation for profit:
profit = -1/4x^2 + 13.5x - 25
To find the value of x that maximizes profit, we need to find the x-coordinate of the vertex of this quadratic equation. The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a, b, and c are the coefficients of the quadratic equation in standard form (ax^2 + bx + c = 0).
In this case, a = -1/4, b = 13.5, and c = -25. Plugging these values into the formula, we can calculate the x-coordinate of the vertex:
x = -13.5 / (2 * (-1/4))
x = -13.5 / (-1/2)
x = -13.5 * (-2/1)
x = 27
So, the daily output to obtain the maximum profit should be 27 radio sets.