Dalco manufacturing estimate that its weakly profit, P, in hundreds of dollars,can be approximated by the formula P= -4x²+8x+7, where x is the number of units produced per week, in thousands.
1)How many units should the company produce per week to earn maximum profit
2)Find the maximum weekly profit.
To find the number of units the company should produce per week to earn maximum profit, we need to determine the value of x that corresponds to the maximum value of the profit function P(x).
1) The maximum or minimum value of a quadratic function occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula: x = -b / (2a), where a and b are the coefficients of the quadratic equation.
In the given profit function P(x) = -4x² + 8x + 7, we identify a = -4 and b = 8.
x = -8 / (2 * (-4)) = -8 / (-8) = 1
Therefore, the company should produce 1,000 units per week (since x represents the number of units produced in thousands) to earn maximum profit.
2) To find the maximum profit, substitute the value of x = 1 into the profit function P(x).
P(1) = -4(1)² + 8(1) + 7 = -4 + 8 + 7 = 11
So, the maximum weekly profit is $11,000.
d(P) = -8x + 8 = 0 for a max/min of P
8x = 8
x = 1
when x = 1
P = -4+ 8+7 = 11
I will leave it up to you to interpret the answer and state the conclusion