Answer:
The number of dishwasher to produce is [tex]x = 688 \ or \ x = 12[/tex]
Step-by-step explanation:
From the question we are told that
The price per dishwasher is [tex]p = 420 - 0.3x[/tex]
The total cost of producing x dishwasher is [tex]C(x) = 5000 + 0.3 x^2[/tex]
Now at the question we can deduce that the price for x dishwasher will be mathematically evaluated as
[tex]P_t(x) = x * p[/tex]
[tex]P_t(x) = x * (420 - 0.30 x)[/tex]
[tex]P_t(x) = 420x - 0.30x^2[/tex]
Now the profit from selling the dishwasher can be mathematically represented as
[tex]P (x)= P_t (x ) - C(x)[/tex]
substituting for [tex]P_t (x ) \ and \ C(x)[/tex]
[tex]P(x) = 420 x -0.3x^2 - [5000 + 0.3x^2][/tex]
[tex]P(x) = 0.6x^2 - 420 x + 5000[/tex]
Solving for x using the quadratic formula we have that
[tex]x_1 = 688 \ or \ x = 12[/tex]
So the for maximum profit and reduce cost price the number of dishwasher the company would produce is [tex]x = 688 \ or \ x = 12[/tex]
