a liquid form of penicillin is sold in bulk at a price of $ 200 per unit.If the total cost for x unit is C(x)=500000+80x+0.003X^2 and the production capacity is at most 30000 at a specified timed, how many of the penicillin must be manufactured and sold in that time to maximize profit?
profit = selling price - cost of making it
=200x - 500000 - 80x - .0003x^2
d(profit) = 120 - .006x
= 0 for a max of profit
x = 120/.006 = appr. 120
check:
at x = 119 , profit = clearly negative
this question makes no sense to me
What is that huge number in C(x) supposed to represent?
x = 120/.006 = 20000
check:
let x = 19000, profit = 697000
let x = 20000, profit = 700000
let x = 21000, profit = 697000
the maximum number they should manufacture is
20000
To find the number of units that must be manufactured and sold in order to maximize profit, we need to find the value of x that maximizes the profit function.
The profit function is given by P(x) = Revenue - Cost.
The revenue is calculated by multiplying the number of units sold by the selling price. In this case, the selling price is $200 per unit. So, the revenue function is R(x) = 200x.
The cost function is given as C(x) = 500000 + 80x + 0.003x^2.
The profit function is then: P(x) = R(x) - C(x) = 200x - (500000 + 80x + 0.003x^2).
To find the quantity that maximizes profit, we need to find the vertex of the profit function, which represents the maximum value. The formula for the x-coordinate of the vertex of a quadratic function in the form ax^2 + bx + c is given by x = -b / (2a).
In this case, a = 0.003, b = -80, and c = 500000. Substituting these values into the formula, we get:
x = -(-80) / (2 * 0.003) = 80 / 0.006 = 13333.33.
Since the maximum production capacity is 30,000 units, and we can only manufacture a whole number of units, the maximum number of units that can be manufactured and sold in that time to maximize profit is 13,333 units.
To find the number of penicillin units that must be manufactured and sold to maximize profit, we need to maximize the profit function, which is the revenue minus the cost.
First, we need to determine the revenue function. Revenue is calculated as the selling price per unit multiplied by the number of units sold. In this case, the selling price per unit is $200, so the revenue function R(x) can be written as:
R(x) = 200x
Next, we can write the profit function P(x) as the revenue function minus the cost function:
P(x) = R(x) - C(x)
Substituting the expressions for R(x) and C(x), we get:
P(x) = 200x - (500000 + 80x + 0.003x^2)
Simplifying further, we have:
P(x) = 200x - 500000 - 80x - 0.003x^2
P(x) = -0.003x^2 + 120x - 500000
To find the number of penicillin units that maximize profit, we need to find the vertex of the parabola defined by the profit function. The x-coordinate of the vertex gives us the optimal number of penicillin units to produce and sell.
The x-coordinate of the vertex of a quadratic function in the form of ax^2 + bx + c can be found using the formula: x = -b / (2a)
In our profit function P(x) = -0.003x^2 + 120x - 500000, a = -0.003 and b = 120. Plugging these values into the formula, we get:
x = -120 / (2 * -0.003)
x = 20000
Therefore, the optimal number of penicillin units to manufacture and sell in the specified time period to maximize profit is 20000 units.