The revenue and cost equations for a product are R=x (50-0.002x) and C= 12x+ 150000. Where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to get a profit of at least 1,650,000

P(x) = R(x)-C(x), so just solve for x in

x(50-.002x) - (12x+150000) = 1650000

I suspect a typo, since that has no real solutions.

To determine the number of units that must be sold to achieve a profit of at least $1,650,000, we need to find the break-even point between revenue and cost. The break-even point occurs when the revenue is equal to the cost.

The revenue equation is given as R = x(50 - 0.002x), where x represents the number of units sold.
The cost equation is given as C = 12x + 150,000.

To find the break-even point, we set the revenue equal to the cost:

x(50 - 0.002x) = 12x + 150,000.

Let's solve this equation step by step:

1. Distribute x to the terms inside the parentheses:
50x - 0.002x^2 = 12x + 150,000.

2. Combine like terms on both sides of the equation:
-0.002x^2 + 50x - 12x = 150,000.

-0.002x^2 + 38x = 150,000.

3. Move all terms to one side of the equation to form a quadratic equation:
-0.002x^2 + 38x - 150,000 = 0.

4. Multiply the entire equation by -1000 (or divide by -0.002) to eliminate the decimal coefficients:
2x^2 - 38000x + 1,500,000 = 0.

5. Solve the quadratic equation using factoring, completing the square, or using the quadratic formula.

Since this equation doesn't easily factor, let's use the quadratic formula, which states that for an equation ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac))/(2a).

For our equation 2x^2 - 38000x + 1,500,000 = 0, we have a = 2, b = -38000, and c = 1,500,000. Plugging these values into the quadratic formula, we get:

x = (-(-38000) ± √((-38000)^2 - 4(2)(1,500,000)))/(2(2)).
= (38000 ± √(1444000000 - 12000000))/(4).
= (38000 ± √(1432000000))/(4).
= (38000 ± 37888.41)/(4).

Now we can calculate the two possible values for x:

x1 = (38000 + 37888.41)/4 ≈ 18972.21/4 ≈ 4743.05.
x2 = (38000 - 37888.41)/4 ≈ 111.59/4 ≈ 27.9.

Since we cannot have a fraction of a unit sold, we must round up to the nearest whole number in order to get a profit of at least $1,650,000. Therefore, we need to sell a minimum of 4744 units to achieve the desired profit.