The revenue and cost equations for a product are

R = x(50 - 0.0002x) and C = 12x + 150,000
where R and C are measured in dollars and x represents the number of units sold. How many units must be sold to obtain a profit of at least $1,650,000? What is the price per unit?

275, 000. $650

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To find the number of units that must be sold to obtain a profit of at least $1,650,000, we need to calculate the profit equation and then solve for x.

The profit equation is given by:

Profit = Revenue - Cost

Substituting the revenue and cost equations given:

Profit = (x(50 - 0.0002x)) - (12x + 150,000)

Now, we can set up the equation and solve for x:

Profit = (x(50 - 0.0002x)) - (12x + 150,000) ≥ 1,650,000

Expanding and simplifying the equation:

50x - 0.0002x² - 12x - 150,000 ≥ 1,650,000

Combining like terms:

-0.0002x² + 38x - 150,000 ≥ 1,650,000

Subtracting 1,650,000 from both sides:

-0.0002x² + 38x - 1,800,000 ≥ 0

To solve this quadratic inequality, we can find the x-intercepts (where the equation equals zero) and determine the intervals where the function is greater than or equal to zero.

Using the quadratic formula, we have:

x = (-b ± √(b² - 4ac)) / 2a

Where a = -0.0002, b = 38, and c = -1,800,000.

Plugging in the values, we get:

x = (-38 ± √(38² - 4(-0.0002)(-1,800,000))) / (2(-0.0002))

Simplifying the equation, we get:

x = (-38 ± √(902.44 + 1440000)) / (-0.0004)

x = (-38 ± √(1,440,902.44)) / (-0.0004)

x = (-38 ± 1,199.257) / (-0.0004)

We can discard the negative root since the number of units sold cannot be negative. Therefore, we have:

x = (-38 + 1,199.257) / (-0.0004)

x = 1,161.257 / (-0.0004)

x ≈ -2,903,142.5

Since the number of units sold cannot be negative, we can round up the value to the nearest whole number:

x ≈ -2,903,142.5 ≈ -2,903,142

Therefore, we need to sell approximately 2,903,142 units to obtain a profit of at least $1,650,000.

To find the price per unit, we can substitute this value back into the revenue equation:

R = x(50 - 0.0002x)

R = 2,903,142(50 - 0.0002(2,903,142))

Simplifying the equation, we get:

R ≈ 2,903,142(50 - 580.6284)

R ≈ 2,903,142(49.4191)

R ≈ 143,705,844.723

Therefore, the price per unit is approximately $49.42.

To find the number of units that must be sold to obtain a profit of at least $1,650,000, we need to equate the revenue and cost functions and solve for x.

First, let's set the revenue (R) equal to the cost (C) and solve for x:

R = C

x(50 - 0.0002x) = 12x + 150,000

Now, let's simplify this equation:

50x - 0.0002x^2 = 12x + 150,000

Rearranging the equation to standard form:

0.0002x^2 + 38x - 150,000 = 0

Now, we can solve the quadratic equation either by factoring, completing the square, or using the quadratic formula. Since this equation doesn't appear to factor easily, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 0.0002, b = 38, and c = -150,000. Plugging in these values into the quadratic formula:

x = (-38 ± √(38^2 - 4(0.0002)(-150,000))) / (2(0.0002))

Simplifying:

x = (-38 ± √(1444 + 1200)) / (0.0004)

x = (-38 ± √(2644)) / (0.0004)

Taking the positive root to obtain a meaningful solution:

x = (-38 + √(2644)) / (0.0004)

Calculating this value using a calculator, we find:

x ≈ 164,891.92

Therefore, approximately 164,892 units must be sold to obtain a profit of at least $1,650,000.

To find the price per unit, we can substitute this value of x back into either the revenue or cost equation. Let's use the revenue equation:

R = x(50 - 0.0002x)

Substituting x = 164,892:

R = 164,892(50 - 0.0002 * 164,892)

Calculating this value using a calculator, we find:

R ≈ $8,440,289.58

Since revenue (R) is the product of price per unit and quantity, we can rearrange the equation to find the price per unit:

Price per unit = R / x

Price per unit = $8,440,289.58 / 164,892

Calculating this value using a calculator, we find:

Price per unit ≈ $51.14