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