The revenue and cost equation for a product are
R = x(75 - 0.0005x) and C = 30x + 250,000
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 $750,000? What is the price per unit?
Profit = Revenue - Cost
subtract the two functions and simplify
you will have a quadratic with a negative x^2 term, so it opens down, thus there will be a max
find the vertex of that parabola
To find the number of units that must be sold to obtain a profit of at least $750,000, we need to set up the profit equation and solve for x.
Profit (P) is calculated by subtracting the cost (C) from the revenue (R):
P = R - C
Substituting the given equations for R and C:
P = x(75 - 0.0005x) - (30x + 250,000)
Simplifying the equation:
P = 75x - 0.0005x^2 - 30x - 250,000
Combining like terms:
P = -0.0005x^2 + 45x - 250,000
We want the profit to be at least $750,000, so we can write the following inequality:
-0.0005x^2 + 45x - 250,000 ≥ 750,000
Rearranging the inequality:
-0.0005x^2 + 45x - 1,000,000 ≥ 0
To solve this quadratic inequality, we can either use factoring or the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation -0.0005x^2 + 45x - 1,000,000 ≥ 0, the coefficients are:
a = -0.0005
b = 45
c = -1,000,000
Plugging these values into the quadratic formula:
x = (-(45) ± √((45)^2 - 4(-0.0005)(-1,000,000))) / (2(-0.0005))
Simplifying:
x = (-45 ± √(2025 - 2000)) / (-0.001)
x = (-45 ± √25) / (-0.001)
x = (-45 ± 5) / (-0.001)
This gives us two possible values for x:
1) x = (-45 + 5) / (-0.001) = 40,000
2) x = (-45 - 5) / (-0.001) = 50,000
Since we are looking for the number of units that must be sold, x must be a positive value. Therefore, the minimum number of units that must be sold to obtain a profit of at least $750,000 is 50,000 units.
To find the price per unit, we can substitute this value of x into the revenue equation:
R = x(75 - 0.0005x)
R = 50,000(75 - 0.0005 * 50,000)
R = 50,000(75 - 25)
R = 50,000 * 50
R = $2,500,000
So, the price per unit is $2,500,000 divided by the number of units sold, which is $2,500,000 / 50,000 = $50 per unit.
To find the number of units that must be sold to obtain a profit of at least $750,000, we need to set up and solve an equation.
Profit is defined as the difference between revenue (R) and cost (C). So, the profit equation is given by:
Profit = R - C
We can substitute the given revenue and cost equations into the profit equation:
Profit = x(75 - 0.0005x) - (30x + 250,000)
Now let's simplify the equation:
Profit = 75x - 0.0005x^2 - 30x - 250,000
Combining like terms:
Profit = -0.0005x^2 + 45x - 250,000
Since we want to find the number of units (x) that result in a profit of at least $750,000, we set up the following inequality:
Profit ≥ $750,000
Substituting the profit equation, we have:
-0.0005x^2 + 45x - 250,000 ≥ $750,000
Rearranging the inequality, we get:
-0.0005x^2 + 45x - 250,000 - $750,000 ≥ 0
Now we can solve this quadratic inequality to find the range of values for x that satisfy the condition. One approach is to solve for x using the quadratic formula and then determine the appropriate range.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -0.0005, b = 45, and c = -250,000 - $750,000.
Plugging those values into the quadratic formula:
x = (-45 ± √(45^2 - 4(-0.0005)(-1,000,000))) / (2(-0.0005))
Simplifying that equation will give you two possible values for x. The larger value represents the number of units that must be sold to obtain a profit of at least $750,000.
To find the price per unit, we can use the revenue equation:
R = x(75 - 0.0005x)
Substitute the value of x you found into the revenue equation to calculate the corresponding value of R. Then, divide that value by the number of units (x) to get the price per unit.