The profit (in millions of dollars) from the sale of x million units of Blue Glue is given by p= .7x-25.5. The cost is given by c= .9x +25.5
(a) Find the revenue equation.
(b) What is the revenue from selling 10 million units?
(c)What is the break-even point?
To find the revenue equation, you need to multiply the number of units sold by the price per unit. In this case, the price per unit is given by the profit equation p = 0.7x - 25.5.
(a) Revenue (r) = Price per unit (p) × Number of units sold (x)
Substituting the value of p from the profit equation:
r = (0.7x - 25.5) × x
= 0.7x^2 - 25.5x
So, the revenue equation is r = 0.7x^2 - 25.5x.
(b) To find the revenue from selling 10 million units, you can substitute x = 10 into the revenue equation:
r = 0.7x^2 - 25.5x
= 0.7(10)^2 - 25.5(10)
= 0.7(100) - 255
= 70 - 255
= -185 million dollars
Therefore, the revenue from selling 10 million units is -185 million dollars.
(c) The break-even point occurs when the revenue (r) equals the cost (c). The cost equation is given by c = 0.9x + 25.5.
Setting r and c equal to each other:
0.7x^2 - 25.5x = 0.9x + 25.5
Simplifying the equation:
0.7x^2 - 26.4x - 0.9x - 25.5 = 0
0.7x^2 - 27.3x - 25.5 = 0
To find the break-even point, you need to solve this quadratic equation. You can either factor it, complete the square, or use the quadratic formula to find the two values of x where the equation equals zero.
Once you find those values of x, they represent the number of units required to break even.
To find the revenue equation, we need to multiply the number of units sold (x) by the selling price per unit.
The selling price per unit is given by p = 0.7x - 25.5.
Therefore, the revenue equation can be expressed as:
R = x * (0.7x - 25.5)
To calculate the revenue from selling 10 million units, we substitute x = 10 into the revenue equation:
R = 10 * (0.7(10) - 25.5)
R = 10 * (7 - 25.5)
R = 10 * (-18.5)
R = -185 million dollars
The break-even point occurs when the revenue equals the cost. So, we need to set the revenue equation equal to the cost equation and solve for x:
x * (0.7x - 25.5) = 0.9x + 25.5
Simplifying the equation, we get:
0.7x^2 - 25.5x = 0.9x + 25.5
Combining like terms, we have:
0.7x^2 - 26.4x - 0.9x - 25.5 = 0
Simplifying further, we get:
0.7x^2 - 27.3x - 25.5 = 0
Now, we need to solve this quadratic equation. We can either use factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula for this example.
The quadratic formula states:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
Applying it to our equation, we have:
x = (-(-27.3) ± sqrt((-27.3)^2 - 4(0.7)(-25.5)))/(2(0.7))
Simplifying further, we get:
x = (27.3 ± sqrt(746.49 + 71.4))/1.4
x = (27.3 ± sqrt(817.89))/1.4
Taking the square root, we have:
x ≈ (27.3 ± 28.6)/1.4
Therefore, the two solutions are:
x ≈ 55.9/1.4 ≈ 39.93
x ≈ -0.7/1.4 ≈ -0.5
Since we can't have a negative number of units sold, the break-even point is approximately 39.93 million units.