The demand equation for a product is p = 60 – 0.0004x, where p is the price per unit and x is the number of units sold. The total revenue for selling x units is shown below.
Revenue = xp = x(40 – 0.0004x)
How many units must be sold to produce a revenue of $260,000? (Round to the nearest unit. Enter your answers from smallest to largest. Enter NONE in any unused answer blanks.)
x =
units (smaller value)
x = units (larger value)
you have the formula. Just plug in your numbers:
260000 = x(40-0.0004x)
now just solve the quadratic equation for x. First, decide whether to use 40 or 60.
To find the number of units that must be sold to produce a revenue of $260,000, we need to set the revenue equation equal to $260,000 and solve for x.
The revenue equation is given as:
Revenue = xp = x(40 – 0.0004x)
Setting this equal to $260,000, we have:
260,000 = x(40 – 0.0004x)
Now, let's solve this equation for x.
Step 1: Multiply x with each term inside the parentheses.
260,000 = 40x - 0.0004x^2
Step 2: Rearrange the equation to bring it in the form of a quadratic equation.
0.0004x^2 - 40x + 260,000 = 0
Step 3: Multiply through by 10,000 to remove the decimal.
0.0004x^2 - 40x + 2,600,000 = 0
Step 4: Divide through by 0.0004 to simplify the equation.
x^2 - 100,000x + 6,500,000,000 = 0
Step 5: Solve the quadratic equation using factoring, completing the square, or using the quadratic formula.
Using the quadratic formula, we have:
x = (-(-100,000) ± √((-100,000)^2 - 4(1)(6,500,000,000))) / (2(1))
Simplifying:
x = (100,000 ± √(10,000,000,000 - 26,000,000,000)) / 2
x = (100,000 ± √(-16,000,000,000)) / 2
The expression inside the square root is negative, which means the equation has no real solutions. Therefore, we cannot find a positive value for x that results in a revenue of $260,000.
Hence, there are no units that must be sold to produce a revenue of $260,000.