A company has determined that the profit, in dollars, it can expect from the manufacture and sale of tennis racquets is given by P = −0.01x^2 + 130x − 180,000. How many racquets should the company manufacture and sell to earn a profit of $174,900? (Enter your answers as a comma-separated list.)
To find out how many racquets the company should manufacture and sell to earn a profit of $174,900, we need to set the profit equation equal to $174,900 and solve for x.
The given profit equation is: P = -0.01x^2 + 130x - 180,000
Setting the equation equal to $174,900:
-0.01x^2 + 130x - 180,000 = 174,900
Rearranging the equation:
-0.01x^2 + 130x - 354,900 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = -0.01, b = 130, and c = -354,900.
Plugging in the values into the quadratic formula:
x = (-130 ± √(130^2 - 4*(-0.01)*(-354,900))) / (2*(-0.01))
Calculating the discriminant, b^2 - 4ac:
Discriminant = 130^2 - 4*(-0.01)*(-354,900)
= 16,900 - 56,976
= -40,076
Since the discriminant is negative, there are no real solutions to the equation. Therefore, the company cannot manufacture and sell a number of racquets to earn a profit of $174,900.
To find the number of racquets the company should manufacture and sell to earn a profit of $174,900, we need to set the profit equation equal to $174,900 and solve for x.
The profit equation is given by:
P = -0.01x^2 + 130x - 180,000
Setting it equal to $174,900:
-0.01x^2 + 130x - 180,000 = 174,900
Rearranging the equation:
0.01x^2 - 130x + 354,900 = 0
Now, we can solve this quadratic equation to find the values of x. There are several methods to solve a quadratic equation, such as factoring, completing the square, or using the quadratic formula.
In this case, let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 0.01, b = -130, and c = 354,900. Plugging these into the quadratic formula, we get:
x = (-(-130) ± √((-130)^2 - 4(0.01)(354,900))) / (2 * 0.01)
Simplifying:
x = (130 ± √(16,900 + 14,196)) / 0.02
x = (130 ± √31,096) / 0.02
Calculating the square root:
x = (130 ± 176.36) / 0.02
Now, we can solve for the two possible values of x:
x1 = (130 + 176.36) / 0.02
x2 = (130 - 176.36) / 0.02
Calculating:
x1 = 15,318
x2 = -2,318
Since manufacturing a negative number of racquets is not possible, the company should manufacture and sell approximately 15,318 racquets to earn a profit of $174,900.