A company has determined that the profit, in dollars, it can expect from the manufacture and sale of tennis racquets is given by P = −0.01x2 + 130x − 180,000. How many racquets should the company manufacture and sell to earn a profit of $174,900?
just solve for x in the equation
−0.01x2 + 130x − 180,000 = 174900
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.
The profit equation is given as P = −0.01x^2 + 130x − 180,000, where P represents the profit in dollars and x represents the number of racquets manufactured and sold.
Set up the equation as follows:
−0.01x^2 + 130x − 180,000 = 174,900
Now, to solve the quadratic equation for x, we can rearrange it to the standard quadratic form:
−0.01x^2 + 130x − 180,000 − 174,900 = 0
Simplifying further:
−0.01x^2 + 130x − 354,900 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = -0.01, b = 130, and c = -354,900.
Plugging in the values:
x = (-(130) ± √((130)^2 - 4(-0.01)(-354,900))) / (2(-0.01))
Simplifying further:
x = (-130 ± √(16,900 + 14,196)) / (-0.02)
x = (-130 ± √31,096) / (-0.02)
Now, calculate the square root:
x = (-130 ± 176.32) / (-0.02)
Now, consider both cases, positive and negative square root:
Case 1: x = (-130 + 176.32) / (-0.02)
x = 23.16 / (-0.02)
x ≈ -1,158
Case 2: x = (-130 - 176.32) / (-0.02)
x = -306.32 / (-0.02)
x ≈ 15,316
Since the number of racquets cannot be negative, we take the positive value of x, which is approximately 15,316.
Therefore, the company should manufacture and sell approximately 15,316 racquets to earn a profit of $174,900.