The quadratic formula and the discriminat.
The daily production of a company is modeled by the function
p=-w^2 + 75w - 1200. The daily production, p, is dependent on the number of workers w, present. If the break-even point is when p=0, what are the least and greatest number of workers the company must have present each day in order to break even?
I used the quadratic formula and my answers are: 23.14 and 51.86. Did I work it correctly and if not, please explain to me how to solve. Thanks.
w^2 + 75w - 1200 = 0
I could not find the answer. It needs to be factors of 1200. Since the third term is negative, one needs to be negative and the other positive to add to 75. For example, 100 and -12, but that sums to 88. I could use 80 and -15, but that gives me 65. Do you have a typo?
The equation is:
-w^2 +75w - 1200
the neg sign was so close to the = sign above. Sorry about that.
Jane, just switch all the signs, (multiply each term by -1)
There is no need for a quadratic to ever start with a negative
w^2 - 75w + 1200 = 0
both of your answers are correct.
To solve the quadratic equation p = -w^2 + 75w - 1200 for the break-even point (when p = 0), you can indeed use the quadratic formula. The quadratic formula is used to solve equations in the form ax^2 + bx + c = 0, where a, b, and c are constants.
In your case, the equation is p = -w^2 + 75w - 1200. To apply the quadratic formula, we can rearrange the equation to the standard form ax^2 + bx + c = 0 by moving all terms to one side of the equation:
-w^2 + 75w - 1200 = 0.
Now we can identify the coefficients:
a = -1 (coefficient of w^2)
b = 75 (coefficient of w)
c = -1200 (constant term)
Applying the quadratic formula, we have:
w = (-b ± √(b^2 - 4ac)) / 2a.
Plugging in the values, we get:
w = (-(75) ± √((75)^2 - 4(-1)(-1200))) / 2(-1).
Simplifying further:
w = (-75 ± √(5625 - 4800)) / (-2).
w = (-75 ± √825) / (-2).
Now, to find the values of w, we consider both the positive and negative solutions for the square root:
w = (-75 + √825) / (-2) ≈ 51.86,
w = (-75 - √825) / (-2) ≈ 23.14.
So, the least and greatest numbers of workers the company must have present each day to break even are approximately 23.14 and 51.86, respectively.
Therefore, your answers of 23.14 and 51.86 are correct.