Pat Kan owns a factory that manufactures souvenir key chains. Her weekly profit (in hundreds of dollars) is given by P(x)=-2x^2+60x-120, where x is the number of cases of key chains sold.
a) what is the largest number of cases she can sell and still make a point?
b) Explain how it is possible for her to lose money if she sells more cases than your answer in part (a)
I've been doing this problem since lastnite and i couldn;t figure out the answer. someone help me please? ireally need to learn how to solve this kind of problem. midterm is coming :[
I don't know what level of math you are taking, is it Calculus?
if you find the vertex (by Calculus or completing the square), you should have had (15, 330)
which means that if she sells 15 cases she makes a maximum profit of 330.
Your function is a parabola which opens downwards, so any other value besides x=15 would produce a smaller value than 330
this parabola crosses the x-axis at appr. 2.1 and 27.8
so it is positive for number of cases between 3 and 27 cases.
For any other number of cases, P(x) would be negative.
e.g. let x = 30 cases
P(30) = -120
thanks for the help really appreciate it! im taking business math. anyway, how did you find out that the parabola crosses the x-axis at appro. 2.1 and 27.8? and how did you come up with 27 cases?
sorry for asking a lotta questions. i just want to understand this problem.
Thanks again
Sorry about not answering last night, but had gone to bed.
I solved
0 = -2x^2+60x-120 using the quadratic formula
since x has to be a whole number, 3 and 27 would be the first and last whole numbers for which the parabola is above the x-axis, that is , for which P is positive.
Try subbing in x = 28 and x = 2, you will get a negative value for P.
To find the largest number of cases Pat Kan can sell and still make a profit (a), we need to determine the maximum value of the profit function P(x) = -2x^2 + 60x - 120. This can be done by finding the x-coordinate of the vertex of the parabola.
The vertex of a parabola in the form of y = ax^2 + bx + c is given by the x-coordinate x = -b/2a. In this case, a = -2, b = 60, and c = -120.
Using the formula, x = -b/2a = -60 / 2(-2) = -30 / -4 = 7.5.
Therefore, the largest number of cases Pat Kan can sell and still make a profit is 7.5 (which is not possible in reality since she can only sell whole cases). So, she would need to sell 7 cases to make a profit.
Now, let's consider how it is possible for her to lose money if she sells more cases than the answer in part (a) (b).
If Pat Kan decides to sell more than 7 cases, she will start incurring losses instead of making profits. This can be seen by analyzing the shape of the parabolic function and its graph.
The profit function P(x) = -2x^2 + 60x - 120 is a downward-opening parabola because the coefficient of x^2 is negative. This means that as the number of cases sold increases, the profit will initially increase until it reaches the maximum point (at 7 cases). After that point, as the number of cases sold continues to increase, the profit will start to decrease.
Therefore, if Pat Kan sells more than 7 cases, the negative coefficient of x^2 will overpower the positive coefficient of x, resulting in a negative profit. In other words, she will lose money.
It's important to note that this is a mathematical model, and in the real world, there may be other factors that could affect the profit of the factory, such as variable costs, market demand, competition, etc.