00:01
So here we have a tour company that's interested in its profits.
00:04
Now the profit function is p of n equal to negative 0 .6n squared plus 36n minus 440, where n is equal to the number of students on the tour.
00:21
And in part a, we want the number of students that gives the maximum profit per student.
00:27
Well, since p of n is a quadratic function, we can simply find the vertex of this quadratic function because we have a parabola that opens downward.
00:37
So our formula says that the n value will be negative b over 2a when the quadratic functions in standard form.
00:45
Well, the b in this case is the coefficient of a linear term, that's 36, so we have negative 36 over 2 times a, which is the coefficient of the quadratic term, 0 .6.
00:56
And this will end up giving us a value of 30.
01:00
So 30 students is the number of students at which the maximum profit per student will be achieved.
01:07
And then in part b, what we'll do is we want to know what is that maximum profit per student.
01:13
Well, then we simply plug 30 into the profit function.
01:16
We find p of 30, which is negative 0 .6 times 30 squared, plus 36 times 30, minus 440, and that will give us $100.
01:29
So the maximum profit of $100 per student will be achieved when we have 30 students on the tour.
01:36
But then we want to know what is the least and greatest numbers of students that should be accepted in order to make a profit? well, here we have a quadratic function.
01:48
So with this quadratic function, we know that it's going to look something like this.
01:52
So in order to make a profit, we're going to have a, to be above the n -axis.
02:00
So we have to know, okay, what are the x -intercepts here, or the n -intercepts? so what we'll do is we'll set the function, negative 0 .6n squared, plus 36n minus 440, equal to 0...