Help with quadratic budget problem

Gov’t Mule is planning a show at the Art Park in Lewiston for next summer. The year before they charged $100 and 3000 people attended the show. Through market research they learned that for every $2 decrease in price they would have another 150 people attend the show.

The band only really needs to earn $67300 in order to pay for all the touring costs. They decide they are willing to earn just $67300 in order to have the maximum number of people attend the concert. How much should they charge for this to happen and how many people attended the show?

The equation I have is y=(100-2x)(3000+150), I just don’t know if that’s right. And i don’t know how I would calculate the part about $63700 and all.

This one was kind of tricky, but I think I have a solution.

So let’s really focus in on two quantities:

(1) p = \text{number of people who attend the NEW concert}
(2) c = \text{cost per person for attending the NEW concert}

We know that when c decreases by 2, p increases by 150. So we have a linear relationship between these two quantities, and the slope of this linear relationship is \frac{\Delta p}{\Delta c} = \frac{150}{-2} = -75.

What’s some other information we know about this linear relationship? We know that one point of the line of the linear relationship when graphed should be (c_0,p_0) = (100, 3000).

So given all this, we have enough information to construct the equation for the line that relates p and c, using the point-slope formula.

p - p_0 = -75(c - c_0)\tag*{}
p - 3000 = -75(c - 100)\tag*{}
p = -75c + 10500 \tag*{}

But also note that the band is willing to earn enough just to cover the costs, so since multiplying p and c gives the total earnings from the concert, we have that

pc = 67300\tag*{}

So now we have two equations with two unknowns, and so we can solve this system by substitution, which should result in a quadratic equation that gives two solutions: (c,p) = (6.773, 9995), (133.3, 505). Since we want to maximize the population, we should go with the solution c=6.773, p = 9995.

Does this clarify things? Let me know!

This clarifies a little bit of the question but I’m still quite confused by the steps to the solution.

Sorry, I probably went too fast. This is a tricky word problem.

So with these word problems, the first thing you always need to do is know what you’re solving for. In this case, we are solving for: “How much should they charge per person given that the band wants to earn $67300?” and “How many people will attend the show given how much the band will charge?”

So we should define the dependent variable p = \{\text{how many people attended the show}\} and the independent variable c = \{\text{how much the band charges per person} \}. Additionally, because c influences p (if c decreases by 2 dollars, then p increases by 150 people), we can write an equation that shows this relationship between the two.

We know that p and c are two quantities that are related by a linear equation (like y=mx+b, where y is p as the independent variable, and x is c as the dependent variable) precisely because when c changes a fixed amount (-2), p also changes a fixed amount (150).

So to find the line that this relationship represents we should use the point-slope formula, where p - p_0 = m(c - c_0), where m is the slope and (c_0,p_0) is a point on the line.

Well, so what is m, the slope? It’s what we talked about earlier: whenever c decreases by 2, then p increases by 150, so the slope must be \frac{150}{-2} = -75.

So now our equation looks like p - p_0 = -75(c - c_0). We’re almost done with finding the equation, but we need to know what the point (c_0,p_0) is. Luckily, at the beginning of the problem, we’re given that the band Gov’t Mule the year before charged $100 per person and 3000 people attended the show, so we know that the point (c_0,p_0) = (100, 3000) has to be on the line.

So now our equation looks like p - 3000 = -75(c - 100). And if we simplify it, it looks like p = -75c + 10500. Great, so now we know how the number of people who attend changes as the price per person changes.

So, just as a quick example, if the band chose to charge 50 dollars per person, then you just substitute 50 for c to get that p = -75(50) + 10500 = 6750 people attended.

So yeah, we have the equation that relates p and c. But we’re not done, because we haven’t answered the question, given that the band only wants to earn $67,300, how much should they charge, and how many people attend if they charge that amount?

So if the band only wants $67,300, then it must be the case that pc = 67300, since pc is the total revenue that the band makes from ticket sales. Think about how if, for example, the concert had 10 people and it was $2 per ticket, the revenue would be 10\cdot 2 = 20 dollars.

So now we have two equations with two unknowns, so we can solve it:

\begin{cases} p = -75c + 10500 ~~~~~(1)\\ pc = 67300 ~~~~~~~~~~~~~~~~~(2) \end{cases}\tag*{}

Take equation (2) and solve for p to get p = \dfrac{67300}{c} and plug this into (1) and solve:

\dfrac{67300}{c} = -75c + 10500\tag*{}
67300 = -75c^2 + 10500c \tag*{}
0 = 75c^2 - 10500c + 67300 \tag*{}
c = 6.773 \text{ or } 133.3\tag*{}

Plug these into either of the equations (1) or (2) to get the corresponding values of p: p = 9950, 505.
Since we want the solution with the largest population, we should choose the solution (6.773, 9950). And so we’re done.

Let me know if you have any other specific questions!

I haven’t learned point-slope formula so is there any other alternative to solve this point?

Okay, that’s fine.

You can still find the right equation that relates p and c without it.

So instead of point-slope, we’ll just use the slope-intercept form.

So p = mc + b, and we know that m = -75, so p = -75c + b.

So we just need to find b.

And we can do this by plugging into the equation the one point we know is on the curve, (c_0,p_0) = (100, 3000), and solve for b:

p = -75c + b \implies (3000) = -75(100) + b \tag*{}
\implies b = 10500\tag*{}

So p = -75c + 10500.

Ok this is something that makes more sense to me. Thank you so much for all your help.

No problem! This was definitely a tougher problem, so don’t sweat it if it seems a little unintuitive

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