Algebra word problem about planning a concert

[Mr Davis 97]

Homework Statement

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?

Relevant Equations

Quadratic equations?

This seems like a simple problem, but I am a little confused by a few things.
For one, what is the use of the piece of information that when they charged $100 per person they got 3000 people to come?

Also, how should I proceed with the information "for every $2 decrease in price they would have another 150 people attend the show." Does this mean that the two quantities are inversely proportional? Like if P is population and C is cost per person, is P = -75C?

 

[fresh_42]

Homework Statement:: 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?
Relevant Equations:: Quadratic equations?

This seems like a simple problem, but I am a little confused by a few things.
For one, what is the use of the piece of information that when they charged $100 per person they got 3000 people to come?

It means that the prize determines the size of the audience: $\$\,100 \,\triangleq \,3,000$ people, $\$\,98 \,\triangleq \,3,150$ people, $\$\,96 \,\triangleq \,3,300$ people, etc.

Also, how should I proceed with the information "for every $2 decrease in price they would have another 150 people attend the show." Does this mean that the two quantities are inversely proportional?

Yes. Although it is strictly speaking no proportion. A proportion is an equation $y=m\cdot x$. Here we have $y=m\cdot x + c\,.$

Like if P is population and C is cost per person, is P = -75C?

Like, yes, exactly, no. You have a linear equation for the number of visitors in dependence of the prize per ticket. It is neither costs, since the money is a yield, nor is it important to know anything per person except for the ticket prize. Costs are the $\,67,300$ for the band, so you should at least gain these costs.

 

[sysprog]

Costs are the $67,300 for the band, so you should at least gain these costs.

$-$ and at most gain that cost, given that your goal is to maximize attendance $\dots$