**I have number 2 and number 4 already answere****d**!!!!

Casey's arty party

The last few years Casey has thrown an “arty party” where peopl

e get together and paint. Since people have very different experiences with paint, Casey has hired a professional to facilitate the party. The painting teacher charges per hour, and provides painting supplies at a per-attendee cost. Last year, 10 people came to the 2 hour party and Casey's bill was $350. The year before, it was 9 people for 2 hours and the bill was $335.

Casey is going to ask this year's guests to chip in for the bill: $35 each. Casey worries whether this is in line with the previous years' expenses. It works fine for last year, but doesn't add up for the year before.

What is wrong with the following explanation of the discrepency: It was $350 for 10 people, so clearly $35 is the right price. For 9 people, Casey would only charge $315 total, saving her party-goers $20 each off the $335. What a bargain!

-Casey's ten year old brother.

1.) Indicate clearly a mistake. Make sure to explain (quantitatively) how much of a difference the mistake actually makes. What happens if her guests ask for the $20 discount?

Casey has the party again this year. Again 10 people show (including Casey), but this time it is $355. When Casey asks the painting teacher, the painting teacher explains the materials costs went up $0.50 each this year, now they are $15.50 each. 10 people, $0.50 more each, so $5 dollars more.

Word got out that the party was super-fun. Five more people want to have their own arty party, and ask if Casey will set it up. Casey likes this idea and decides to go into business setting up arty parties charging $40 per person.

2.)How much profit would Casey make on a two hour, 10 person arty party, if each person paid Casey $40?

The five people pay Casey the $40 each, $200 total, and Casey sets up the second arty party. Casey is pretty excited to be making $22.50 profit, but that excitement is short-lived. Unfortunately, the bill is $277.50, while the five people have only given Casey $200 total. Casey asks the painting teacher if there was a mistake, but the painting teacher says,

The painting supplies are $15.50 each, and you had five fewer people, so $77.50 less than the last one, $355.00 - $77.50 is $277.50. Happy painting!

-Painting teacher with a calculator

3.) Does this make sense? How much did Casey assume it would cost for the five person arty party? Why are the numbers so different?

4.)Are arty parties a real thing? Are there businesses that facilitate a party with painting or similar activities? Find a reliable source describing such a business, including their pricing model. How much does an arty party really cost?

5.)Give a mathematical formula for the total cost of an arty party from Casey's painting teacher that lasts H hours for P people. Show how it works for five and ten people parties that last 2 hours, as well as a few other examples.

**Make sure to indicate how the formula changed when the cost of materials changed from $15 each, to $15.50 each.

6.) How much should Casey charge per person to break even arranging arty parties?

How much should Casey charge per person to make a $50 profit each time?

How does the mathematical concept of domain and/or piecewise function help improve this answer?