Algebraic Word Problem

I know it's a lot, but Im desperate and frustrated...any help would be greatly appreciated

Given: assume the number of items sold (n) depends linearly on the price charged for that item (x). From this, you will show that profit (p) depends on price charged.

A college bookstore has been selling tshirts for a year. During the fall term, 100 were sold at $5 each. During the Spring term, the price was raised to $7 and only 80 were sold.

a. write an equation showing the dependence of price charged on the number of shirts sold. what does this equation tell you in general? graph.

b. using the relationship in part A, how many tshirts would be sold during a term at $3.50 each? What should the cost of a shirt be in order to sell 45 in a term?

c. suppose it cost the bookstore $3 to buy each shirt. calculate the total profit where they may charge $5 $6, $7, $8, $9 and $10. Then calculate average rate of change in total profit between charging $5 & $6, $6 & $7, $7 & $8, $8 & $9, $9 & $10. Use this data to explain why profit does not depend linearly on price. Explain in general why profit should realistically depend quadratically on price.

d. generate a formular describing how P depends on x (profit = number sold * price per shirt). Graph. Find P intercept; what does this value mean?

e. calculate breakeven points.

f. find price that will maximize profit. what will the profit be?

g. explain why this information is important to the bookstore.

Please do not double post.

a.

x=5, y=100
x=7. y=80

First find the slope of the line which is given by (change in y)/(change in x)

m= 20/-2 = -10

Then solve for the form y=mx+b for one point to find b

100=-10*5+b

b=150

so we know that the equation for the line is y = -10x + 150
What we learn from the graph is that as price is increased t-shirt sales decrease. At a certain price no one will want to buy the shirts and if you gave them away (price = 0) you could "sell" 150 of them

b. using the relationship in part A, how many tshirts would be sold during a term at $3.50 each? What should the cost of a shirt be in order to sell 45 in a term?

y = -10x + 150

y=-10 * 3.5 +150

y=-35+150

y=115 shirts at $3.50

second part

45=-10x+150

-105=-10x

x=$10.50

c. suppose it cost the bookstore $3 to buy each shirt. calculate the total profit where they may charge $5 $6, $7, $8, $9 and $10. Then calculate average rate of change in total profit between charging $5 & $6, $6 & $7, $7 & $8, $8 & $9, $9 & $10. Use this data to explain why profit does not depend linearly on price. Explain in general why profit should realistically depend quadratically on price.

y=-10x+150
Profit = (number of shirts sold * selling price) - (number of shirts sold * cost)
Profit = (y*x) - (3*y)
P=(-10x+150)*x - 3*(-10x+150)
P=-10x^2 + 150x + 30x - 450
P=-10x^2 + 180x - 450

Then plug in 5, 6, 7, 8, 9, and 10 for x to get profit. example with 5
-10*(5)^2 + 180*5 - 450 = $200

Here is the graph of the profit.

P intercept shows how much money you would lose if you gave the shirts away.

The break even points are where the graph crosses the x-axis

The max of the graph is where you earn the most money.