Question

1. mp problem solving you decide to make and sell bracelets. the cost of your materials is $84.00. you charge $3.50 for each bracelet.

a. write a function that represents the profit $p$ for selling $b$ bracelets.
b. which variable is independent? dependent? explain.
c. you will break even when the cost of your materials equals your income. how many bracelets must you sell to break even?

1. mp modeling real life a furniture store is having a sale where everything is 40% off.

a. write and graph a function that represents the amount of discount on an item at regular price.
b. you buy a bookshelf that has a regular price of $85. what is the sale price of the bookshelf?

1. mp reasoning you want to take a two-hour airboat tour. which is a better deal, snake tours or gator tours? use functions to justify your answer.

snake tours
$25 per hour
all rates are per person.
gator tours
$35 boarding fee plus $5 each 1/2 hour
all rates are per person.

Explanation:

Problem 36

Step1: Define profit formula

Profit = Total Income - Total Cost
Total income from selling \(b\) bracelets is \(3.50b\), fixed cost is \$84.00.
\(P = 3.50b - 84\)

Step2: Identify independent/dependent

Independent variable: \(b\) (number of bracelets sold, it is controlled freely). Dependent variable: \(P\) (profit depends on the number of bracelets sold).

Step3: Calculate break-even point

Set profit \(P=0\) (cost = income):
\(0 = 3.50b - 84\)
Solve for \(b\):
\(3.50b = 84\)
\(b = \frac{84}{3.50} = 24\)

Step1: Define discount function

Let regular price = \(p\), discount = \(d\). 40% off means discount is 40% of regular price.
\(d = 0.40p\)
(Graph: Linear line through origin, slope = 0.40, x-axis = regular price, y-axis = discount)

Step2: Calculate sale price

Sale price = Regular price - Discount
Discount = \(0.40 \times 85 = 34\)
Sale price = \(85 - 34 = 51\)

Step1: Define cost function for Snake Tours

Cost = Hourly rate × Hours
\(C_s = 25t\), where \(t=2\) hours
\(C_s = 25 \times 2 = 50\)

Step2: Define cost function for Gator Tours

1/2 hour rate = \$5, so hourly rate = \$10. Cost = Boarding fee + (Hourly rate × Hours)
\(C_g = 35 + 10t\), where \(t=2\) hours
\(C_g = 35 + 10 \times 2 = 55\)

Step3: Compare costs

\(50 < 55\), so Snake Tours is cheaper.

Answer:

a. \(P = 3.50b - 84\)
b. Independent: \(b\) (number of bracelets, set freely); Dependent: \(P\) (profit depends on \(b\))
c. 24 bracelets

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Problem 37