Question

amelia is raising money for her school dance. she has already raised some money by selling bracelets, and now she is selling boxes of cookies to raise more money. the table below shows the total amount of money amelia will have raised after she sells 6, 8, 10, or 15 boxes of cookies.

boxes of cookies sold (c)	total money raised (r)
8	$130
10	$136
15	$151

complete the equation to represent the situation.

$r = \square c + \square$

complete the sentences to make true statements.

$\\$118$ $\text{number of boxes sold}$ $\\$106$ $\\$3$ $\text{initial value}$ $\text{rate of change}$

the amount of money amelia has already raised by selling bracelets is represented by the $\underline{\quad}$, which is $\underline{\quad}$. the additional amount amelia raises with each box of cookies she sells is represented by the $\underline{\quad}$, which is $\underline{\quad}$ per box.

Explanation:

Step1: Find the slope (rate of change)

The slope \( m \) of a linear equation \( R = mc + b \) is calculated using two points \((c_1, R_1)\) and \((c_2, R_2)\) with the formula \( m=\frac{R_2 - R_1}{c_2 - c_1} \). Let's take the points \((6, 124)\) and \((8, 130)\).
\( m=\frac{130 - 124}{8 - 6}=\frac{6}{2}=3 \).

Step2: Find the y-intercept (initial value)

Use the slope \( m = 3 \) and one point, say \((6, 124)\), in the equation \( R = mc + b \). Substitute \( c = 6 \), \( R = 124 \), and \( m = 3 \):
\( 124 = 3(6)+b \)
\( 124 = 18 + b \)
Subtract 18 from both sides: \( b = 124 - 18 = 106 \).

Step3: Interpret the slope and y-intercept

• The y-intercept (\( b = 106 \)) represents the initial money raised (before selling cookies), so it's the initial value.
• The slope (\( m = 3 \)) represents the rate of change, i.e., the money raised per box of cookies.

Answer:

The equation is \( R = \boldsymbol{3}c + \boldsymbol{106} \).
The amount of money Amelia has already raised by selling bracelets is represented by the \(\boldsymbol{\text{initial value}}\), which is \(\boldsymbol{\$106}\). The additional amount Amelia raises with each box of cookies she sells is represented by the \(\boldsymbol{\text{rate of change}}\), which is \(\boldsymbol{\$3}\) per box.