Question

pret rate of change and tial valueuse the information to answer problems 1-2.the following table shows the cost to order muffins to be delivered from a bakery.| muffins | 1 | 5 | 10 | 50 | 100 || ---- | ---- | ---- | ---- | ---- | ---- || cost ($) | 17 | 25 | 35 | 115 | 215 |1. use structure what is the rate of change, and what does it represent?2. what is the initial value, and what does it represent?use the information to answer problems 3-5.the following linear function represents the cost of a home printer plus ink cartridges where $x$ represents the number of ink cartridges purchased and $y$ represents the total cost.$y = 25x + 115$3. what is the rate of change, and what does it represent?4. how much does the printer itself cost?5. how much will the purchaser spend buying the printer and 6 ink cartridges?6. open ended write a scenario that can be modeled by the linear function $y = 25x + 40$.lesson 6.3more practice/homeworkonlinevideo tutorials andinteractive examples$15 delivery fee muffins

Explanation:

Step1: Calculate rate of change (Q1)

Use $\text{Rate of Change} = \frac{\Delta \text{Cost}}{\Delta \text{Muffins}}$. Take points (1,17) and (5,25):
$\text{Rate of Change} = \frac{25-17}{5-1} = \frac{8}{4} = 2$

Step2: Interpret rate of change (Q1)

The rate is the cost per muffin.

Step3: Find initial value (Q2)

Use linear form $y=mx+b$. Substitute $m=2$, $x=1$, $y=17$:
$17 = 2(1) + b \implies b=15$

Step4: Interpret initial value (Q2)

The initial value is the fixed delivery fee.

Step5: Identify rate of change (Q3)

From $y=25x+115$, rate is the coefficient of $x$.

Step6: Interpret rate of change (Q3)

The rate is cost per ink cartridge.

Step7: Find printer cost (Q4)

Initial value (b) in $y=25x+115$ is the printer cost.

Step8: Calculate total cost (Q5)

Substitute $x=6$ into $y=25x+115$:
$y=25(6)+115=150+115=265$

Step9: Create scenario (Q6)

Map $y=25x+40$ to a real linear cost scenario.

Answer:

1. The rate of change is $\$2$ per muffin. It represents the cost of each individual muffin.
2. The initial value is $\$15$. It represents the fixed delivery fee for the order.
3. The rate of change is $\$25$ per ink cartridge. It represents the cost of one ink cartridge.
4. $\$115$
5. $\$265$
6. Example scenario: A gym charges a $\$40$ initial membership fee, plus $\$25$ for each month of access. Here, $x$ is the number of months, and $y$ is the total cost of membership.