Question

1. a teacher read part of a book to a class. the graph shows the number of pages read by the teacher over the next several days. find and interpret the rate of change and the initial value. (example 1)
2. a water park charges a rental fee plus $1.50 per hour to rent inflatable rafts. the total cost to rent a raft for 6 hours is $15. assume the relationship is linear. find and interpret the rate of change and the initial value. (example 2)
3. a teacher already had a certain number of canned goods for the food drive. each day of the food drive, the class plans to bring in 10 cans. the total number of canned goods for day 10 is 205. assume the relationship is linear. find and interpret the rate of change and the initial value. (example 2)
4. melissa frosted some cupcakes in the morning for a party. the table shows the total number of cupcakes frosted after she starts up after lunch. assume the relationship between the two quantities is linear. find and interpret the rate of change and the initial value. (example 1)

Explanation:

Problem 1

Step1: Calculate rate of change

Use two points, e.g., $(3,450)$ and $(4,550)$.
Rate of change = $\frac{550 - 450}{4 - 3} = 100$

Step2: Find initial value

Use point-slope form $y - y_1 = m(x - x_1)$. Substitute $m=100$, $(3,450)$:
$y - 450 = 100(x - 3)$
$y = 100x + 150$
Initial value is $150$.

Step3: Interpret values

Rate of change: 100 pages per day. Initial value: 150 pages read before day 1.

Step1: Define linear equation

Let $y = mx + b$, where $m=1.50$ (hourly rate), $y=15$, $x=6$.

Step2: Solve for initial value $b$

$15 = 1.50(6) + b$
$15 = 9 + b$
$b = 15 - 9 = 6$

Step3: Interpret values

Rate of change is the hourly cost. Initial value is the rental fee.

Step1: Define linear equation

Let $y = mx + b$, where $m=10$ (cans per day), $y=205$, $x=10$.

Step2: Solve for initial value $b$

$205 = 10(10) + b$
$205 = 100 + b$
$b = 205 - 100 = 105$

Step3: Interpret values

Rate of change is daily cans added. Initial value is starting cans.

Answer:

Rate of change: 100 pages per day (the teacher reads 100 additional pages each day). Initial value: 150 pages (the number of pages the teacher had already read before the tracked days began).

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Problem 2