Question

ryan and taylor are both saving money to buy new video game equipment. ryan’s savings plan can be modeled by the function $s = 15m + 50$ where $m$ represents the number of months since ryan started his plan, and $s$ represents the amount of money saved in dollars. the table shown represents taylor’s savings plan.

taylors savings plan

number of months	amount saved ($)
1	45
2	65
3	85

select the options that correctly complete the sentences.
ryan’s rate of change is dropdown taylor’s rate of change.
ryan’s starting balance is dropdown taylor’s starting balance.

Explanation:

Step1: Find Ryan's rate of change

Ryan's savings function is \( s = 15m + 50 \). In a linear function \( y = mx + b \), the slope \( m \) is the rate of change. So Ryan's rate of change is \( 15 \) dollars per month.

Step2: Find Taylor's rate of change

To find Taylor's rate of change, we use the formula for slope \( \frac{\Delta s}{\Delta m} \). Using the table, when \( m = 0 \), \( s = 25 \); when \( m = 1 \), \( s = 45 \). So \( \Delta s = 45 - 25 = 20 \), \( \Delta m = 1 - 0 = 1 \). Thus, Taylor's rate of change is \( \frac{20}{1} = 20 \) dollars per month.

Step3: Compare Ryan's and Taylor's rate of change

Ryan's rate is \( 15 \), Taylor's is \( 20 \). So Ryan's rate of change is less than Taylor's.

Step4: Find Ryan's starting balance

For Ryan's function \( s = 15m + 50 \), when \( m = 0 \) (starting time), \( s = 50 \). So Ryan's starting balance is \( 50 \) dollars.

Step5: Find Taylor's starting balance

From Taylor's table, when \( m = 0 \), \( s = 25 \). So Taylor's starting balance is \( 25 \) dollars.

Step6: Compare Ryan's and Taylor's starting balance

Ryan's starting balance (\( 50 \)) is greater than Taylor's (\( 25 \)).

Answer:

Ryan's rate of change is less than Taylor's rate of change.
Ryan's starting balance is greater than Taylor's starting balance.