Algebra I: Descriptive Modeling

Distance between cities (such as Atlanta, GA and Athens, GA) is most appropriately measured in miles.

General Form: $y = mx + b$ or $C = bx + a$ , representing constant additive changes.
Carpet Cleaner Cost: Given by $C = 20 + 25x$ , where $C$ is total cost in dollars and $x$ is the number of days.
- Cost for $11$ days: $C = 20 + 25(11) = 295$ dollars.
- Days rented for $170$ dollars: 170 = 20 + 25x \times 1 \tag*{} \rightarrow 150 = 25x \rightarrow x = 6 days.
Net Profit: Given by $p = 15d - 20$ , where $p$ is net profit and $d$ is dogs walked per day.
- The constant $20$ represents the daily operating cost in dollars.
Gym Membership: An initial fee of $50$ dollars plus $25$ dollars per month is best modeled by a linear equation.
Jet Ski Rental: Total cost of $250$ dollars for $6$ hours, with a base rate of $150$ dollars for the first $2$ hours.
- Model equation: $4h + 150 = 250$
- Additional hourly rate $h$ : $4h = 100 \rightarrow h = 25$ dollars per hour.

General Form: $y = a(b)^t$
- $a$ : Initial value or starting amount.
- $b$ : Growth factor ( $b > 1$ ) or decay factor ( $b < 1$ ).
Investment Growth: $V(t) = 30\,000(1.125)^t$
- The value $1.125$ represents the growth factor of the investment.
Car Value Depreciation: A $38\,000$ dollar car decreasing in value by $12\%$ per year is best modeled by an exponential equation.
Magazine Subscriptions: $y = 42\,000(0.96)^t$
- The value $42\,000$ represents the initial number of subscribers.
Bacteria Population: $y = 500(1.16)^t$
- The value $500$ represents the initial number of bacteria.