Algebra Quiz: Interpreting Parameters In Linear Exponential Models

A streaming service charges a flat monthly fee plus a per-movie rental charge. The total cost $T$ (in dollars) for renting $n$ movies in a month is $T = 4n + 12$. What does the 4 represent in this context?

A fitness tracker estimates calories burned during a walk using $C = 60w + 20$, where $C$ is calories and $w$ is the number of miles walked. What do the parameters 60 and 20 represent in this context?

A rideshare company charges a flat booking fee plus a per-mile charge. The total cost $C$ (in dollars) for a ride of $m$ miles is $C = 2.25m + 4.50$. What does the 2.25 represent in this context?

A streaming service charges a base fee plus a cost per movie rented. The total cost $C$ (in dollars) for renting $n$ movies is $C = 3n + 12$. What does the parameter $3$ represent in this context?

The amount of a medicine in the bloodstream is modeled by $M(t) = 60(0.9)^t$, where $t$ is time in hours and $M$ is measured in milligrams. What is the percent decay rate per hour?

A taxi fare is modeled by $F = 2.50d + 4$, where $F$ is the fare (in dollars) and $d$ is the distance traveled (in miles). In this function, what is the meaning of the 4?

A town's population is modeled by $P(t) = 25{,}000(1.02)^t$, where $t$ is the number of years since 2026. What is the initial value of the population in this model?

The value of a laptop after $t$ years is modeled by $V(t)=900(0.85)^t$, where $V$ is in dollars. What does the 0.85 represent in this context?

A savings account balance is modeled by $A(t) = 600(1.05)^t$, where $t$ is the number of years and $A(t)$ is in dollars. What is the annual interest rate?

A town's population is modeled by $P(t) = 15{,}000(1.02)^t$, where $t$ is years and $P(t)$ is the population. What is the percent growth rate per year?

A car's value depreciates according to $V(t) = 28000(0.85)^t$, where $t$ is years after purchase. The owner plans to sell the car when its value drops to exactly half the original price. In this context, what information would you need to determine when this occurs, and what role does 0.85 play in that calculation?

The number of active users $U$ (in millions) on a social media platform $t$ years after launch is modeled by $U(t) = 0.5(4)^{t/2}$. The platform's growth team is analyzing user acquisition patterns. What does the expression $\frac{t}{2}$ in the exponent indicate about the platform's user growth cycle compared to a model with just $4^t$?

A radioactive substance decays according to $N(t) = 800(0.5)^{t/12}$, where $N$ is the number of grams remaining after $t$ hours. A researcher studying this substance wants to know when only 100 grams remain. In this decay model, what specific information does the fraction $\frac{t}{12}$ in the exponent provide about the substance's half-life?

The population of bacteria in a petri dish is modeled by $P(t) = 250 \cdot 3^{0.5t}$, where $t$ is time in hours. A scientist observes that after 4 hours, the population has grown significantly. What does the value 0.5 in the exponent most directly represent?

The temperature $T$ (in °F) of a cooling soup $t$ minutes after being removed from heat follows $T(t) = 72 + 128(0.92)^t$. A food safety expert notes that the soup should not be consumed once it reaches room temperature. What does the parameter 72 represent, and how does it relate to the cooling process?

A subscription service charges according to $C(n) = 15n + 8$, where $n$ is the number of premium features activated and $C$ is the monthly cost in dollars. The company is considering restructuring to $C(n) = 12n + 20$. Comparing these two models, what does the change in the coefficient of $n$ indicate about the pricing strategy?

A gym membership has an initial enrollment fee and charges a monthly rate. The total cost $C$ (in dollars) after $t$ months is modeled by $C(t) = 45 + 35t$. If the gym decides to offer a 20% discount on the monthly rate but keeps the enrollment fee unchanged, what does the coefficient of $t$ represent in the new cost function?

A water tank contains water modeled by $W(t)=500(0.95)^t$, where $W$ is the amount of water (in liters) after $t$ hours. What is the percent decay rate per hour?

The profit $P$ (in thousands of dollars) from selling $x$ units of a product follows $P(x) = -2 + 1.5x$. The company's financial analyst notes that they need to sell a minimum number of units to avoid losses. What does the $y$-intercept $-2$ represent, and how does it affect the break-even analysis?

An investment account grows according to $A(t) = 5000(1.06)^t$ where $t$ is years and $A$ is the account value in dollars. An investor notes that after 10 years, they want to withdraw half the money. What does the base 1.06 tell us about the investment's performance relative to an account that grows by $A(t) = 5000(1.04)^t$?