Final answer:
Two formulas represent the quantity of air-freshener remaining after t days: (a) a linear decay Q = 30 - 2t for a decrease of 2 grams a day, and (b) an exponential decay Q = 30 × (0.88)^t for a daily decrease of 12%.
Step-by-step explanation:
The student has asked us to create a formula for the quantity Q grams of air-freshener remaining t days after it starts to evaporate at two different rates: (a) 2 grams a day, and (b) 12% a day.
Case (a): Decrease by 2 grams a day
For the first scenario where the air-freshener decreases by 2 grams a day, we can model the remaining quantity with a linear equation:
Q = 30 - 2t
Here, 30 grams is the initial amount, and 2t represents the total amount that has evaporated after t days.
Case (b): Decrease by 12% a day
For the second scenario where the air-freshener decreases by 12% each day, we need an exponential decay model:
Q = 30 × (0.88)^t
The term 0.88 comes from subtracting the daily decrease percentage (12% or 0.12) from 1.
In both equations, 't' is the time in days, and 'Q' is the quantity of the air-freshener remaining.
To graph these equations, for case (a) we would draw a straight line starting at (0,30) and declining with a slope of -2. For case (b), we would draw an exponential decay curve that starts at (0,30) and drops off more rapidly as time increases. The graphs visually represent the rate at which the air-freshener is evaporating over time.