Question

An air freshener starts with 25 grams and evaporates. In each of the following cases, write a formula for the quantity, Q grams, of air freshener remaining t days after the start.
(a) The decrease is 4 grams per day.
(b) The decrease is 9% per day.
(c) The decrease is at a continuous rate of 12% per day.

Answer 1

a) Decrease is a constant rate. Therefore the function will be linear.

Q = 25 - 4t

b) Decrease is exponential. You will have a exponential function.

Q = 25(1-.09)^t  = 25(.91)^t

c) Decrease is exponential. You will have a continuous exponential function.

Q = 25e^(-.12t)

Answer 2

Initial number of grams of air freshener = 25 grams.

We need to take final quantity by Q, number of days after start t.

Let us write equations:

(a) The decrease is 4 grams per day.

Q = Intial quantity - 4 × number of days.

Q = 25 - 4t.

(b) The decrease is 9% per day.

9% could be written in decimals as 0.09.

$Q = 25(1-0.09)^t.$

(c) The decrease is at a continuous rate of 12% per day.

We need apply formula of continuous exponential function for continuous decrease.

Therefore,

$Q = 25e^{-0.12t}$