The scent of a certain air freshener evaporates at a rate proportional to the amount of the air freshener present. Half of the air freshener evaporates within $2$ hours of being sprayed. If the scent of the air freshener is undetectable once $80\%$ has evaporated, how long will the scent of the air freshener last?

• A. $t=1hr$
• B. $t=2hrs$
• C. $t=4.64hrs$
• D. $t=1.83hrs$

Answer: (C)

Verified step by step guidance

1. Step 1: Recognize that the problem describes exponential decay, where the rate of evaporation is proportional to the amount of air freshener present. The general formula for exponential decay is given by: $y=y_0\cdot e^{-}$, where $y_0$ is the initial amount, $k$ is the decay constant, and $t$ is time.
2. Step 2: Use the information that half of the air freshener evaporates in 222 hours to find the decay constant $k$. Substitute $y=\frac{y_0}{2}$ and $t=222$ into the formula: $\frac{y_0}{2}=y_0\cdot e^{-}$. Simplify to solve for $k$.
3. Step 3: Simplify the equation from Step 2 by dividing both sides by $y_0$, then take the natural logarithm of both sides to isolate $k$. The equation becomes: $-k\cdot 222=ln\left(\frac{1}{2}\right)$. Solve for $k$.
4. Step 4: Use the decay constant $k$ to determine the time it takes for 80% of the air freshener to evaporate. If 80% has evaporated, then 20% remains, so $y=0.2y_0$. Substitute this into the formula: $0.2y_0=y_0\cdot e^{-}$. Simplify to solve for $t$.
5. Step 5: Simplify the equation from Step 4 by dividing both sides by $y_0$, then take the natural logarithm of both sides to isolate $t$. The equation becomes: $t=\frac{-}{ln}$. Substitute the value of $k$ from Step 3 to find the time.