Problem 89 - Intermediate Algebra

Question

Disinfectants. The exponential function \(A(t)=2,000,000 e^{-0.588 t}\) approximates the number of germs on a table top, \(t\) minutes after disinfectant was sprayed on it. Estimate the germ count on the table 5 minutes after it is sprayed.

Step-by-Step Solution

Answer

After 5 minutes, the estimated germ count is approximately 104,120.

Step 1: Identify the Given Formula and Values

We are given the exponential function \(A(t) = 2,000,000 e^{-0.588 t}\) which estimates the number of germs, \(A(t)\), on a table top after \(t\) minutes. We need to find the germ count after 5 minutes, so \(t = 5\).

Step 2: Substitute the Given Value in the Formula

Substitute \(t = 5\) into the function: \(A(5) = 2,000,000 e^{-0.588 \times 5}\).

Step 3: Calculate the Exponent

Multiply the exponent: \(-0.588 \times 5 = -2.94\).

Step 4: Evaluate the Exponential Expression

Substitute into the function: \(A(5) = 2,000,000 e^{-2.94}\). Use a calculator to find \(e^{-2.94}\) which is approximately 0.05206.

Step 5: Multiply to Find the Germ Count

Finally, calculate the germ count by multiplying the values: \(A(5) = 2,000,000 \times 0.05206 \approx 104,120\).

Key Concepts

Germ Count Estimation

Estimating the germ count is a practical application of exponential functions in real-world scenarios, like maintaining hygiene and cleanliness. In our context, we're using the function \(A(t) = 2,000,000 e^{-0.588 t}\) to approximate the number of germs remaining on a table after a disinfectant has been applied. Germ estimation helps to predict and control microbial activity, ensuring effective sanitization.

When estimating germs, follow these steps:

• Identify known values: The initial germ count before disinfectant application was 2,000,000.
• Calculate how germ counts change over time: Using the decay factor of \(-0.588\), which indicates how rapidly germs die.
• Compute estimated germ count at a specific time, like 5 minutes after application.

Accurate germ estimation is crucial for public health and helps manage sanitation processes efficiently.

Disinfectant Effectiveness

Disinfectant effectiveness is key to ensuring that cleaning efforts are successful and that harmful microorganisms are significantly reduced. The effectiveness of a disinfectant is measured by how quickly and thoroughly it reduces germs. In the given problem, we observe the action of the disinfectant through the exponential decay in the function \(A(t) = 2,000,000 e^{-0.588 t}\).

Disinfectant effectiveness relies on:

• Concentration of the active ingredient: Stronger solutions might lead to quicker germ reduction.
• Contact time: Sufficient time is necessary for the disinfectant to exert its full effectiveness.
• Type of microorganisms: Different germs may have different susceptibility levels to disinfectants.

Understanding disinfectant effectiveness is important in selecting the right cleaning products and ensuring the sanitization protocols are followed to prevent microbial growth.

Exponential Decay

Exponential decay describes a process where quantities decrease rapidly at a rate proportional to the current amount. This principle is crucial for understanding the mathematics behind processes like microbial reduction through sanitizers.

The key characteristics of exponential decay include:

• A negative exponent indicates the rate of decay, such as \(-0.588t\) in the formula.
• The initial value, here 2,000,000 germs, begins the decay process.
• Decay results in the rapid drop in number but slows as the quantity decreases.

Exponential decay models are used in various fields, from physics to economics, but are especially relevant in biology to understand how disinfectants reduce germ counts over time. Appreciating this concept allows us to design effective cleaning strategies by predicting how long it takes for disinfectants to achieve desired cleanliness levels.