Question

In a certain fabric mill, cloth is removed from a dye bath and is then observed to dry exponentially at the rate of \(24 \%\) per hour. What percent of the original moisture will still be present after 5 h?

Short answer

After 5 hours, approximately \( A_0 e^{-1.2} \) or \( A_0 * 0.3012 \) of the original moisture is left, which is about \(30.12\%\) of the original moisture.

Step-by-step solution

Understanding Exponential Decay

Exponential decay can be described by the formula \( A(t) = A_0 e^{-kt} \), where \( A_0 \) is the initial amount, \( A(t) \) is the amount at time \( t \), and \( k \) is the decay rate. Here, the decay rate is given as \(24\%\) per hour, so \( k = 0.24 \).

Transforming the Percentage Decay Rate into Decimal Form

Convert the given percentage decay rate into decimal form by dividing by 100. Thus, \(24\%\) becomes \(0.24\).

Setting Up the Exponential Decay Formula

Since the decay is 24% per hour, the formula adjusting for our scenario is \( A(t) = A_0 e^{-0.24t} \).

Calculating the Remainder of Moisture after 5 Hours

To find the remaining moisture after 5 hours, substitute \( t = 5 \) into the formula to get \( A(5) = A_0 e^{-0.24 * 5} \).

Completing the Calculation

Compute the value using a calculator or any computational tool to evaluate \( A(5) \), which represents the percentage of original moisture left.

Key concepts

Exponential Decay Formula

The exponential decay formula is crucial for understanding how quantities decrease over time at a rate proportional to their current value. It's represented as
\( A(t) = A_0 e^{-kt} \),
where \( A_0 \) is the initial quantity, \( A(t) \) is the quantity at time \( t \), and \( k \) is the decay constant, which represents the decay rate.
For example, if something decreases in value by a certain percentage each period, this formula can predict what its future value will be after any given amount of time. This has practical applications in various fields including science, engineering, and finance.
In our scenario of the fabric mill, the cloth's moisture decreases by a fixed percentage every hour. Visualizing this, picture a graph that starts at a point representing the full moisture content and steeply falls as time increases, reflecting the rapid loss of moisture. The steeper the 'curve,' the faster the moisture is lost. The exponential decay formula helps us capture this changing rate in a precise mathematical expression.

Decay Rate

The decay rate in an exponential decay process is a constant value that quantifies how quickly the quantity is decreasing. It's represented by the variable \( k \) in the exponential decay formula.
Understanding the decay rate is essential for interpreting or predicting the behavior of a decaying quantity. It is the factor by which the quantity diminishes over a standard interval of time.
In the provided exercise, the decay rate is given as 24% per hour, reminding us that each hour, the cloth dries and loses 24% of its moisture from the previous hour. This rate determines how sharply the value of moisture falls over the time on our imagined graph. If we imagine a different scenario where the decay rate is lower, the graph curve would be less steep, depicting a slower drying process. It's this rate that allows us to predict how much moisture remains in the cloth at any given time.

Percentage to Decimal Conversion

Percentage to decimal conversion is a basic mathematical skill that's used in many different applications, including calculating exponential decay. A percentage represents a part per hundred, and converting it into a decimal involves dividing it by 100.
For instance, in our exercise, the decay rate of 24% needs to be converted to a decimal to be used in the exponential decay formula. The conversion is straightforward:
\[ 24\% = \frac{24}{100} = 0.24 \]
This conversion is important because the decay rate \( k \) in the formula is not expressed as a percentage but as a decimal. So, when you encounter a percentage in problems involving growth or decay, remember to convert it to a decimal before proceeding with any calculations. It's a simple step, but it's critical for ensuring the accuracy of your work. This foundational math skill underpins effective problem-solving in many disciplines, from finance to pharmacology.