To solve this problem, we can use the concept of exponential decay, which describes how a quantity decreases at a rate proportional to its current value. In this case, the moisture content of the sheet is what we're focusing on. Let's break it down step by step.
Understanding the Problem
We know that the sheet loses half of its moisture in the first hour. This means that if we start with an initial moisture content of \( M_0 \), after one hour, the moisture content \( M(t) \) will be:
- After 1 hour: \( M(1) = \frac{M_0}{2} \)
Setting Up the Equation
The rate of moisture loss can be modeled by the differential equation:
\(\frac{dM}{dt} = -kM\)
where \( k \) is a positive constant that represents the rate of moisture loss. The solution to this equation is:
\( M(t) = M_0 e^{-kt} \)
Finding the Rate Constant \( k \)
Since we know that the moisture content halves in one hour, we can use this information to find \( k \). Plugging in the values:
\( M(1) = M_0 e^{-k \cdot 1} = \frac{M_0}{2} \)
Dividing both sides by \( M_0 \) gives:
\( e^{-k} = \frac{1}{2} \)
Taking the natural logarithm of both sides results in:
\( -k = \ln\left(\frac{1}{2}\right) \)
Thus, we find:
\( k = -\ln\left(\frac{1}{2}\right) = \ln(2) \)
Calculating the Time for 95% Moisture Loss
Now, we want to find the time \( t \) when the moisture content is reduced to 5% of its original value (which means 95% loss). So we set:
\( M(t) = 0.05M_0 \)
Substituting into our moisture equation gives:
\( 0.05M_0 = M_0 e^{-kt} \)
Dividing both sides by \( M_0 \) leads to:
\( 0.05 = e^{-kt} \)
Taking the natural logarithm of both sides results in:
\( \ln(0.05) = -kt \)
Substituting \( k = \ln(2) \) into the equation gives:
\( t = -\frac{\ln(0.05)}{\ln(2)} \)
Calculating the Final Time
Now we can compute the value of \( t \):
- First, calculate \( \ln(0.05) \approx -2.9957 \)
- Next, calculate \( \ln(2) \approx 0.6931 \)
- Now, substitute these values into the equation:
\( t \approx -\frac{-2.9957}{0.6931} \approx 4.32 \) hours
Final Thoughts
Therefore, under the same weather conditions, the sheet will lose 95% of its moisture in approximately 4.32 hours. This illustrates how exponential decay works in real-world scenarios, such as moisture loss in porous materials. Understanding these principles can help in various fields, from environmental science to engineering.
