Math 112 Section:
4. Tuberculosis (TB) is one of the top 10 causes of death worldwide. According to the World Health Organization (WHO), in the United States, TB deaths have fallen by an average of $3.7 \%$ per year since 1995. In the year 1995 , there were approximately 40,000 deaths from TB in the United States. Use the given data to create an exponential decay function model the number of TB death cases in the United States in the year $t$, where $t=0$, corresponds to the year 1995. Use this function to approximate the year in which the number of TB deaths in the United States was 1,033. (Note that the actual data shows this happened in 2013.)

Step 1 ：Define the exponential decay function: \( N(t) = N_0 \cdot e^{(-kt)} \)

Step 2 ：Set \( N_0 = 40,000 \) deaths for the year 1995 (\( t = 0 \))

Step 3 ：Convert the average percentage decrease per year to a decimal: \( k = \frac{3.7}{100} = 0.037 \)

Step 4 ：Write the exponential decay function with the given values: \( N(t) = 40,000 \cdot e^{(-0.037t)} \)

Step 5 ：Set \( N(t) \) to 1,033 to find the year when the number of TB deaths is approximately 1,033: \( 1,033 = 40,000 \cdot e^{(-0.037t)} \)

Step 6 ：Divide both sides by 40,000: \( \frac{1,033}{40,000} = e^{(-0.037t)} \)

Step 7 ：Calculate the left side of the equation: \( 0.025825 = e^{(-0.037t)} \)

Step 8 ：Take the natural logarithm of both sides: \( \ln(0.025825) = \ln(e^{(-0.037t)}) \)

Step 9 ：Simplify the equation using the property of logarithms: \( \ln(0.025825) = -0.037t \)

Step 10 ：Solve for \( t \): \( t = \frac{\ln(0.025825)}{-0.037} \)

Step 11 ：Calculate \( t \) using a calculator: \( t \approx \frac{-3.6542}{-0.037} \)

Step 12 ：Round \( t \) to two decimal places: \( t \approx 98.76 \)

Step 13 ：Add \( t \) to the base year 1995 to find the approximate year: \( 1995 + 98.76 \approx 1995 + 99 \)

Step 14 ：\(\boxed{\text{Year} \approx 2094}\)