The exponential function given by \( \mathrm{H}(\mathrm{t})=80,036.66(1.0488)^{\mathrm{t}} \), where t is the number of years after 2008, can be used to project the number of centenarians in a certain country. Use this function to project the centenarian population in this country in 2013 and in 2038 . The centenarian population in 2013 is approximately (Round to the nearest whole number.) The centenarian population in 2038 is approximately (Round to the nearest whole number.)

To project the centenarian population using the function \( \mathrm{H}(\mathrm{t})=80,036.66(1.0488)^{\mathrm{t}} \), we need to determine the value of \( t \) for the years 2013 and 2038. 1. **For the year 2013:** - The year 2013 is 5 years after 2008, so \( t = 2013 - 2008 = 5 \). - Now we can substitute \( t = 5 \) into the function: \[ \mathrm{H}(5) = 80,036.66(1.0488)^{5} \] - First, calculate \( (1.0488)^{5} \): \[ (1.0488)^{5} \approx 1.270678 \] - Now, substitute this value back into the equation: \[ \mathrm{H}(5) \approx 80,036.66 \times 1.270678 \approx 101,000.00 \] - Rounding to the nearest whole number, the centenarian population in 2013 is approximately **101,000**. 2. **For the year 2038:** - The year 2038 is 30 years after 2008, so \( t = 2038 - 2008 = 30 \). - Now we can substitute \( t = 30 \) into the function: \[ \mathrm{H}(30) = 80,036.66(1.0488)^{30} \] - First, calculate \( (1.0488)^{30} \): \[ (1.0488)^{30} \approx 4.292202 \] - Now, substitute this value back into the equation: \[ \mathrm{H}(30) \approx 80,036.66 \times 4.292202 \approx 343,000.00 \] - Rounding to the nearest whole number, the centenarian population in 2038 is approximately **343,000**. ### Summary: - The centenarian population in 2013 is approximately **101,000**. - The centenarian population in 2038 is approximately **343,000**.