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.)

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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**.
The centenarian population in 2013 is approximately 101,000. The centenarian population in 2038 is approximately 343,000.

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