1. **State the problem:** We have an exponential function $$B(x) = 137(1.164)^x$$ where $$x$$ is the number of years since 2015, and $$B(x)$$ estimates the number of people (in thousands) participating in an ornithology science project. 2. **Find $$B(0)$$ and explain its meaning:** - Substitute $$x=0$$ into the function: $$B(0) = 137(1.164)^0 = 137 \times 1 = 137$$ - This means in 2015 (when $$x=0$$), 137 thousand people participated. 3. **Estimate participation in 2020:** - For 2020, $$x = 2020 - 2015 = 5$$. - Calculate $$B(5)$$: $$B(5) = 137(1.164)^5$$ - Compute the power: $$1.164^5 \approx 2.136$$ - Multiply: $$B(5) \approx 137 \times 2.136 = 292.632 \approx 292.7$$ thousand people. 4. **Project participation in 2034:** - For 2034, $$x = 2034 - 2015 = 19$$. - Calculate $$B(19)$$: $$B(19) = 137(1.164)^{19}$$ - Compute the power: $$1.164^{19} \approx 22.04$$ - Multiply: $$B(19) \approx 137 \times 22.04 = 3019.48 \approx 3019.5$$ thousand people. 5. **Summary:** - $$B(0) = 137$$ means in 2015, 137 thousand people participated. - In 2020, about 292.7 thousand participated. - In 2034, about 3019.5 thousand (or 3,019,500) people are projected to participate. **Final answers:** - a. $$B(0) = 137$$; This represents that in 2015, the number of people who participated worldwide in the ornithology science project was 137 thousand. - b. In 2034, about 3019.5 thousand people will participate.