1. **Stating the problem:** We need to evaluate the function $$a(t) = 5,000,000 \cdot e^{\int_0^2 (0.09 + 0.0006t^2) dt + \int_9^{15} (0.1836 - 0.005t) dt + \int_{15}^{17} 0.1836 dt}$$ 2. **Recall the integral rules:** - The integral of a sum is the sum of the integrals. - For constants, $$\int_a^b c \, dt = c(b - a)$$. - For powers, $$\int t^n dt = \frac{t^{n+1}}{n+1} + C$$. 3. **Calculate each integral separately:** - First integral: $$\int_0^2 (0.09 + 0.0006t^2) dt = \int_0^2 0.09 dt + \int_0^2 0.0006 t^2 dt$$ Calculate each part: $$\int_0^2 0.09 dt = 0.09 \times (2 - 0) = 0.18$$ $$\int_0^2 0.0006 t^2 dt = 0.0006 \times \frac{2^3}{3} = 0.0006 \times \frac{8}{3} = 0.0016$$ Sum: $$0.18 + 0.0016 = 0.1816$$ - Second integral: $$\int_9^{15} (0.1836 - 0.005t) dt = \int_9^{15} 0.1836 dt - \int_9^{15} 0.005t dt$$ Calculate each part: $$\int_9^{15} 0.1836 dt = 0.1836 \times (15 - 9) = 0.1836 \times 6 = 1.1016$$ $$\int_9^{15} 0.005t dt = 0.005 \times \frac{15^2 - 9^2}{2} = 0.005 \times \frac{225 - 81}{2} = 0.005 \times 72 = 0.36$$ Sum: $$1.1016 - 0.36 = 0.7416$$ - Third integral: $$\int_{15}^{17} 0.1836 dt = 0.1836 \times (17 - 15) = 0.1836 \times 2 = 0.3672$$ 4. **Sum all integrals:** $$0.1816 + 0.7416 + 0.3672 = 1.2904$$ 5. **Calculate the exponential:** $$e^{1.2904} \approx 3.634$$ 6. **Calculate $$a(t)$$:** $$a(t) = 5,000,000 \times 3.634 = 18,170,000$$ **Final answer:** $$a(t) \approx 18,170,000$$