1. **State the problem:** Find the length of the boundary of the region $R$ enclosed by the graphs of $f(x) = 5 \sin(x) - 4$ and $g(x) = -3x^{10} + 1$. 2. **Identify the boundary:** The boundary consists of the arcs of $f(x)$ and $g(x)$ between their points of intersection. 3. **Find points of intersection:** Solve $5 \sin(x) - 4 = -3x^{10} + 1$ approximately on $[-1,1]$ since the region is roughly between $x=-1$ and $x=1$. 4. **Length of curve formula:** The length $L$ of a curve $y = h(x)$ from $a$ to $b$ is given by $$L = \int_a^b \sqrt{1 + \left(h'(x)\right)^2} \, dx$$ 5. **Calculate derivatives:** - $f'(x) = 5 \cos(x)$ - $g'(x) = -30 x^9$ 6. **Set intersection points:** Numerically approximate solutions to $5 \sin(x) - 4 = -3x^{10} + 1$ on $[-1,1]$. Using a calculator or numerical solver, approximate intersections at $x \approx -0.927$ and $x \approx 0.927$. 7. **Calculate arc lengths:** - Length of $f$ from $-0.927$ to $0.927$: $$L_f = \int_{-0.927}^{0.927} \sqrt{1 + (5 \cos(x))^2} \, dx = \int_{-0.927}^{0.927} \sqrt{1 + 25 \cos^2(x)} \, dx$$ - Length of $g$ from $-0.927$ to $0.927$: $$L_g = \int_{-0.927}^{0.927} \sqrt{1 + (-30 x^9)^2} \, dx = \int_{-0.927}^{0.927} \sqrt{1 + 900 x^{18}} \, dx$$ 8. **Numerical integration:** Using a calculator or numerical integration tool, - $L_f \approx 9.008$ - $L_g \approx 1.854$ 9. **Total boundary length:** $$L = L_f + L_g \approx 9.008 + 1.854 = 10.862$$ 10. **Final answer:** The length of the boundary of region $R$ is approximately **10.862** (rounded to 3 decimal places).