1. **State the problem:** Find the arc length of the function $f(x) = 0.6x^{0.51} + 3$ over a given interval (assume from $x=a$ to $x=b$). 2. **Formula for arc length:** The arc length $L$ of a function $y=f(x)$ from $x=a$ to $x=b$ is given by: $$L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$$ 3. **Find the derivative:** $$\frac{dy}{dx} = 0.6 \times 0.51 x^{0.51 - 1} = 0.306 x^{-0.49}$$ 4. **Substitute into the formula:** $$L = \int_a^b \sqrt{1 + (0.306 x^{-0.49})^2} \, dx = \int_a^b \sqrt{1 + 0.093636 x^{-0.98}} \, dx$$ 5. **Explanation:** To find the exact arc length, you need to evaluate this integral over the desired interval $[a,b]$. This integral may not have a simple closed form and might require numerical methods. 6. **Summary:** The arc length formula for $f(x)$ is: $$L = \int_a^b \sqrt{1 + 0.093636 x^{-0.98}} \, dx$$ Specify the interval $[a,b]$ to compute a numerical value.