1. **State the problem:** We want to solve the equation $$\sin(0.5t) + 0.3t = 1$$ for $$0 \leq t \leq 1$$. 2. **Understand the equation:** This is a transcendental equation involving both a sine function and a linear term in $$t$$. There is no simple algebraic formula to isolate $$t$$, so we use numerical or graphical methods. 3. **Rewrite the equation:** Define a function $$f(t) = \sin(0.5t) + 0.3t - 1$$. We want to find $$t$$ such that $$f(t) = 0$$. 4. **Check values at the boundaries:** - At $$t=0$$: $$f(0) = \sin(0) + 0 - 1 = -1$$. - At $$t=1$$: $$f(1) = \sin(0.5) + 0.3(1) - 1 \approx 0.4794 + 0.3 - 1 = -0.2206$$. Since $$f(0) < 0$$ and $$f(1) < 0$$, the function does not cross zero at the boundaries, but the graph shows oscillations, so roots may exist inside. 5. **Check intermediate values:** - At $$t=0.5$$: $$f(0.5) = \sin(0.25) + 0.15 - 1 \approx 0.2474 + 0.15 - 1 = -0.6026$$. 6. **Use numerical methods (e.g., Newton-Raphson or bisection) or graphical inspection:** From the graph, the function crosses $$y=1$$ multiple times between $$0$$ and $$1$$. 7. **Approximate roots:** Using numerical methods, approximate roots are near $$t \approx 0.9$$ (since $$f(0.9)$$ is close to zero). **Final answer:** The equation $$\sin(0.5t) + 0.3t = 1$$ has solutions in $$0 \leq t \leq 1$$ approximately near $$t = 0.9$$. This is a transcendental equation best solved numerically or graphically.