I solved it!~

:D

Here's the official solution!~

Thank you so much!~

First, we should convert the given information about the real-world context into mathematical terms of the sinusoidal function and its graph.

Then, we should use the given information to find the amplitude, midline, and period of the function's graph.

Finally, we should find aaa, b, and d in the expression \(a*cos(b*t)+d\) by considering the features we found.

At t=0, the swing is 1m, behind the center. This means the graph of the function passes through (0,-1).

We are given that this is the farthest point behind the center, which corresponds to a minimum point of the graph.

\(\frac{\pi}{6}\) seconds later the distance is 0m. This corresponds to the point \((\frac{\pi}{6},0)\).

We are given that this is the middle of the swing, which corresponds to the midline of the graph.

In conclusion, the graph has a minimum point at (0,-1) and then intersects its midline at \((\frac{\pi}{6},0)\).

The midline intersection is at y=0, so this is the midline.

The minimum point is 1 unit below the midline, so the amplitude is 1.

The minimum point is \((\frac{\pi}{6})\) units to the left of the midline intersection, so the period is \(4*\frac{\pi}{6} \\ \\=\frac{2\pi}{3}\).

The amplitude is -1.

The midline is y=0.

The period is 3.

The answer:

\(D(t)=-cos(3t)\)

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