1. **State the problem:** We need to find the sinusoidal function $D(t) = a \cdot \sin(b \cdot t) + d$ that models the boat's distance from the lake floor over time $t$ in radians. 2. **Given information:** - At $t=0$, the boat is at the middle of its oscillation, so $D(0) = d = 1$ m. - The maximum height is $1.2$ m, reached at $t = \frac{\pi}{4}$ seconds. 3. **Understand the sinusoidal function:** - The amplitude $a$ is the distance from the middle to the maximum or minimum. - The vertical shift $d$ is the middle value of the oscillation. - The function reaches its maximum when $\sin(b \cdot t) = 1$. 4. **Use the maximum height to find $a$ and $b$:** - Maximum height: $D\left(\frac{\pi}{4}\right) = a \cdot \sin\left(b \cdot \frac{\pi}{4}\right) + d = 1.2$ - Since $d=1$, substitute: $$1.2 = a \cdot \sin\left(b \cdot \frac{\pi}{4}\right) + 1$$ - Simplify: $$a \cdot \sin\left(b \cdot \frac{\pi}{4}\right) = 0.2$$ 5. **At $t=0$, $\sin(0) = 0$, so $D(0) = d = 1$ is consistent.** 6. **Since the maximum occurs at $t=\frac{\pi}{4}$, the sine term must be 1 at that time:** $$\sin\left(b \cdot \frac{\pi}{4}\right) = 1$$ - The sine function equals 1 at $\frac{\pi}{2}$, so: $$b \cdot \frac{\pi}{4} = \frac{\pi}{2}$$ - Solve for $b$: $$b = \frac{\frac{\pi}{2}}{\frac{\pi}{4}} = 2$$ 7. **Find amplitude $a$ using the value of $b$:** $$a \cdot 1 = 0.2 \implies a = 0.2$$ 8. **Write the final function:** $$D(t) = 0.2 \cdot \sin(2t) + 1$$ This function models the boat's distance from the lake floor over time $t$ in radians.