1. **State the problem:** We want to find the sinusoidal function $S(t) = a \cdot \sin(b \cdot t) + d$ modeling the stock price, where $t$ is in radians and $S(t)$ is the stock price at day $t$. 2. **Given information:** - Average value (midline) is $3.47$ at $t=0$. - Minimum value is $1.97$ at $t=91.25$ days. 3. **Identify parameters:** - The midline $d$ is the average value: $$d = 3.47$$ - The amplitude $a$ is the distance from the midline to the minimum or maximum. Since minimum is $1.97$, amplitude is: $$a = 3.47 - 1.97 = 1.5$$ 4. **Form of the function:** $$S(t) = 1.5 \cdot \sin(b \cdot t) + 3.47$$ 5. **Find $b$ using the minimum point:** - The sine function reaches minimum at $\sin(\theta) = -1$. - Since minimum occurs at $t = 91.25$, we have: $$\sin(b \cdot 91.25) = -1$$ - The sine function equals $-1$ at angles of the form: $$b \cdot 91.25 = \frac{3\pi}{2} + 2\pi k, \quad k \in \mathbb{Z}$$ - Choose $k=0$ for the first minimum: $$b = \frac{3\pi/2}{91.25} = \frac{3\pi}{2 \times 91.25}$$ 6. **Simplify $b$:** $$b = \frac{3\pi}{182.5}$$ 7. **Final function:** $$S(t) = 1.5 \cdot \sin\left(\frac{3\pi}{182.5} t\right) + 3.47$$ This function models the stock price with the given conditions.