1. **State the problem:** We want to find the function $A(t)$ modeling the accordion's length in cm as a function of time $t$ in radians, given it follows a sinusoidal form: $$A(t) = a \cdot \cos(b \cdot t) + d.$$ 2. **Given information:** - At $t=0$, $A(0) = 15$ cm (shortest length). - At $t=1.5$ seconds, $A(1.5) = 21$ cm (average length). 3. **Identify parameters:** - The shortest length is the minimum of the cosine wave: $d - a = 15$. - The average length is the midline: $d = 21$. 4. **Find amplitude $a$:** From $d - a = 15$ and $d=21$, we get $$a = d - 15 = 21 - 15 = 6.$$ 5. **Write the function so far:** $$A(t) = 6 \cdot \cos(b t) + 21.$$ 6. **Use the second condition $A(1.5) = 21$:** $$21 = 6 \cdot \cos(b \cdot 1.5) + 21.$$ Subtract 21 from both sides: $$0 = 6 \cdot \cos(1.5 b) \implies \cos(1.5 b) = 0.$$ 7. **Solve for $b$:** The cosine is zero at angles $$\frac{\pi}{2}, \frac{3\pi}{2}, \ldots$$ The smallest positive solution is $$1.5 b = \frac{\pi}{2} \implies b = \frac{\pi}{3}.$$ 8. **Final function:** $$\boxed{A(t) = 6 \cdot \cos\left(\frac{\pi}{3} t\right) + 21}.$$ This function models the accordion length in cm as a function of time $t$ in radians.