1. **Problem statement:** We have a board of length 14.5 cm leaning against a vertical wall 10.5 cm high. The angle between the ground and the board is $a$ (in degrees). We want to find the piecewise function for $x(a)$, the derivative $\frac{dx}{da}$ at $a=76^\circ$, and estimate the change in $a$ for a small change in $x$ using linear approximation. 2. **Piecewise function for $x(a)$:** - When the board leans against the wall, the top point $P$ has coordinates $(x,y)$. - For $0^\circ \leq a < 46.397^\circ$, the board top is on the wall, so $x=0$. - For $46.398^\circ \leq a \leq 90^\circ$, the board top extends beyond the wall, so $$x(a) = 14.5 - 14.5 \cos(a)$$ 3. **Derivative $\frac{dx}{da}$ for $46.398^\circ \leq a \leq 90^\circ$:** Using the formula, $$x(a) = 14.5 - 14.5 \cos(a)$$ Differentiate with respect to $a$ (in degrees): $$\frac{dx}{da} = -14.5 \times (-\sin(a)) \times \frac{\pi}{180} = \frac{14.5 \pi}{180} \sin(a)$$ Note: We multiply by $\frac{\pi}{180}$ to convert degrees to radians for the sine function. 4. **Evaluate $\frac{dx}{da}$ at $a=76^\circ$:** Calculate $\sin(76^\circ)$: $$\sin(76^\circ) \approx 0.970295$$ Then, $$\frac{dx}{da}\bigg|_{a=76^\circ} = \frac{14.5 \times \pi}{180} \times 0.970295 \approx 0.774 \text{ cm/deg}$$ 5. **Linear approximation to find change in $a$ for $\Delta x = -0.06$ cm:** Using linear approximation, $$\Delta x \approx \frac{dx}{da} \Delta a$$ Rearranged, $$\Delta a \approx \frac{\Delta x}{\frac{dx}{da}} = \frac{-0.06}{0.774} \approx -0.0775^\circ$$ Since $x$ decreases, $a$ should decrease by about $0.0775^\circ$. **Final answers:** - Piecewise function: $$x(a) = \begin{cases} 0 & 0^\circ \leq a < 46.397^\circ \\ 14.5 - 14.5 \cos(a) & 46.398^\circ \leq a \leq 90^\circ \end{cases}$$ - Derivative at $76^\circ$: $$\frac{dx}{da}\bigg|_{a=76^\circ} \approx 0.774 \text{ cm/deg}$$ - Change in $a$ for $\Delta x = -0.06$ cm: $$\Delta a \approx 0.078^\circ \text{ decrease}$$