The fence

I'm building a cloth (board) fence. The boards are rounded in a semicircle at the top. The tops of the boards between the columns should copy an imaginary circle. The tip of the first and last board forms the chord of a circle whose radius is unknown. The length of the circle string is 180cm. The arc height "in the middle" of the string is 23cm. The boards are 16, and their axes are 12 cm apart. Please calculate the heights of board numbers 2 to 8, i.e., half the arc.

Correct answer:

r =  187.6 cm
y₂ =  170.6 cm
y₃ =  175.6 cm
y₄ =  179.6 cm
y₅ =  182.8 cm
y₆ =  185.2 cm
y₇ =  186.7 cm
y₈ =  187.5 cm

Step-by-step explanation:

$$\begin{gathered} t=180\text{cm} \\ {r}^2=(t/2{)}^2+(r-23{)}^2 \\ r=(8100+529)/46=187.6\text{cm} \end{gathered}$$

$$\begin{gathered} (12-90{)}^2+y_2^2=r^2 \\ {y}_2=\sqrt{r^2-(12-90{)}^2}=\sqrt{187.587^2-(12-90{)}^2}=170.6\text{cm} \end{gathered}$$

$$\begin{gathered} (24-90{)}^2+y_3^2=r^2 \\ {y}_3=\sqrt{r^2-(24-90{)}^2}=\sqrt{187.587^2-(24-90{)}^2}=175.6\text{cm} \end{gathered}$$

$$\begin{gathered} (36-90{)}^2+y_4^2=r^2 \\ {y}_4=\sqrt{r^2-(36-90{)}^2}=\sqrt{187.587^2-(36-90{)}^2}=179.6\text{cm} \end{gathered}$$

$$\begin{gathered} (48-90{)}^2+y_5^2=r^2 \\ {y}_5=\sqrt{r^2-(48-90{)}^2}=\sqrt{187.587^2-(48-90{)}^2}=182.8\text{cm} \end{gathered}$$

$$\begin{gathered} (60-90{)}^2+y_6^2=r^2 \\ {y}_6=\sqrt{r^2-(60-90{)}^2}=\sqrt{187.587^2-(60-90{)}^2}=185.2\text{cm} \end{gathered}$$

$$\begin{gathered} (72-90{)}^2+y_7^2=r^2 \\ {y}_7=\sqrt{r^2-(72-90{)}^2}=\sqrt{187.587^2-(72-90{)}^2}=186.7\text{cm} \end{gathered}$$

$$\begin{gathered} (84-90{)}^2+y_8^2=r^2 \\ {y}_8=\sqrt{r^2-(84-90{)}^2}=\sqrt{187.587^2-(84-90{)}^2}=187.5\text{cm} \end{gathered}$$

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