An Elizabethan collar

An Elizabethan collar is used to prevent an animal from irritating a wound. The angle between the opening (diameter 6 inches) with a 16-inch diameter and the side of the collar is 53 degrees. Find the surface area of the collar shown.

Correct answer:

A =  216.3535 in²

Step-by-step explanation:

$$\begin{gathered} d_1=6\text{in} \\ {d}_2=16\text{in} \\ \alpha =53{}^{\circ } \\ {r}_1=d_1/2=6/2=3\text{in} \\ {r}_2=d_2/2=16/2=8\text{in} \\ \tan \alpha =(r_2-r_1):h \\ h=\frac{r_2-r_1}{\tan \alpha }=\frac{r_2-r_1}{\tan 53°}=\frac{8-3}{\tan 53°}=\frac{8-3}{1.327045}=3.76777\text{in} \\ \cos \alpha =h:s \\ s=h/\cos \alpha =h/\cos 53°=3.7678/\cos 53°=3.7678/0.601815=6.26068\text{in} \\ A=\pi \cdot (r_1+r_2)\cdot s=3.1416\cdot (3+8)\cdot 6.2607=216.3535{\text{in}}^2 \end{gathered}$$

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