Respuesta :
Answer:
The circumference of the circle is [tex]C=45.346 \:units[/tex].
Step-by-step explanation:
A sector is the part of a circle enclosed by two radii of a circle and their intercepted arc. A pie-shaped part of a circle.
The area of a circle is given by [tex]A_{circle}=\pi r^2[/tex]
The formula used to calculate the area of a sector of a circle is:
[tex]A_{sector}=\frac{central \:angle}{360} \cdot {Area \:of \:whole \:circle}\\\\A_{sector}=\frac{\theta}{360} \cdot \pi r^2[/tex]
The circumference of a circle is the distance around the outside of the circle and its given by
[tex]C=2\pi r[/tex]
We know the central angle [tex]\theta[/tex] = 110º and the area of the sector 50 units squared.
First, we use the formula to calculate the area of a sector to find the radius.
[tex]50=\frac{110}{360} \cdot \pi r^2\\\\\frac{110}{360}\pi r^2=50\\\\\frac{11\pi }{36}r^2=50\\\\11\pi r^2=1800\\\\r^2=\frac{1800}{11\pi }\\\\\mathrm{For\:}x^2=f\left(a\right)\mathrm{\:the\:solutions\:are\:}x=\sqrt{f\left(a\right)},\:\:-\sqrt{f\left(a\right)}\\\\r=\sqrt{\frac{1800}{11\pi }},\:r=-\sqrt{\frac{1800}{11\pi }}[/tex]
The radius can't be negative. Therefore,
[tex]r=\sqrt{\frac{1800}{11\pi }}\approx 7.217 \:units[/tex]
Next, we apply the formula for the circumference of a circle.
[tex]C=2\pi (7.217)=45.346 \:units[/tex]
