Answer: radius = 5.83 cm height = 8.12 cm
Step-by-step explanation:
First, let's find the circumference of the 210° section of the circle.
[tex]C=2\pi r\bigg(\dfrac{\theta}{360^o}\bigg)\\\\\\C=2\pi(10)\bigg(\dfrac{210^o}{360^o}\bigg)\\\\\\C=\dfrac{35}{3}\pi[/tex]
The circumference of the the cone is [tex]\dfrac{35}{3}\pi[/tex] . We can use this to find the radius .
[tex]C=2\pi r\\\\\dfrac{35}{3}\pi=2\pi r\\\\\\\dfrac{35\pi}{3\cdot 2\pi}=r\\\\\\\dfrac{35}{6}=r\\\\\\5.83=r[/tex]
When you fold the 210° section into a cone, the slant height is the original radius of 10. We can use the radius and slant height of the cone to form a right triangle with the height. Use the Pythagorean Theorem to find the height.
radius² + height² = slant height²
[tex]\dfrac{35}{6}^2\ +\ h^2=10^2\\\\\\.\qquad \quad h^2=10^2-\bigg(\dfrac{35}{6}\bigg)^2\\\\.\qquad \quad h=\sqrt{\dfrac{6^2(10)^2-35^2}{6^2}}\\\\\\.\qquad \quad h=\sqrt{\dfrac{2375}{36}}\\\\\\.\qquad \quad h=8.12[/tex]
