Answer:
Radius = 4.5 cm
Height = 7.8 cm
Step-by-step explanation:
The area of the sector is equal to the lateral surface area of the cone.
The formula for the area of a sector is:
[tex]\boxed{\begin{array}{l}\underline{\sf Area \;of\; a\; sector}\\\\A=\dfrac12 r^2 \theta\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$A$ is the area.}\\ \phantom{ww}\bullet\;\textsf{$r$ is the radius.}\\ \phantom{ww}\bullet\;\textsf{$\theta$ is the angle measured in radians.}\end{array}}[/tex]
Given that r = 9 cm and θ = 2π/2, the area of the sector is:
[tex]\textsf{Area of sector}=\dfrac{1}{2}(9)^2 \cdot \dfrac{2\pi}{2}[/tex]
[tex]\textsf{Area of sector}=\dfrac{1}{2}\cdot 81 \cdot \pi[/tex]
[tex]\textsf{Area of sector}=\dfrac{81}{2}\pi\; \sf cm^2[/tex]
The formula for the lateral surface area of a cone is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Lateral Surface Area of a Cone}}\\\\LSA=\pi r l\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$LSA$ is the lateral surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$l$ is the slant height.}\end{array}}[/tex]
The radius of the sector is equal to the slant height (l) of the cone.
Given that the radius of the sector is 9 cm, then l = 9 cm.
Therefore:
[tex]\begin{aligned}\textsf{Lateral surface area of cone}&=\textsf{Area of sector}\\\\\pi r (9)&=\dfrac{81}{2}\pi\\\\9r&=\dfrac{81}{2}\\\\r&=\dfrac{9}{2}\; \sf cm\\\\r&=4.5\; \sf cm\end{aligned}[/tex]
So, the radius of the circular base of the cone is 4.5 cm.
The radius, height, and slant height of the cone form a right triangle.
Given that we have the radius (r = 4.5) and the slant height (l = 9), we can calculate the height (h) by using Pythagoras Theorem:
[tex]\boxed{\begin{array}{l}\underline{\sf Pythagoras \;Theorem} \\\\a^2+b^2=c^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a$ and $b$ are the legs of the right triangle.}\\\phantom{ww}\bullet\;\textsf{$c$ is the hypotenuse (longest side) of the right triangle.}\\\end{array}}[/tex]
Therefore:
[tex]\begin{aligned}r^2+h^2&=l^2\\\\4.5^2+h^2&=9^2\\\\20.25+h^2&=81\\\\h^2&=81-20.25\\\\h^2&=60.75\\\\h&=\sqrt{60.75}\\\\h&=7.794228634...\\\\h&=7.8\; \sf cm\;(nearest\;tenth)\end{aligned}[/tex]
So, the height of the cone is 7.8 cm (rounded to the nearest tenth).
