Answer: [tex]17.68cm[/tex]
Step-by-step explanation:
Using the area formula of a cone, find the height first.
[tex]A=\pi r(r+\sqrt{h^2+r^2})[/tex]
Solve for h,
Begin by dividing by [tex]\pi r[/tex]
[tex]\frac{A}{\pi r}=r+\sqrt{h^2+r^2}[/tex]
Subtract r.
[tex]\frac{A}{\pi r}-r=\sqrt{h^2+r^2}[/tex]
Square both sides.
[tex](\frac{A}{\pi r}-r)^2=(\sqrt{h^2+r^2})^2[/tex]
[tex](\frac{A}{\pi r}-r)^2=h^2+r^2[/tex]
Subtract [tex]r^2[/tex]
[tex](\frac{A}{\pi r}-r)^2-r^2=h^2[/tex]
Extract the square root.
[tex]\sqrt{(\frac{A}{\pi r}-r)^2-r^2 } =\sqrt{h^2}[/tex]
[tex]\sqrt{(\frac{A}{\pi r}-r)^2-r^2 } =h[/tex]
Plug in your values.
[tex]\sqrt{[\frac{670cm^2}{(3.14)(8cm)}-(8cm)]^2-(8cm)^2 } =h[/tex]
Solve;
[tex]\sqrt{[\frac{670cm^2}{25.12cm}-(8cm)]^2-(8cm)^2 } =h[/tex]
[tex]\sqrt{[26.67cm-(8cm)]^2-(8cm)^2 } =h[/tex]
[tex]\sqrt{(18.67cm)^2-(8cm)^2 } =h[/tex]
[tex]\sqrt{348.57cm^2-64cm^2}=h[/tex]
[tex]\sqrt{284.57cm^2}=h[/tex]
[tex]15.77cm=h[/tex]
------------------------------------------------------------------
Now, to find the slant height use this formula: [tex]l=\sqrt{h^2+r^2}[/tex]
[tex]l=\sqrt{(15.77cm)^2+(8cm)^2}\\l=\sqrt{248.69cm^2+64cm^2}\\ l=\sqrt{312.69cm^2}\\ l=17.68cm[/tex]
