Find the diameter of a right cone with a slant height of \(18cm\) and a surface area of \(208\pi cm^2\).

In advance, thank you!

TheXSquaredFactor
May 22, 2017

#1**+3 **

Find the diameter of a right cone with a slant height 18cm of and a surface area of 208 pi cm^{2}.

Omi67
May 22, 2017

#3**0 **

Thank you. I knew that the surface area equals the circumference of the base multiplied by the slant height and then add the area of the base. I didn't realize that I could setup an equation with that information, though...

TheXSquaredFactor
May 22, 2017

#1**0 **

**Find the diameter of a right cone with a slant height of \(18\ cm\) and a surface area of \(208 \pi \ cm^2 \).**

Surface area of cone \( A = \pi r^2+\pi rl\)

\(\begin{array}{|rcll|} \hline \pi r^2+ \pi rl &=& A \quad & | \quad : \pi \\ r^2+rl &=& \frac{A}{\pi} \\ r^2+rl -\frac{A}{\pi} &=& 0 \\ r &=& \frac{-l\pm \sqrt{l^2-4\cdot (-\frac{A}{\pi}) } } {2} \\ r &=& \frac{-l\pm \sqrt{l^2+ \frac{4A}{\pi} } } {2} \\ 2r = d &=& -l\pm \sqrt{l^2+ \frac{4A}{\pi} } \quad & | \quad A = 280\pi \qquad l = 18 \\ d &=& -18\pm \sqrt{18^2+ 4\cdot 208 } \\ d &=& -18\pm \sqrt{1156 } \\ d &=& -18\pm 34 \\ d &=& -18 + 34 \\ d &=& 16 \\ \hline \end{array}\)

The diameter is** 16 cm**

heureka
May 22, 2017

#3**0 **

Thank you. I knew that the surface area equals the circumference of the base multiplied by the slant height and then add the area of the base. I didn't realize that I could setup an equation with that information, though...

TheXSquaredFactor
May 22, 2017