Find the slant height $ l $ of a cone if radius $r = 6$ and volume $V = 542.87$.

$l = 15.6001$

STEP 1: find height $ h $

To find height $ h $ use formula:

$$ V = \dfrac{ r ^{ 2 } \cdot h \cdot \pi}{ 3 } $$

After substituting $ V = 542.87 $ and $ r = 6 $ we have:

$$ 542.87 = \dfrac{ 6 ^{ 2 } \cdot h \cdot \pi}{ 3 } $$$$ 542.87 \cdot 3 = 6 ^{ 2 } \cdot h \cdot \pi $$$$ 1628.61 = 36 \cdot h \cdot \pi $$$$ h = \dfrac{ 1628.61 }{ 36 \, \pi} $$$$ h = 14.4001 $$

STEP 2: find side $ l $

To find side $ l $ use Pythagorean Theorem:

$$ r^2 + h^2 = l^2 $$

After substituting $ r = 6 $ and $ h = 14.4001 $ we have:

$$ 6^2 + 14.4001^2 = l^2 $$ $$ 36 + 207.3621 = l^2 $$ $$ l^2 = 243.3621 $$ $$ l = \sqrt{ 243.3621 } $$$$ l = 15.6001 $$

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