Find the length $ l $ of a triangular prism if height $h = 13$ and volume $V = 3640$.

$l = \frac{ 140 \sqrt{ 3}}{ 13 }$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ h = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$

After substituting $ h = 13 $ we have:

$$ 13 = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 13 \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 26 $$ $$ a = \dfrac{ 26 }{ \sqrt{ 3 } } $$ $$ a = \frac{ 26 \sqrt{ 3}}{ 3 } $$

STEP 2: find length $ l $

To find length $ l $ use formula:

$$ V = \dfrac{ \sqrt{ 3 } \cdot a ^{ 2 } \cdot l }{ 2 } $$

After substituting $ V = 3640 $ and $ a = \frac{ 26 \sqrt{ 3}}{ 3 } $ we have:

$$ 3640 = \dfrac{ \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l }{ 2 } $$$$ 3640 \cdot 2 = \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l $$$$ 7280 = \sqrt{ 3 } \cdot \frac{ 26 \sqrt{ 3}}{ 3 } ^{ 2 } \cdot l $$$$ 7280 = \frac{ 676 \sqrt{ 3}}{ 3 } \cdot l $$$$ l = \dfrac{ 7280 }{ \frac{ 676 \sqrt{ 3}}{ 3 } } $$$$ l = \frac{ 140 \sqrt{ 3}}{ 13 } $$

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