Find the area $ A $ of a triangular prism if volume $V = 1.7918$ and base area $B = 3.4$.

$A = 9.0151$

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ B = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$

After substituting $ B = 3.4 $ we have:

$$ 3.4 = \dfrac{ \sqrt{ 3 } \cdot a^2 }{ 4 }$$ $$ 3.4 \cdot 4 = \sqrt{ 3 } \cdot a^2 $$ $$ 13.6 = \sqrt{ 3 } \cdot a^2 $$ $$ a^2 = \dfrac{ 13.6 }{ \sqrt{ 3 } } $$ $$ a^2 = 7.852 $$ $$ a = \sqrt{ 7.852 } $$$$ a \approx 2.8021 $$

STEP 2: find length $ l $

To find length $ l $ use formula:

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

After substituting $ V = 1.7918 $ and $ a = 2.8021 $ we have:

$$ 1.7918 = \dfrac{ \sqrt{ 3 } \cdot 2.8021 ^{ 2 } \cdot l }{ 2 } $$$$ 1.7918 \cdot 2 = \sqrt{ 3 } \cdot 2.8021 ^{ 2 } \cdot l $$$$ 3.5836 = \sqrt{ 3 } \cdot 2.8021 ^{ 2 } \cdot l $$$$ 3.5836 = 13.6 \cdot l $$$$ l = \dfrac{ 3.5836 }{ 13.6 } $$$$ l = 0.2635 $$

STEP 3: find area $ A $

To find area $ A $ use formula:

$$ A = \frac{ a^2 \sqrt{3}}{2} + 3 a l $$

After substituting $ a = 2.8021 $ and $ l = 0.2635 $ we have:

$$ A = \frac{ 2.8021^2 \sqrt{3}}{2} + 3 \cdot 2.8021 \cdot 0.2635 $$$$ A = \frac{ 7.852 \sqrt{3}}{2} + 3 \cdot 0.7384 $$$$ A = \frac{ 13.6}{2} + 2.2151 $$$$ A = 6.8 + 2.2151 $$$$ A = 9.0151 $$

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