Find the height $ h $ of a pyramid if base radius $r = 5$ and total surface $A = 125$.

Problem has no solutions.

STEP 1: find side $ a $

To find side $ a $ use formula:

$$ r = \dfrac{ a }{ 2 } $$

After substituting $ r = 5 $ we have:

$$ 5 = \dfrac{ a }{ 2 } $$ $$ a = 5 \cdot 2 $$ $$ a = 10 $$

STEP 2: find base area $ B $

To find base area $ B $ use formula:

$$ B = a^2 $$

After substituting $ a = 10 $ we have:

$$ B = 10^2 $$ $$ B = 100 $$

STEP 3: find lateral surface $ L $

To find lateral surface $ L $ use formula:

$$ A = B + L $$

After substituting $ A = 125 $ and $ B = 100 $ we have:

$$ 125 = 100 + L $$ $$ L = 125 - 100 $$ $$ L = 25 $$

STEP 4: find slant height $ s $

To find slant height $ s $ use formula:

$$ L = 2 \cdot a \cdot s $$

After substituting $ L = 25 $ and $ a = 10 $ we have:

$$ 25 = 2 \cdot 10 \cdot s $$$$ 25 = 20 \cdot s $$$$ s = \dfrac{ 25 }{ 20 } $$$$ s = \frac{ 5 }{ 4 } $$

STEP 5: find height $ h $

To find height $ h $ use Pythagorean Theorem:

$$ h^2 + \frac{ a^2 }{ 4 } = s^2 $$

After substituting $ a = 10 $ and $ s = \frac{ 5 }{ 4 } $ we have:

$$ h ^ {\,2} + \frac{ 10^2 }{ 4 } = \left(\frac{ 5 }{ 4 }\right)^2 $$ $$ h ^ {\,2} + \frac{ 100 }{ 4 } = \left(\frac{ 5 }{ 4 }\right)^2 $$ $$ h ^ {\,2} + 25 = \left(\frac{ 5 }{ 4 }\right)^2 $$ $$ h ^ {\,2} = \frac{ 25 }{ 16 } - 25 $$ $$ h ^ {\,2} = -\frac{ 375 }{ 16 } $$

This equation has no solution $ \Longrightarrow $ The problem has no solution.

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