Find the height $ h $ of a cone if radius $r = 5.13$ and area $A = 98$.

Problem has no solutions.

STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

$$ AB = r^2 \cdot \pi $$

After substituting $ r = 5.13 $ we have:

$$ AB = 5.13^2 \cdot \pi $$ $$ AB = 26.3169 \cdot \pi $$

STEP 2: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

$$ A = AB + CSA $$

After substituting $ A = 98 $ and $ AB = 26.3169\pi $ we have:

$$ 98 = 26.3169\pi + CSA $$ $$ CSA = 98 - 26.3169 $$ $$ CSA = 71.6831 $$

STEP 3: find slant height $ l $

To find slant height $ l $ use formula:

$$ CSA = r \cdot l \cdot \pi $$

After substituting $ CSA = 71.6831 $ and $ r = 5.13 $ we have:

$$ 71.6831 = 5.13 \cdot l \cdot \pi $$$$ l = \dfrac{ 71.6831 }{ 5.13 \, \pi} $$$$ l = 4.4478 $$

STEP 4: find side $ h $

To find side $ h $ use Pythagorean Theorem:

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

After substituting $ r = 5.13 $ and $ l = 4.4478 $ we have:

$$ 5.13^2 + h^2 = 4.4478^2 $$ $$ h^2 = 4.4478^2 - 5.13^2 $$ $$ h^2 = 19.7833 - 26.3169 $$ $$ h^2 = -6.5336 $$

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

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