Find the height $ h $ of a cone if radius $r = 6.54$ and area $A = 70$.

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 = 6.54 $ we have:

$$ AB = 6.54^2 \cdot \pi $$ $$ AB = 42.7716 \cdot \pi $$

STEP 2: find Curved Surface Area $ CSA $

To find Curved Surface Area $ CSA $ use formula:

$$ A = AB + CSA $$

After substituting $ A = 70 $ and $ AB = 42.7716\pi $ we have:

$$ 70 = 42.7716\pi + CSA $$ $$ CSA = 70 - 42.7716 $$ $$ CSA = 27.2284 $$

STEP 3: find slant height $ l $

To find slant height $ l $ use formula:

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

After substituting $ CSA = 27.2284 $ and $ r = 6.54 $ we have:

$$ 27.2284 = 6.54 \cdot l \cdot \pi $$$$ l = \dfrac{ 27.2284 }{ 6.54 \, \pi} $$$$ l = 1.3252 $$

STEP 4: find side $ h $

To find side $ h $ use Pythagorean Theorem:

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

After substituting $ r = 6.54 $ and $ l = 1.3252 $ we have:

$$ 6.54^2 + h^2 = 1.3252^2 $$ $$ h^2 = 1.3252^2 - 6.54^2 $$ $$ h^2 = 1.7563 - 42.7716 $$ $$ h^2 = -41.0153 $$

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

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