Find the height $ h $ of a cone if area $A = 1507.9645\, \text{cm}$ and Curved Surface Area $CSA = 706.8583\, \text{cm}$.

Problem has no solutions.

STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

$$ A = AB + CSA $$

After substituting $A = 1507.9645\, \text{cm}$ and $CSA = 706.8583\, \text{cm}$ we have:

$$ 1507.9645\, \text{cm} = AB + 706.8583\, \text{cm} $$ $$ 1507.9645\, \text{cm} = AB + 706.8583\, \text{cm} $$ $$ 1 AB = 1507.9645\, \text{cm} - 706.8583\, \text{cm} $$ $$ 1 AB = 801.1061\, \text{cm} $$ $$ AB = 801.1061\, \text{cm} $$

STEP 2: find radius $ r $

To find radius $ r $ use formula:

After substituting $AB = 801.1061\, \text{cm}$ we have:

$$ AB = r^2 \cdot \pi $$ $$ r^2 \cdot \pi = 801.1061\, \text{cm} $$ $$ r = \sqrt{ \frac{ 801.1061\, \text{cm} }{\pi} } $$ $$ r \approx 15.9687 $$

STEP 3: find slant height $ l $

To find slant height $ l $ use formula:

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

After substituting $CSA = 706.8583\, \text{cm}$ and $r = 15.9687\, \text{cm}^0$ we have:

$$ 706.8583\, \text{cm} = 15.9687 \cdot l \cdot \pi $$$$ l = \dfrac{ 706.8583\, \text{cm}}{ 15.9687 \, \pi } $$$$ l \approx 14.09\, \text{cm} $$

STEP 4: find side $ h $

To find side $ h $ use Pythagorean Theorem:

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

After substituting $r = 15.9687\, \text{cm}^0$ and $l = 14.09\, \text{cm}$ we have:

$$ 15.9687 + h^2 = \left( 14.09\, \text{cm} \right)^{2} $$ $$ h^2 = \left( 14.09\, \text{cm} \right)^{2} - 15.9687 $$ $$ h^2 = 198.5289\, \text{cm}^2 - 254.9994 $$ $$ h^2 = -56.4705\, \text{cm}^2 $$

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

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