STEP 1: find base area $ AB $

To find base area $ AB $ use formula:

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

After substituting $r = 3.9\, \text{cm}$ we have:

$$ AB = \left( 3.9\, \text{cm} \right)^{2} \cdot \pi $$ $$ AB = 15.21\, \text{cm}^2 \cdot \pi $$

STEP 2: find lateral area $ AL $

To find lateral area $ AL $ use formula:

$$ A = 2 AB + AL $$

After substituting $A = 291.4548\, \text{cm}$ and $AB = 15.21\pi\, \text{cm}^2$ we have:

$$ 291.4548\, \text{cm} = 2 \cdot 15.21\pi\, \text{cm}^2 + AL $$ $$ 291.4548\, \text{cm} = 30.42\, \text{cm}^2 + AL $$ $$ AL = 291.4548\, \text{cm} - 30.42\, \text{cm}^2 $$ $$ AL = 261.0348\, \text{cm} $$

STEP 3: find height $ h $

To find height $ h $ use formula:

$$ AL = 2 \cdot h \cdot r \cdot \pi$$

After substituting $AL = 261.0348\, \text{cm}$ and $r = 3.9\, \text{cm}$ we have:

$$ 261.0348\, \text{cm} = 2 \cdot h \cdot \left( 3.9\, \text{cm} \right)^{4} \cdot \pi$$$$ 261.0348\, \text{cm} = 7.8\, \text{cm} \cdot h \cdot \pi $$$$ h = \dfrac{ 261.0348\, \text{cm}}{ 7.8\, \text{cm} \, \pi } $$$$ h \approx 10.6525 $$

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