Question

Find the Curved Surface Area $CS A$ of a cone if height $h = 21 cm$ and volume $V = 252 π cm$ .

Answer

$\frac{36 2417}{49} π cm^{-} 2$

Explanation

STEP 1: find radius $r$

To find radius $r$ use formula:

V = \frac{h ^{2} \cdot r \cdot π}{3}

After substituting $V = 252 π cm$ and $h = 21 cm$ we have:

252 π cm = \frac{21 cm ^{2} \cdot r \cdot π}{3}

252 π cm \cdot 3 = 21 cm^{2} \cdot r \cdot π

756 π cm = 441 cm^{2} \cdot r \cdot π

r = \frac{756 π cm}{441 cm ^{2} π}

r = \frac{12}{7} cm^{-} 1

STEP 2: find side $l$

To find side $l$ use Pythagorean Theorem:

r^{2} + h^{2} = l^{2}

After substituting $r = \frac{12}{7} cm^{-} 1$ and $h = 21 cm$ we have:

(\frac{12}{7} cm^{-} 1)^{2} + (21 cm)^{2} = l^{2}

\frac{144}{49} cm^{-} 2 + 441 cm^{2} = l^{2}

l^{2} = \frac{21753}{49} cm^{-} 2

l = \frac{21753}{49} cm^{-} 2

l = \frac{3 2417}{7} cm^{-} 1

STEP 3: find Curved Surface Area $CS A$

To find Curved Surface Area $CS A$ use formula:

CS A = l \cdot r \cdot π

After substituting $r = \frac{12}{7} cm^{-} 1$ and $l = \frac{3 2417}{7} cm^{-} 1$ we have:

CS A = \frac{3 2417}{7} cm^{-} 1 \cdot \frac{12}{7} cm^{-} 1 \cdot π

CS A = \frac{3 2417}{7} cm^{-} 1 \cdot \frac{12}{7} cm^{-} 1 \cdot π

CS A = \frac{36 2417}{49} π cm^{-} 2

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