Conical area

A right-angled triangle has sides a=12 and b=19 at the right angle. The hypotenuse is c. If the triangle rotates on the c side as an axis, find the volume and surface area of the conical area created by this rotation.

Correct answer:

V =  2422.4377
S =  988.0996

Step-by-step explanation:

$$\begin{gathered} a=12 \\ b=19 \\ c=\sqrt{a^2+b^2}=\sqrt{12^2+19^2}=\sqrt{505}\doteq 22.4722 \\ {c}_1=a^2/c=12^2/22.4722\doteq 6.4079 \\ {c}_2=b^2/c=19^2/22.4722\doteq 16.0643 \\ h=\sqrt{c_1\cdot c_2}=\sqrt{6.4079\cdot 16.0643}\doteq 10.1459 \\ {S}_1=\pi \cdot h^2=3.1416\cdot 10.1459^2\doteq 323.3912 \\ V=(c_1+c_2)\cdot S_1/3=(6.4079+16.0643)\cdot 323.3912/3=2422.4377 \end{gathered}$$

$S=\pi \cdot h\cdot (a+b)=3.1416\cdot 10.1459\cdot (12+19)=988.0996$

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