How does this work?????

Mathematics

1 answer:

ankoles [38]3 years ago

3 0

Answer:

x = 4 $\sqrt{5}$

Step-by-step explanation:

Since the triangle is right, solve for x using Pythagoras' identity, that is

x² = 8² + 4² = 64 + 16 = 80

Take the square root of both sides

x = $\sqrt{80}$ = $\sqrt{16(5)}$ = $\sqrt{16}$ × $\sqrt{5}$ = 4 $\sqrt{5}$ ← exact value

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Find the volume of the solid that lies inside both the cylinder x 2 y 2 = 1 and the sphere x 2 y 2 z 2 = 16

IrinaVladis [17]

The volume of the solid that lies inside both the cylinder x² + y² = 1 and the sphere x² + y² + z² = 16 is 24.74 cubic units.

<h3>What is the volume of the solid?</h3>

Find the volume of the solid that lies inside both the cylinder x² + y² = 1 and the sphere x² + y² + z² = 16

From the sphere and cylinder, the cylindrical coordinate will be

$z = \pm \sqrt{16 - r^2}$

And the radius of the cylinder is |r| < 1 and the 0 ≤ θ ≤ 2π.

Then the volume will be given as

$\rm V = \int _0^{2\pi} \int _0^1 \int _{- \sqrt{16 - r^2}}^{\sqrt{16-r^2}} \left ( r \ dz \ dr \ d\theta \right ) \\\\\\V = 2\pi \int_0^1 \left ( 2r\sqrt{16-r^2} \right ) \ dr\\\\\\V = 4\pi \left [ -\dfrac{1}{3} (16 - r^2)^{3/2} \right ]\\\\\\V = \dfrac{4\pi}{3} \left ( 16^{3/2} - 15^{3/2} \right )\\\\\\V = 24.74$