You want to find the volume inside the hemisphere $x^2+y^2+z^2=4$ (i.e. inside the sphere but above the plane $z=0$ ) and outside the cylinder $x^2+y^2=1$ . Call this region $R$ .

In cylindrical coordinates, we have

$\displaystyle\iiint_R\mathrm dV=\int_0^{2\pi}\int_1^2\int_0^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta$

$\displaystyle=2\pi\int_1^2 r\sqrt{4-r^2}\,\mathrm dr$

$\displaystyle=-\pi\int_3^0\sqrt u\,\mathrm du$

(where $u=4-r^2$ )

$\displaystyle=\pi\int_0^3\sqrt u\,\mathrm du$

$=\dfrac{2\pi}3u^{3/2}\bigg|_0^3=2\sqrt3\,\pi$

7 0

4 years ago

Solve for the missing side to the nearest tenth. 19 X 51

telo118 [61]

Answer:

x = 30.2 units

Step-by-step explanation:

Trigonometric Ratios

The ratios of the sides of a right triangle are called trigonometric ratios.

Selecting any of the acute angles, it has an adjacent side and an opposite side. The trigonometric ratios are defined upon those sides and the hypotenuse.

The given right triangle has an angle of measure 51° and its adjacent leg has a measure of 19 units. It's required to calculate the hypotenuse of the triangle.

We use the cosine ratio to calculate x:

$\displaystyle \cos\theta=\frac{\text{adjacent leg}}{\text{hypotenuse}}$

$\displaystyle \cos 51^\circ=\frac{19}{x}$

Solving for x:

$\displaystyle x=\frac{19}{\cos 51^\circ}$

$\displaystyle x=\frac{19}{0.6293}$

x = 30.2 units

3 0

3 years ago

I live 9 (rectangular) blocks from Tech, and I walk each morning. The trip is 3 blocks north and 6 blocks west; and I have to fo

Gnoma [55]

Answer:

We conclude that he have 84 possible complete trips to Tech.

Step-by-step explanation:

We know that he live 9 (rectangular) blocks from Tech. The trip is 3 blocks north and 6 blocks west. We calculate number of possible complete trips to Tech. We know that

$C^9_3=C^9_6$

We calculate

$C^9_{3}=\frac{9!}{3!(9-3)!}=\frac{9\cdot 8\cdot 7\cdot 6!}{3\cdot 2\cdot 1\cdot 6!} =84$

We conclude that he have 84 possible complete trips to Tech.

6 0

4 years ago

Josie measured the diameter of the base of her

ra1l [238]

Answer:

6cm

Step-by-step explanation:

The radius is half of the diameter. Being 12 inches, the radius will be 6 inches.

7 0

3 years ago