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3 years ago

What is 21 > 15 + 2a

Mathematics

1 answer:

raketka [301]3 years ago

5 0

My bests answer I got was a<3

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Cylindrical hole of radius a is bored through a sphere of radius 2a. the surface of the hole passes through the center of the sp

r-ruslan [8.4K]

The amount of material removed is the volume of the region within the sphere bounded by the cylinder. Consider a sphere of radius $2a$ centered at the origin; this sphere has equation

$x^2+y^2+z^2=4a^2\iff z^2=4a^2-x^2-y^2$

The given cylinder has equation

$x^2+y^2=a^2$

The volume of the region of interest $\mathcal D$ is given by

$\displaystyle\iiint_{\mathcal D}\mathrm dV$

Converting to cylindrical coordinates, setting

$x=r\cos\theta$

$y=r\sin\theta$

$z=\zeta$

we have

$z^2=4a^2-r^2$

and

$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\left|\dfrac{\partial(x,y,z)}{\partial(r,\theta,\zeta)}\right|\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$

$\implies\mathrm dV=r\,\mathrm dr\,\mathrm d\theta\,\mathrm d\zeta$

where $\dfrac{\partial(x,y,z)}{\partial(r,\theta,\zeta)}$ is the Jacobian of the transformation from $(x,y,z)$ to $(r,\theta,\zeta)$ . The region $\mathcal D$ is described by the set

$\left\{(r,\theta,\zeta)\,:\,0\le r\le a\land0\le\theta\le2\pi\land-\sqrt{4a^2-r^2}\le\zeta\le\sqrt{4a^2-r^2}\right\}$

The integral is then

$\displaystyle\iiint_{\mathcal D}\mathrm dV=\int_{\theta=0}^{\theta=2\pi}\int_{r=0}^{r=a}\int_{\zeta=-\sqrt{4a^2-r^2}}^{\zeta=\sqrt{4a^2-r^2}}r\,\mathrm d\zeta\,\mathrm dr\,\mathrm d\theta$

The integral with respect to $\zeta$ is symmetric about $\zeta=0$ , so we instead compute twice the integral from $\zeta=0$ to $\zeta=\sqrt{4a^2-r^2}$ , and we can immediately compute the integral with respect to $\theta$ :

$=\displaystyle4\pi\int_{r=0}^{r=a}r\sqrt{4a^2-r^2}\,\mathrm dr$

Now, let $s=4a^2-r^2$ , so that $\mathrm ds=-2r\,\mathrm dr$ :

$=\displaystyle-2\pi\int_{r=0}^{r=a}-2r\sqrt{4a^2-r^2}\,\mathrm dr=-2\pi\int_{s=4a^2}^{s=3a^2}\sqrt s\,\mathrm ds$

$=-2\pi\cdot\dfrac23s^{3/2}\bigg|_{s=4a^2}^{s=3a^2}$

$=\dfrac{4\pi}3\left((4a^2)^{3/2}-(3a^2)^{3/2}\right)$

$=\dfrac{4\pi(8-3^{3/2})a^3}3$

6 0

4 years ago

Does anyone know this?

Ilia_Sergeevich [38]

Answer:

a is 7

Step-by-step explanation:

counting all of the 90° angles and then g and h are in the middle so you add the middle also

3 0

2 years ago

How do you factor x2-49 ?

expeople1 [14]

--------------------------------------------------
Question
---------------------------------------------------
x² - 49

--------------------------------------------------
Form into 2 perfect squares
---------------------------------------------------
x² - 7²

--------------------------------------------------
Formula
---------------------------------------------------
a² - b² = (a + b) (a - b)

--------------------------------------------------
Apply the formula
---------------------------------------------------
x² - 7² = (x + 7)(x - 7)

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Answer: (x + 7)(x - 7)
---------------------------------------------------

4 0

3 years ago

Read 2 more answers

How do you solve this question /

Alinara [238K]

Answer:

5/18

Step-by-step explanation:

Parentheses next to each other mean to multiply. So, this problem means you need to multiply -4/9 and -5/8. To multiply fractions, you just have to multiply across. To make it easier to understand what I mean...

-4 x -5 = 20 (when you multiply two negatives, it makes a positive)

9 x 8 = 72

20/72 simplifies to 5/18.

Hope this helps!! Have a wonderful day C:

8 0

3 years ago

Please help!!

crimeas [40]

$1)~3x+12=3(x+4)\\\\ 2)~7y-7=7(y-1)\\\\ 3)~5x+30y=5(x+6y)\\\\ 4)~8m+36n=4(2m+9n)\\\\ 5)~6a^2+27=3(2a^2+9)\\\\ 6)~4y^2-24y=4y(y-6)\\\\ 7)~21cd-3d=3d(7d-1)\\\\ 8)~14gh-18h=2h(7g-9)\\\\ 9)~15a^2b-30ab=15ab(a-2)\\\\ 10)~16bc^2+24bc=8bc(2c+3)$

3 0

3 years ago