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melamori03 [73]
3 years ago
7

How many ways can 6 students stand in a line if there are 10 children to choose from?

Mathematics
1 answer:
Gre4nikov [31]3 years ago
4 0

Answer:

Around 720.

Step-by-step explanation:

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Rewrite the following integral in spherical coordinates.​
lora16 [44]

In cylindrical coordinates, we have r^2=x^2+y^2, so that

z = \pm \sqrt{2-r^2} = \pm \sqrt{2-x^2-y^2}

correspond to the upper and lower halves of a sphere with radius \sqrt2. In spherical coordinates, this sphere is \rho=\sqrt2.

1 \le r \le \sqrt2 means our region is between two cylinders with radius 1 and \sqrt2. In spherical coordinates, the inner cylinder has equation

x^2+y^2 = 1 \implies \rho^2\cos^2(\theta) \sin^2(\phi) + \rho^2\sin^2(\theta) \sin^2(\phi) = \rho^2 \sin^2(\phi) = 1 \\\\ \implies \rho^2 = \csc^2(\phi) \\\\ \implies \rho = \csc(\phi)

This cylinder meets the sphere when

x^2 + y^2 + z^2 = 1 + z^2 = 2 \implies z^2 = 1 \\\\ \implies \rho^2 \cos^2(\phi) = 1 \\\\ \implies \rho^2 = \sec^2(\phi) \\\\ \implies \rho = \sec(\phi)

which occurs at

\csc(\phi) = \sec(\phi) \implies \tan(\phi) = 1 \implies \phi = \dfrac\pi4+n\pi

where n\in\Bbb Z. Then \frac\pi4\le\phi\le\frac{3\pi}4.

The volume element transforms to

dx\,dy\,dz = r\,dr\,d\theta\,dz = \rho^2 \sin(\phi) \, d\rho \, d\theta \, d\phi

Putting everything together, we have

\displaystyle \int_0^{2\pi} \int_1^{\sqrt2} \int_{-\sqrt{2-r^2}}^{\sqrt{2-r^2}} r \, dz \, dr \, d\theta = \boxed{\int_0^{2\pi} \int_{\pi/4}^{3\pi/4} \int_{\csc(\phi)}^{\sqrt2} \rho^2 \sin(\phi) \, d\rho \, d\phi \, d\theta} = \frac{4\pi}3

4 0
2 years ago
Emily and Lily are reading the same copy of the same book for a summer reading assignment.
AnnyKZ [126]

Answer:

Step-by-step explanation:

emily : 16 + 22x....with x being the number of days

Lily : 24 + 18x

16 + 22x = 24 + 18x

22x - 18x = 24 - 16

4x = 8

x = 8/4

x = 2 <==== they will be on the same page in 2 days

16 + 22x = 16 + 22(2) = 16 + 44 = 60

24 + 18x = 24 + 18(2) = 24 + 36 = 60

and they will both be on page 60 <===

7 0
3 years ago
How do you calculate a score
dem82 [27]

Answer:

You divide what you got by how many questions there are. For example, if you got 18/20, you'd divide 18 by 20 with a calculator. Which would be 0.9, then you move the decimal over 2 times making it 90, so you got 90%

Step-by-step explanation:

3 0
2 years ago
Across a horizontal distance of 25 feet, a roller coaster has a steep drop. The heig
liberstina [14]
56 to the depth of the feet that is divide by the tens place and the 1/4 places
Of the 1 so overall it is 56
8 0
3 years ago
Gregory brought 11 pumpkins to school for pumpkin-carving contest. Each pumpkin
Zigmanuir [339]

Answer:

Step-by-step explanation:

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5 0
3 years ago
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