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1 year ago

Just do these two problems for me and I’ll be so happy

Mathematics

1 answer:

Nady [450]1 year ago

7 0

Answer:

See below

Step-by-step explanation:

5) y = x + 7

6) y = -x + 1

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A torus is formed by rotating a circle of radius r about a line in the plane of the circle that is a distance R (> r) from th

jeyben [28]

Consider a circle with radius $r$ centered at some point $(R+r,0)$ on the $x$ -axis. This circle has equation

$(x-(R+r))^2+y^2=r^2$

Revolve the region bounded by this circle across the $y$ -axis to get a torus. Using the shell method, the volume of the resulting torus is

$\displaystyle2\pi\int_R^{R+2r}2xy\,\mathrm dx$

where $2y=\sqrt{r^2-(x-(R+r))^2}-(-\sqrt{r^2-(x-(R+r))^2})=2\sqrt{r^2-(x-(R+r))^2}$ .

So the volume is

$\displaystyle4\pi\int_R^{R+2r}x\sqrt{r^2-(x-(R+r))^2}\,\mathrm dx$

Substitute

$x-(R+r)=r\sin t\implies\mathrm dx=r\cos t\,\mathrm dt$

and the integral becomes

$\displaystyle4\pi r^2\int_{-\pi/2}^{\pi/2}(R+r+r\sin t)\cos^2t\,\mathrm dt$

Notice that $\sin t\cos^2t$ is an odd function, so the integral over $\left[-\frac\pi2,\frac\pi2\right]$ is 0. This leaves us with

$\displaystyle4\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}\cos^2t\,\mathrm dt$

Write

$\cos^2t=\dfrac{1+\cos(2t)}2$

so the volume is

$\displaystyle2\pi r^2(R+r)\int_{-\pi/2}^{\pi/2}(1+\cos(2t))\,\mathrm dt=\boxed{2\pi^2r^2(R+r)}$

6 0

2 years ago

Question down below¡

lutik1710 [3]

Pattern A starts with 20 and substracts 3
First number = 20
Second number = 20 - 3 = 17
Third number = 17 - 3 = 14
Fourth number = 14 - 3 = 11

Pattern B starts with 10 and adds 2
First number = 10
Second number = 10 + 2 = 12
Third number = 12 + 2 = 14
Fourth number = 14 + 2 = 16

Now join pattern A and B
First combination = (20,10)
Second combination = (17,12)
Third combination = (14, 14)
Fourth combination = (11, 16)

The answer is d. (20,10), (17,12), (14,14), (11,16)

3 0

2 years ago

What is ratio of 28 red pens and 84 black pens

Setler [38]

Ratio

28:84

If you want it simplified
it would be
1:3

8 0

3 years ago

Read 2 more answers

List the factor pair of numbers for 48 and 56

levacccp [35]

gcf(48,56) = 8
The factors of 48 are 48, 24, 16, 12, 8, 6, 4, 3, 2, 1
The factors of 56 are 56, 28, 14, 8, 7, 4, 2, 1
The matching factors for 48 and 56 are 1, 2, 4, and 8.

3 0

2 years ago

If the diameter of the circle below is 16 in., what is the area of the circle?

mote1985 [20]

Answer:

64π inches² (approximately 201.06 inches²)

Step-by-step explanation:

$A=\pi r^2$ where $r$ is the radius