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3 years ago

Write a system of two equations.

1 answer:

8 0

A system of linear equation consists of two or more linear equations with the two or more same variables. Such equations are called system of equations (or) simultaneous equations (or) a pair of equations.

Example:

x + y = 5 is one linear equation.

2x – y = 4 is another linear equation.

But x + y = 5, 2x – y = 4 together they form system of linear equation.

The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.

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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

3 years ago

A white label wrapped around a rectangular box with square bases covers 80 percent of the lateral surface area of the box. Find

erastova [34]

80% = 6 in
100%=?

100/x 80/6 (proportional relationship)

then cross multiply

600 = 80x
600/80= 7.5

x=7.5

4 0

3 years ago

Read 2 more answers

Mrs. James has $40.50 to spend at the store. She wants to buy a dress that costs thirty dollars and necklaces that cost $3.50. H

Cloud [144]

Answer:

3 necklaces

Step-by-step explanation:

$40.50-$30.00=$10.50 Subtract her total money and cost of dress

$10.50-$3.50=$6.00 subtract each necklace price from remaining money

$6.00-$3.50=$3.50 keep subtracting price of necklace

$3.50-$3.50=$0 that was 3 necklaces

8 0

2 years ago

The audience thinks

stellarik [79]

Answer:

3/10 or 30/100

8 0

3 years ago

Read 2 more answers

The circle P has a center at (0, 0) and a point on the circle at (0, 4). If it is dilated by a factor of 4, what is the distance

galina1969 [7]

Answer:

A. 32

Step-by-step explanation:

If the center is (0, 0) and a point is (0, 4) then the distance from the center to that point is 4 units. That distance is the radius. If you are dilating by a factor of 4, multiply the radius by 4 and you get 16. The new radius is 16 and the diameter= radius*2.

16*2=32

7 0

2 years ago