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3 years ago

Find the length of the arc of a circle with radius of 20 centimeters intercepted by a central angle of 45 degrees

Mathematics

1 answer:

oksian1 [2.3K]3 years ago

8 0

If the radius is 20cm, the circumference is

$C = 2\pi r = 40\pi$ centimeters.

Now, 45° is one eighth of a whole turn, so the arc intercepted by a 45° angle is one eighth of the whole circumference, so the answer is

$\dfrac{40\pi}{8} = 5\pi$

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Find all the solutions for the equation:

Contact [7]

$2y^2\,\mathrm dx-(x+y)^2\,\mathrm dy=0$

Divide both sides by $x^2\,\mathrm dx$ to get

$2\left(\dfrac yx\right)^2-\left(1+\dfrac yx\right)^2\dfrac{\mathrm dy}{\mathrm dx}=0$

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{2\left(\frac yx\right)^2}{\left(1+\frac yx\right)^2}$

Substitute $v(x)=\dfrac{y(x)}x$ , so that $\dfrac{\mathrm dv(x)}{\mathrm dx}=\dfrac{x\frac{\mathrm dy(x)}{\mathrm dx}-y(x)}{x^2}$ . Then

$x\dfrac{\mathrm dv}{\mathrm dx}+v=\dfrac{2v^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=\dfrac{2v^2-v(1+v)^2}{(1+v)^2}$

$x\dfrac{\mathrm dv}{\mathrm dx}=-\dfrac{v(1+v^2)}{(1+v)^2}$

The remaining ODE is separable. Separating the variables gives

$\dfrac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=-\dfrac{\mathrm dx}x$

Integrate both sides. On the left, split up the integrand into partial fractions.

$\dfrac{(1+v)^2}{v(1+v^2)}=\dfrac{v^2+2v+1}{v(v^2+1)}=\dfrac av+\dfrac{bv+c}{v^2+1}$

$\implies v^2+2v+1=a(v^2+1)+(bv+c)v$

$\implies v^2+2v+1=(a+b)v^2+cv+a$

$\implies a=1,b=0,c=2$

Then

$\displaystyle\int\frac{(1+v)^2}{v(1+v^2)}\,\mathrm dv=\int\left(\frac1v+\frac2{v^2+1}\right)\,\mathrm dv=\ln|v|+2\tan^{-1}v$

On the right, we have

$\displaystyle-\int\frac{\mathrm dx}x=-\ln|x|+C$

Solving for $v(x)$ explicitly is unlikely to succeed, so we leave the solution in implicit form,

$\ln|v(x)|+2\tan^{-1}v(x)=-\ln|x|+C$

and finally solve in terms of $y(x)$ by replacing $v(x)=\dfrac{y(x)}x$ :

$\ln\left|\frac{y(x)}x\right|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\ln|y(x)|-\ln|x|+2\tan^{-1}\dfrac{y(x)}x=-\ln|x|+C$

$\boxed{\ln|y(x)|+2\tan^{-1}\dfrac{y(x)}x=C}$

7 0

3 years ago

23)

Kaylis [27]

Answer:

90 degrees

Step-by-step explanation:

Step 1: State what is known

Notice that the 2 lines cut the square in half and create 4 congruant triangles. We know a full circle is 360 degrees and the intersection is a perfect circle formed by 4 angles. We can find the angle AEB by dividing 360 by 4 which gives you 90 degrees

8 0

2 years ago

Determine whether each given value of x satisfies the inequality x+1<x/3

Vinil7 [7]

$x+1 < \dfrac{x}{3}\ \ \ \ |\cdot3\\\\3x+3 < x\ \ \ \ |-3\\\\3x < x-3\ \ \ \ |-x\\\\2x < -3\ \ \ \ |:2\\\\x < -1.5$

Answer: (c) x = -4

5 0

3 years ago

Each letter of the word ASSESSMENT is written on a card and placed in a bag. You are equally likely to get any card.

NARA [144]

Answer:

I would say getting four S would be more percent than the other letters in the assessment word. They have more chance, and more possibility.

Step-by-step explanation: Brainliest please

6 0

3 years ago

What is the value of X^2-4y/2 if x=4 and y=-3

Dominik [7]

To solve this, we just need to plug in the numbers.

4^2-4*3/2

Now solve!

4^2-4*3/2

16-4*9

16-36

-20

There's your answer!

Any questions?

3 0

3 years ago