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2 years ago

What is the perimeter of a square which has the same area as a circle with the circumference of 4pi.

Mathematics

1 answer:

Hoochie [10]2 years ago

6 0

Answer:

perimeter = 8 pi

Step-by-step explanation:

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Use Stokes' Theorem to evaluate C F · dr F(x, y, z) = xyi + yzj + zxk, C is the boundary of the part of the paraboloid z = 1 − x

Serggg [28]

I assume $C$ has counterclockwise orientation when viewed from above.

By Stokes' theorem,

$\displaystyle\int_C\vec F\cdot\mathrm d\vec r=\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S$

so we first compute the curl:

$\vec F(x,y,z)=xy\,\vec\imath+yz\,\vec\jmath+xz\,\vec k$

$\implies\nabla\times\vec F(x,y,z)=-y\,\vec\imath-z\,\vec\jmath-x\,\vec k$

Then parameterize $S$ by

$\vec r(u,v)=\cos u\sin v\,\vec\imath+\sin u\sin v\,\vec\jmath+\cos^2v\,\vec k$

where the $z$ -component is obtained from

$1-(\cos u\sin v)^2-(\sin u\sin v)^2=1-\sin^2v=\cos^2v$

with $0\le u\le\dfrac\pi2$ and $0\le v\le\dfrac\pi2$ .

Take the normal vector to $S$ to be

$\vec r_v\times\vec r_u=2\cos u\cos v\sin^2v\,\vec\imath+\sin u\sin v\sin(2v)\,\vec\jmath+\cos v\sin v\,\vec k$

Then the line integral is equal in value to the surface integral,

$\displaystyle\iint_S(\nabla\times\vec F)\cdot\mathrm d\vec S$

$=\displaystyle\int_0^{\pi/2}\int_0^{\pi/2}(-\sin u\sin v\,\vec\imath-\cos^2v\,\vec\jmath-\cos u\sin v\,\vec k)\cdot(\vec r_v\times\vec r_u)\,\mathrm du\,\mathrm dv$

$=\displaystyle-\int_0^{\pi/2}\int_0^{\pi/2}\cos v\sin^2v(\cos u+2\cos^2v\sin u+\sin(2u)\sin v)\,\mathrm du\,\mathrm dv=\boxed{-\frac{17}{20}}$

6 0

2 years ago

Solve for c 4c+9d=16

Ad libitum [116K]

1 plus 1 =2

2+2 4 3+3 6 77÷11 7

8 0

2 years ago

Is 11,13,17 a right triangle

11Alexandr11 [23.1K]

Answer:

Step-by-step explanation:

We can use the Pythagorean theorem to check

a^2 +b^2 = c^2

11^2+13^2 = 17^2

121+169 =289

410 does not equal 289

This is not a right triangle

3 0

3 years ago

Read 2 more answers

A 30-60-90 triangle's short leg has a length of 25. What is the length of the longest leg?

astraxan [27]

Answer:

25 $\sqrt{3}$

Step-by-step explanation:

30-60-90 Triangles

short leg = x

long leg = x $\sqrt{3}$

hypotenuse = 2x

5 0

2 years ago

180 °<br> 35 °<br> X °<br><br> X = ? °

just olya [345]

Answer:

x=145°

Step-by-step explanation:

180 = 35 + x

x = 180-35

x= 145°

3 0

2 years ago