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Anuta_ua [19.1K]

3 years ago

12

David bought a 5-gallon container of paint to paint his house. After he finished painting, he had 2 quarts of paint left over. H

ow many quarts of paint did David use?

1 answer:

Deffense [45]3 years ago

6 0

Answer:3 1/2

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

3 years ago

Given f(x)=-3x+3solve for x when f(x)=6

natali 33 [55]

Answer:

x = -1

Step-by-step explanation:

Given: $f(x) = - 3x + 3$

Solve for x, When: $f(x) = 6$

Step by step:

$- 3x + 3 = 6$

$- 3x = 6 - 3$

$- 3x = 3$

$x = 3 \div - 3$

$\boxed{\green{x = - 1}}$

3 0

3 years ago

I’ll give Brainliest please help

Strike441 [17]

Answer:

Sirius will be 20 years old and his father will be 40 years old. ratio form: 20:40

Step-by-step explanation:

hope this helps

4 0

3 years ago

Read 2 more answers

Tom and Bob have a total of 49 toys if Bob has 5 more toys than Tom, how many toys does each one have

pickupchik [31]

49 + 5 = 54

So the answer is 54

5 0

3 years ago

Read 2 more answers

HELPPPPPPPPPPPPPPPPPPPPPPPPPP

ss7ja [257]

Answer:

Step-by-step explanation:

+(+) Two like signs become a positive sign 3+(+2) = 3 + 2 = 5

−(−) 6−(−3) = 6 + 3 = 9

+(−) Two unlike signs become a negative sign 7+(−2) = 7 − 2 = 5

−(+) 8−(+2) = 8 − 2 = 6

7 0

3 years ago