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2 years ago

Need help with this problem on my hw

Mathematics

1 answer:

Andrew [12]2 years ago

6 0

Answer:

157

Step-by-step explanation:

C = $\pi$ x diameter.

C = 3.14 x 50

C = 157 units

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For each of the following vector fields F , decide whether it is conservative or not by computing curl F . Type in a potential f

Phantasy [73]

The key idea is that, if a vector field is conservative, then it has curl 0. Equivalently, if the curl is not 0, then the field is not conservative. But if we find that the curl is 0, that on its own doesn't mean the field is conservative.

$\mathrm{curl}\vec F=\dfrac{\partial(5x+10y)}{\partial x}-\dfrac{\partial(-6x+5y)}{\partial y}=5-5=0$

We want to find $f$ such that $\nabla f=\vec F$ . This means

$\dfrac{\partial f}{\partial x}=-6x+5y\implies f(x,y)=-3x^2+5xy+g(y)$

$\dfrac{\partial f}{\partial y}=5x+10y=5x+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=10y\implies g(y)=5y^2+C$

$\implies\boxed{f(x,y)=-3x^2+5xy+5y^2+C}$

so $\vec F$ is conservative.

$\mathrm{curl}\vec F=\left(\dfrac{\partial(-2y)}{\partial z}-\dfrac{\partial(1)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x)}{\partial z}-\dfrac{\partial(1)}{\partial z}\right)\vec\jmath+\left(\dfrac{\partial(-2y)}{\partial x}-\dfrac{\partial(-3x)}{\partial y}\right)\vec k=\vec0$

Then

$\dfrac{\partial f}{\partial x}=-3x\implies f(x,y,z)=-\dfrac32x^2+g(y,z)$

$\dfrac{\partial f}{\partial y}=-2y=\dfrac{\partial g}{\partial y}\implies g(y,z)=-y^2+h(y)$

$\dfrac{\partial f}{\partial z}=1=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=z+C$

$\implies\boxed{f(x,y,z)=-\dfrac32x^2-y^2+z+C}$

so $\vec F$ is conservative.

$\mathrm{curl}\vec F=\dfrac{\partial(10y-3x\cos y)}{\partial x}-\dfrac{\partial(-\sin y)}{\partial y}=-3\cos y+\cos y=-2\cos y\neq0$

so $\vec F$ is not conservative.

$\mathrm{curl}\vec F=\left(\dfrac{\partial(5y^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial y}\right)\vec\imath+\left(\dfrac{\partial(-3x^2)}{\partial z}-\dfrac{\partial(5z^2)}{\partial x}\right)\vec\jmath+\left(\dfrac{\partial(5y^2)}{\partial x}-\dfrac{\partial(-3x^2)}{\partial y}\right)\vec k=\vec0$

Then

$\dfrac{\partial f}{\partial x}=-3x^2\implies f(x,y,z)=-x^3+g(y,z)$

$\dfrac{\partial f}{\partial y}=5y^2=\dfrac{\partial g}{\partial y}\implies g(y,z)=\dfrac53y^3+h(z)$

$\dfrac{\partial f}{\partial z}=5z^2=\dfrac{\mathrm dh}{\mathrm dz}\implies h(z)=\dfrac53z^3+C$

$\implies\boxed{f(x,y,z)=-x^3+\dfrac53y^3+\dfrac53z^3+C}$

so $\vec F$ is conservative.

4 0

3 years ago

charlie has 2 fair coins. he repeatedly tosses the pair of coins simultaneously (i.e., two tosses at a time), until he has seen

Serggg [28]

By using the trial method we get a total number of trials taken by Charlie to see both sides of both the coins is 4.

<h3>What is probability?</h3>

Probability is the name of the area of mathematics that deals with the examination of random events. The ratio of favorable occurrences to the total number of events is used to calculate an event's probability.

P(E) = F(E)/T (E)P(E)

It stands for the probability that an event will occur.

F(E) = Amount of favorable occurrences

Total number of trials (T(E))

Given that Charlie has 2 fair coins.

If he tosses the pair of coins simultaneously, then the number of samples can be HH, HT, TH, TT.

So to see both sides of both the coins he should toss the coin four times.

To know more about probability, visit:

brainly.com/question/12629667

#SPJ4

7 0

9 months ago

Can someone plz help me with Algebra 2....... :) Will give Brainliest!!

Anarel [89]

Using logarithms property of log(x)+log(y)=log(xy)
so here, you can sum the equation to;
$log((x+6)*(x-6))=2$
so you can simply say that;
$log_{8}((x+6)( x-6))=2$
and by multiplying (x+6)*(x-6)
$log_{8}(x^2-36)=2$
and as you know also that;
$a^{b}=c$ is same as $log _{a}c=b$
so you can simply state it as;
$8^2=x^2-36
64=x^2-36
64+36=x^2
x^2=100
x=10$
And you can check your work by substituting with 10 instead of x in the original function.
Hope this helps!

6 0

3 years ago

Select the correct measurement of the angle shown on the protractor. a protractor showing an angle with one side lined up with t

Mumz [18]

Answer:

110°

Step-by-step explanation:

Each tick line in a protractor shows 10°. So the next is 110° inclined.

6 0

3 years ago

Read 2 more answers

DOUBLE CHECKING: I got X=12, and you have to solve for measure K so I got k=42, is this correct? I’ll provide a picture and my e

iragen [17]

×=12

4×+7+6×+9=118

4×+6×=118*7+9

10×=120

×=120/10

3 0

3 years ago