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Volgvan
2 years ago
7

Consider F and C below. F(x, y) = x4y5i + x5y4j, C: r(t) = t3 − 2t, t3 + 2t , 0 ≤ t ≤ 1 (a) Find a function f such that F = ∇f.

f(x, y) = (b) Use part (a) to evaluate C F · dr along the given curve C.
Mathematics
1 answer:
vovikov84 [41]2 years ago
5 0
\mathbf f(x,y)=x^4y^5\,\mathbf i+x^5y^4\,\mathbf j

We want a scalar function f(x,y) whose gradient is equivalent to the vector field. That means

\dfrac{\partial f}{\partial x}=x^4y^5\implies f(x,y)=\dfrac{x^5y^5}5+g(y)
\dfrac{\partial f}{\partial y}=x^5y^4\implies x^5y^4=x^5y^4+\dfrac{\mathrm dg}{\mathrm dy}
\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=C

So (a)

f(x,y)=\dfrac{x^5y^5}5+C

which means (b)

\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(\mathbf r(1))-f(\mathbf r(0))=f(-1,3)-f(0,0)=-\dfrac{243}5-0=-\dfrac{243}5
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Find the GCF of 36, 42​
Ipatiy [6.2K]

Answer:

Answer is 6

Step-by-step explanation:

So the greatest common factor 36 and 42 is 6.

I hope it's helpful!!

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2 years ago
Please help me and take a picture and show your work​
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48

Step-by-step explanation:

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6 * 8 is 48.

3 0
2 years ago
The circumference of the flotation device above is 62.8 inches . If three flotation devices are lined up end-to-end , then what
Karo-lina-s [1.5K]

Answer: 60 in

Step-by-step explanation:

Hi, to answer this question, first we have to calculate the diameter of one flotation device:

Circumference of a circle C = (3.14) (diameter)

Replacing with the values given:

62.8 =3.14 d

Solving for d (diameter)

62.8 /3.14 =d

20in = d

Now, since the three flotation devices are identical, we simply have to multiply the diameter of one device by 3 , to obtain length of the flotation devices lined up.

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6 0
3 years ago
Molly read 50 out of the 200 pages in her fiction book. Joel is also reading a fiction book, but his book has 280 pages in total
katrin [286]

Answer:

x = 70

Step-by-step explanation:

Given

Molly: 50 out of 200

Joel: x out of 280

Required

Find x

Start by representing the giving parameters as ration

Molly= 50 : 200

Joel = x : 280

From the question, the ratios are equivalent. So,

50 : 200 = x : 280

Convert to fractions

\frac{50}{200} = \frac{x}{280}

Solve for x

x = \frac{50}{200} * 280

x = \frac{1}{4} * 280

x = \frac{280}{4}

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5 0
2 years ago
Over the last three evenings, Yoko received a total of 73 phone calls at the call center. The third evening, she received 3 time
stealth61 [152]
The first evening she received x calls.
The second evening she received 7 fewer calls than the first evening, so she received x-7 calls.
The third evening she received 3 times as many calls as the first evening, so she received 3x calls.
Over the three evenings she received 73 calls altogether.

x+x-7+3x=73 \\
5x-7=73 \\
5x=73+7 \\
5x=80 \\
x=\frac{80}{5} \\
x=16 \Leftarrow \hbox{the first evening} \\ \\ \hbox{the second evening:} \\
x-7=16-7=9 \\ \\ \hbox{the third evening:} \\
3x=3 \times 16=48

The first evening she received 16 calls, the second evening she received 9 calls, the third evening she received 48 calls.
3 0
3 years ago
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