1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
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
You might be interested in
Please help me with this problem please
11111nata11111 [884]

9514 1404 393

Answer:

  • shortest: 11 1/8 in
  • longest: 11 3/8 in

Step-by-step explanation:

It should be no surprise that the shortest allowable length is the nominal length less the full allowed tolerance:

  11 1/4 - 1/8 = 11 1/8 . . . inches

__

Likewise, the longest allowable length is the nominal length plus the maximum tolerance:

  11 1/4 + 1/8 = 11 3/8 . . . inches

_____

Of course, you know that 1/4 = 2/8, so adding or subtracting 1/8 gives the fractions shown above.

5 0
2 years ago
Here are the first four terms of a sequence.
Nataly_w [17]

Answer:

  • aₙ = 2n² + n + 1

Step-by-step explanation:

We observe that the difference of terms is 4, 7 and 15 and next level difference is 4. It means the sequence is quadratic.

<u>We can compare this with simple quadratic sequence </u>

  • 1, 4, 9, 16

<u>and find out that doubling each term gives us </u>

  • 2, 8, 18, 32

This is close to our sequence, write the terms as follows to find exact rule

<u>The first term: </u>

  • a₁ = 4  = 2*1² + 2 = 2*1² + 1 + 1

<u>The second term</u>

  • a₂ = 11  = 2*2² + 3 = 2*2² + 2 + 1

<u>The third term:</u>

  • a₃ = 22 = 2*3² + 4 = 2*3² + 3 + 1

<u>The fourth term:</u>

  • a₄ = 37 = 2*4² + 5 = 2*4² + 4 + 1

<u>The nth term as per observation above is:</u>

  • aₙ = 2n² + n + 1
8 0
2 years ago
A person bought some cosmetics from wholesale market at the rate of Rs 360 per dozen he sells it at Rs 80 a pair find the gain p
bagirrra123 [75]

Given,

CP of cosmetics = Rs 360 per dozen

SP of a pair of cosmetics = Rs 80

To find,

The gain percent

Solution,

1 dozen = 12 items

CP of 1 cosmetic = 360/12 = Rs 30

SP of 1 cosmetic = 80/2 = Rs 40

Profit = SP-CP

= Rs 40 - Rs 30

= Rs 10

Profit percentage is given by :

\%=\dfrac{profit}{CP}\times 100\\\\=\dfrac{10}{30}\times 100\\\\=33.34\%

So, the profit percentage is 33.34%.

5 0
2 years ago
Two numbers have theses properties . both numbers are greater than 6 . their hcf = 6 . their lcm = 60. what are the two numbers
aleksandrvk [35]
Let the numbers be x and y.
x*y=HCF*LCM=6*60=360
thus
y=360/x
next we find the list of combinations of x and y and test if they satisfy the conditions above:
(6,60),(12,30),(18,20),(24,15)
out of the above, only (6,60) and (12,30) satisfy both conditions. Thus our answer is:
(6,60) or (12,30)
6 0
2 years ago
A person is on the outer edge of a carousel with a radius of 20 feet that is rotating counterclockwise around a point that is ce
lubasha [3.4K]
<h2>Answer:</h2>

\boxed{(5(-\sqrt{2}+\sqrt{6}),5(\sqrt{2}+\sqrt{6}))}

<h2>Step-by-step explanation:</h2>

The origin of the coordinate system is the center of the circle. So we have an angle that measures 5\pi/12=75^{\circ}. so the x-coordinate and y-coordinate can be found, by using trigonometry as follows:

x=20cos(5\pi/12)=5\sqrt{6}-5\sqrt{2} \\ \\ Arranging:\\ \\x=5(-\sqrt{2}+\sqrt{6}) \\ \\y=20sin(5\pi/12)=5\sqrt(6)+5\sqrt{2} \\ \\ Arranging:\\ \\y=5(\sqrt{2}+\sqrt{6})

Finally, the exact value of the position of the rider after the carousel rotates 5\pi/12 radians is:

\boxed{(5(-\sqrt{2}+\sqrt{6}),5(\sqrt{2}+\sqrt{6}))}

3 0
2 years ago
Other questions:
  • Abigail's mom said she will be home from work in 656565 minutes. It is 4{:}27\text{ p.M.}4:27 p.M.4, colon, 27, start text, spac
    14·2 answers
  • Multiply 3/5 and 3/4
    9·1 answer
  • Find the original price of each item round your answer to the nearest dollar ? ​
    5·1 answer
  • Which answer lists these numbers in order from least to greatest? –11.8, 125, 1.63, –3.982, 49.2
    6·1 answer
  • Determine whether each ordered pair is a solution of y = 3x-5. (2,2) (4,7) (6,13) (5,10)
    13·1 answer
  • Compute the amount of interest earned in the following simple interest problem.
    12·2 answers
  • How does the area of triangle ABC Compare to the area of parallelogram ghjk
    8·2 answers
  • A graph passes through the points (0, 2), (2, 6), and (3, 12). Does this graph represent a linear function or a non-linear funct
    15·1 answer
  • Daniel is baking cookies but does not have all of the he needs. The ratio of flour:sugar in his recipe is 3:2cups. If he only ha
    13·1 answer
  • Both questions//Congruence
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!