Convert to polar coordinates, in which the circle's equation becomes
[tex]\left(\dfrac x5\right)^2+\left(\dfrac y5\right)^2=1\implies x^2+y^2=5^2\implies r^2=5^2\implies r=5[/tex]
where [tex]x=5\cos\theta[/tex] and [tex]y=5\sin\theta[/tex], and we get the part of the circle in the first quadrant with [tex]0\le \theta\le\frac\pi2[/tex].
So the integral is
[tex]\displaystyle\int_C(8x-3y)\,\mathrm ds=\int_0^{\frac\pi2}(8x(\theta)-3y(\theta))\sqrt{\left(\dfrac{\mathrm dx}{\mathrm d\theta}\right)^2+\left(\dfrac{\mathrm dy}{\mathrm d\theta}\right)^2}\,\mathrm d\theta[/tex]
[tex]=\displaystyle\int_0^{\frac\pi2}(40\cos\theta-15\sin\theta)\sqrt{25\cos^2\theta+25\sin^2\theta}\,\mathrm d\theta[/tex]
[tex]=\displaystyle25\int_0^{\frac\pi2}(8\cos\theta-3\sin\theta)\,\mathrm d\theta[/tex]
[tex]=25(8\sin\theta+3\cos\theta)\bigg|_0^{\frac\pi2}=200-75=\boxed{125}[/tex]
Line integral involves integrating a function along a curve
The value of the line integral is 125
The equation is given as:
[tex]\mathbf{(\frac{x}{5})^2 + (\frac{y}{5})^2 = 1}[/tex]
Expand
[tex]\mathbf{\frac{x^2}{5^2} + \frac{y^2}{5^2} = 1}[/tex]
Multiply through by 5^2
[tex]\mathbf{x^2 + y^2 = 5^2}[/tex]
The equation of a circle is represented as:
[tex]\mathbf{(x - a)^2 + (y - b)^2 = r^2}[/tex]
So, by comparison
[tex]\mathbf{r^2 = 5^2}[/tex]
[tex]\mathbf{r = 5}[/tex]
Where:
[tex]\mathbf{x = rcos\theta}[/tex]
[tex]\mathbf{y = rsin\theta}[/tex]
So, we have:
[tex]\mathbf{\int_c (8x - 3y)ds}[/tex]
Because it is in the first quadrant (i.e. 0 to pi/2), the integrand becomes
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_0 (8x - 3y)rd\theta}[/tex]
Convert to polar forms
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_0 (8x - 3y)\sqrt{x^2 + y^2}d\theta}[/tex]
Substitute [tex]\mathbf{x = rcos\theta}[/tex] and [tex]\mathbf{y = rsin\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c (8rcos(\theta) - 3rsin(\theta))\sqrt{(rcos(\theta))^2 + ( rsin(\theta))^2}d\theta}[/tex]
Substitute 5 for r
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c (40cos(\theta) - 15sin(\theta))\sqrt{25cos^2\theta + 25sin^2\theta}\ d\theta}[/tex]
Factor out 5
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c 5(8cos(\theta) - 3sin(\theta))\sqrt{25cos^2\theta + 25sin^2\theta}\ d\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c 5(8cos(\theta) - 3sin(\theta))\sqrt{25(cos^2\theta + sin^2\theta)}\ d\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c 5(8cos(\theta) - 3sin(\theta))5\sqrt{(cos^2\theta + sin^2\theta)}\ d\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds = \int\limits^{\frac{\pi}{2}}_c 25(8cos(\theta) - 3sin(\theta))\sqrt{(cos^2\theta + sin^2\theta)}\ d\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =25 \int\limits^{\frac{\pi}{2}}_c (8cos(\theta) - 3sin(\theta))\sqrt{(cos^2\theta + sin^2\theta)}\ d\theta}[/tex]
In trigonometry
[tex]\mathbf{cos^2\theta + sin^2\theta = 1}[/tex]
So, we have:
[tex]\mathbf{\int_c (8x - 3y)ds =25 \int\limits^{\frac{\pi}{2}}_c (8cos(\theta) - 3sin(\theta))\sqrt{1}\ d\theta}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =25 \int\limits^{\frac{\pi}{2}}_0 (8cos(\theta) - 3sin(\theta))\ d\theta}[/tex]
Integrate
[tex]\mathbf{\int_c (8x - 3y)ds =25 \times [ (8sin(\theta) + 3cos(\theta))\ } ]|\limits^{\frac{\pi}{2}}_0}[/tex]
Expand
[tex]\mathbf{\int_c (8x - 3y)ds =25 \times ([ 8sin(\frac{\pi}{2}) + 3cos(\frac{\pi}{2})] - ([ (8sin(0) + 3cos(0)])}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =25 \times ([ 8\times 1 + 3\times 0)] - ([ (8\times 0 + 3\times 1])}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =25 \times ([ 8] - [ 3])}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =25 \times 5}[/tex]
[tex]\mathbf{\int_c (8x - 3y)ds =125}[/tex]
Hence, the value of the line integral is 125
Read more about line integrals at:
https://brainly.com/question/16571667
Find the ratio y:x, if 3x=11y,
Answer:
3:11
Step-by-step explanation:
3x=11y
Rearrange to the format of the ratio.
11y = 3x
Let's find y/x. y/x is y:x.
11y = 3x
Divide both sides by 11.
11y/11 = 3x/11
y = 3x/11
NOW LET'S DIVIDE BOTH SIDES BY x TO GET x AS THE DENOMINATOR OF y.
y = 3x/11
y/x = (3x/11)/x
y/x = (3x/11) * 1/x
y/x = 3/11
Therefore, y:x = 3:11
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A 7 volume numbered set of books is placed randomly on a shelf. What is the probability that the books will be numbered in the correct order from left to right?
Answer:
The probability of them being in the right order would be a fraction of 1/7
The system shown below is
.
x + 3y = 4
3x + 9y = 12
Answer:
Consistent and Dependent
Step-by-step explanation:
Just did the question
The diameter of the Sun is approximately 1,392,000,000 meters. What is this length
expressed in scientific notation
Answer:
1392 x 10 m
Step-by-step explanation:
Answer:
1.392 x 10⁹ m
Step-by-step explanation:
when you put numbers in scientific notation, there is one (1) non-zero number before the decimal
1.392
now count how many place you would have to move the decimal to the right in order to make 1,392,000,000
9 place
this means your second half looks like x 10⁹
now put it all together
1.392 x 10⁹ m
Evaluate the expression when a=-7 and x=2.
a-6x
.
х
5
?
Answer: 15
-7 - 6 - 2
This is the same as 7 + 6 +2, but negative due to the sign.
So, -15.
Answer:
-19
Step-by-step explanation:
a-6x
(-7)-6(2)
-7-12
-19
The power generated by an electrical circuit (in watts) as a function of its current x (in amperes) is modeled by: P(x)=-12x^2+120x What current will produce the maximum power?
Answer:
A current of 5 amperes will produce the maximum power.
Step-by-step explanation:
Let be [tex]p(x) = -12\cdot x^{2}+120\cdot x[/tex], where [tex]p(x)[/tex] is measured in watts and [tex]x[/tex] in amperes. At first we must obtain the first and second derivatives of the function to determine the current associated with maximum power. That is:
First derivative
[tex]p'(x) = -24\cdot x + 120[/tex]
Second derivative
[tex]p''(x) = -24\cdot x[/tex]
Now, we equalize the first derivative to zero and solve it afterwards: (First Derivative Test)
[tex]-24\cdot x + 120 = 0[/tex]
[tex]x = 5\,A[/tex]
The only critical point is [tex]x = 5\,A[/tex].
As next step we need to assure that critical point leads to an absolute maximum by evaluating the critical point found above in the second derivative: (Second Derivative Test)
[tex]p(5)'' = -24\cdot (5)[/tex]
[tex]p''(5) = -120[/tex]
Which indicates that critical point leads to an absolute maximum.
A current of 5 amperes will produce the maximum power.
Answer: 5
Step-by-step explanation:
khan
5. Francisco enjoys scuba diving. He descends 7 feet below sea level. He descends the same
distance 5 more times. What is Francisco's final elevation? Explain using 2-3 sentences how you
got to your answer.
Answer:
35
Step-by-step explanation:
I got my answer by multiplying the sea level he descended to the number of time he descended.
The table of values below represents a linear function and shows Marco’s progress as he is pumping gas into his car. What is the output for the initial value? Gas in Marco’s Car Seconds Spent Pumping Gas 0 12 24 36 48 Gallons of Gas in Car 3 5 7 9 11 0 gallons 2 gallons 3 gallons 6 gallons
Answer:
C
Step-by-step explanation:
The initial value obtained using the slope-intercept relation is 3 gallons.
We need to obtain the rate of change which gives the amount of gas entering into his car per second :
Rate of change = Rise / RunRate of change = - 11 - 3) / (48 - 0)
Rate of change = 8/48 = 1/6 gallons
Using the slope intercept relation :
y = bx + c b = slope ; c = initial amount of gasChoosing any pair of (x, y) point on the table :
(0, 3)
3 = 1/6x + c
3 = 1/6(0) + c
3 = 0 + c
c = 3
Therefore, the initial value is 3 gallons.
Learn more : https://brainly.com/question/23105648
unit rate of the given quantities
16 skiers on 4 boats
Answer: 16/4
Step-by-step explanation: all you do is convert it into a fraction
what are two fractions whose product is greater than 1.
Answer:
2/3 x 4/2
Step-by-step explanation:
Which best describes the association shown in the scatter plot?
Answer:
A- Strong Positive
Step-by-step explanation:
hopes this helps
Answer:
strong positive
Step-by-step explanation:
Which diagram shows plane C and plane D intersecting at ←XY→?
Answer: The bottom left option.
Step-by-step explanation:
←XY→ is the line that connects the points X and Y.
The fact that the planes C and D intersect at this linea, means that all the points where the planes are intersecting are points in the given line.
Then the correct option is the bottom left one, because the planes D and C are intersecting in a line, and the line passes through the points X and Y, so that line is called ←XY→.
QUESTION 1 1 POINT
Factor: 27m3
64.
Provide your answer below:
Content attribution
Answer:
[tex] \boxed{ \bold{ \boxed{ \sf{(3m - 4)(9 {m}^{2} + 12m + 16)}}}}[/tex]
Step-by-step explanation:
[tex] \sf{27 {m}^{3} - 64}[/tex]
[tex] \dashrightarrow{ \sf{( {3m)}^{3} - {(4)}^{3} }}[/tex]
Use the formula of a³ - b³ = ( a - b) ( a² + ab + b² )
[tex] \longrightarrow{ \sf{(3m - 4)(3 {m}^{2} + 3m \times 4 + {4}^{2} }})[/tex]
[tex] \longrightarrow{ \sf{(3m - 4)( {9m}^{2} + 12m + 16}})[/tex]
Hope I helped!
Best regards! :D
Which of the following is the midpoint of a line segment with endpoints -4 and 7? Options are -3 0 1.5 5.5 thanks <3
Answer:
1.5
Step-by-step explanation:
Answer:
1.5
Explanation:
-4 -3 0 1.5 5.5 7
this is because the median is 1.5 and if you cross out every number from each number u'll find urself with 1.5.
ex:
-4 -3 -2 -1 0 1 2 3 4 5 6 7
-3 -2 -1 0 1 2 3 4 5 6
-2 -1 0 1 2 3 4 5
-1 0 1 2 3 4
0 1 2 3
1 2
and in between 1 and 2 is 1.5 so 1.5 is ur answer
i hope this helped <3
Which symbol correctly compares the two fractions?
Use >, <, or =.
Enter your answer in the box.
5/12
3/6
Answer:
5/12<3/6
Step-by-step explanation:
3/6 is 1/2 and 5/12 is smaller than 1/2
Answer:
5/12 < 3/6
Step-by-step explanation:
3/6 is equivalent to 6/12 because 3 times 2 is 6 and 6 times two is 12. 6 (the new numerator of 3/6) is greater than the numerator of 5/12- 5. 5<6. They have the same denominator so 5/12< 6/12 which is the same as 5/12 < 3/6. Hope that helped!
find the l.c.m of 2,4,6 and 8
Answer:
24.
Step-by-step explanation:
2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24
4 , 8 , 12, 16, 20, 24
6, 12, 18, 24
8, 16, 24
Looking at the multiples of these numbers, we choose the multiple they all have in common. I usually do this for the two biggest numbers, (8 and 6;) and check my work with the others. If you would like more examples from me or need extra help, feel free to ask! I am on Brainly replying to questions and comments from 1 pm - 2 pm EST. Hope this helped!
"759,993 rounded to the nearest hundread
Answer:
760,000
Step-by-step explanation:
759,993
we round 993 up to 1,000
y = 1/9 * x - 8 what is the slope ?
Answer:
1/9
Step-by-step explanation:
the slope is m in y=mx+b
hope this helps!
Jennifer is a runner and ran a 10k last weekend. The function t=7k represents the relationship between kilometers, k, and the total running time, t. What is a reasonable range for this situation?
Answer:
We know that Jennifer ran 10k last weekend.
Now, we have the function:
t = f(k) = 7*k
(notice that this is a linear function)
First, let's define what is the range.
The range is the possible values that the outcome, t, can take.
To see the range, we first must see the domain (the possible values of of the input)
In this case, we know that the minimum value of k is 0 (the time to run 0 kilometers)
t(0) = 7*0 = 0
Here we found the minimum of the range.
Now, the maximum value in the domain is 10 (because she ran 10km, we can not have a value of k larger than 10)
The time is:
t(10) = 7*10 = 70
This will be the maximum of the range.
Then the range can be written as:
R = [0, 70]
How do u calculate these numbers from least to greatest 0.02, 0, 6.98, 2 1/16, 2.2, 6.89, 2.01
Answer:
0, 0.02, 2.01, 2 1/16, 2.2, 6.89, 6.98
Step-by-step explanation:
First thing is to turn 2 1/16 into decimal form which is 2.0625
Then look at the digit to the left of the decimal, 0 is the least, then 2, then 6
Hello i need help.
Answer:
477,900
Step-by-step explanation:
how would three billion nine hundred six thousand twelve be written in standard form 3976012 3000976012 3009760012
Answer:
30906012
Step-by-step explanation:
Solve the problem.
7/8 - 6/7 =
Answer:
1/56
Step-by-step explanation:
Answer:
1/56
Step-by-step explanation
First, find the common denominator
8 x 7 = 56
7 x 8 = 56
Your new equation is now:
7/56 - 6/56 = 1/56
Ned took a test with 25 questions. He lost 4 points for each of the 6 questions he got wrong and earned an additional 10 points for answering a bonus question correctly. How many points did Ned receive or lose overall?
Answer:
86 is his final grade. He lost 24 points and gained 10 points .
Step-by-step explanation:
There are 25 questions total. Assuming that answering all 25 questions correctly earns you a 100.
Is this answer correct?
Answer:
YES
Step-by-step explanation:
PLEASE MARK AS BRAINLIST ANSWER
Test Review that I cannot understand
Answer:
I will help you with the first one
Step-by-step explanation:
x= -3 -2 -1
y= -4 -3 -2.59
of the following options, what could be a possible first step in solving the equation x + 6 = 3x -2
Step-by-step explanation:
X + 6 = 3x - 2
Bringing like terms on one side
6 + 2 = 3x - x
8 = 2x
8/2 = x
4 = x
Answer:
subtracting 6 from both sides
Step-by-step explanation:
Mrs Perry shares out 21 biscuits between
Gemma and Zak in the ratio 1 : 6
How many biscuits does each child get?
biscuits.
Gemma gets
biscuits and Zak gets
Answer:
Gemma gets 3 biscuits
Zak gets 18 biscuits
Step-by-step explanation:
1:6 = 1+6
= 7
1/7 * 21 = 3
6/7 * 21 = 18
Step-by-step explanation:
x+6x=21
7x=21
x=3
6x=3×6= 18
Gemma gets 3 and Zak got 18
I solved it but I am getting 30 % another friend got 40% please explain to me how and which is right. work: 4/10 =.4 .... .4 x 100 = 40 % Formula percentile over rank
Answer: C) 40th percentile
=========================================================
Explanation:
The original list is {827,691,616,740,651,772,868,557,891,937} which has 10 items inside it.
The list sorts to {557,616,651,691,740,772,827,868,891,937}
We see that there are 3 items below 691. So one way to calculate the percentile is to do 3/10 = 30% like you did for your answer.
However, the term "percentile" is ambiguous as there is another way to calculate percentile. The other method involves the term in question. So we have 4 terms in the sublist {557,616,651,691} which is how your friend got 4/10 = 40%
It's unfortunate why this term is not well defined. Ideally your teacher would state instructions on how to deal with this dilemma, or post a formula to use.
The good news is at least "30th percentile" isn't listed as an answer choice, so we can rule out the first method above.
Can anyone answer these? (BRAINLIEST)
Answer:
55, 116, and 120
Step-by-step explanation:
5. a. alternate exterior b. 55
6. a. corresponding b. 116
7. a. alternate interior b. 120
I haven't done this kinda math in a while, but im positive it's right. Good luck! :)
