Complete Question:
Justify the rationale behind Ms Calcutta's decision to use the circumcenter
Answer:
Check below for the answer and explanations.
Step-by-step explanation:
Ms Calcutta wants every occupant of the three office buildings to have equal easy access to the recycling station. The circumcenter of the three three office buildings, where the recycling station is located, has equal distance from the three offices, this means that all of Ms. Calcutta's workers in the three buildings can have to the recycling station equally without causing unnecessary inconvenience for occupants of any of the buildings.
In ΔCDE, the measure of ∠E=90°, the measure of ∠D=38°, and DE = 12 feet. Find the length of CD to the nearest tenth of a foot.
Given:
Given that the triangle CDE is a right triangle.
The measure of ∠E is 90° and ∠D is 38°
The length of DE is 12 feet.
We need to determine the length of CD.
Length of CD:
The length of CD can be determined using the trigonometric ratio.
Thus, we have;
[tex]cos \ \theta=\frac{adj}{hyp}[/tex]
where [tex]\theta=D[/tex] and the side adjacent to angle D is ED and hypotenuse is CD.
Substituting these values, we get;
[tex]cos \ D=\frac{ED}{CD}[/tex]
where ∠D = 38°, ED = 12 feet.
Substituting, we get;
[tex]cos \ 38^{\circ}=\frac{12}{CD}[/tex]
Simplifying, we have;
[tex]CD=\frac{12}{cos \ 38^{\circ}}[/tex]
[tex]CD=\frac{12}{0.788}[/tex]
Dividing, we get;
[tex]CD=15.2[/tex]
Thus, the length of CD is 15.2 feet.
Answer:
51.6
Step-by-step explanation:
its because you have to use cos toh and all of the calculator functions
What is the surface area of a pyramid that has a base of 27mm and the slanted height is 35mm?
The scatterplot shows the time it takes for a new hot plate to boil various amounts of water, and one lab group’s attempt at a line of best fit. What feedback should another lab group give during a peer review of the work? The line of best fit is reasonable because it does not go through any points. The line of best fit is reasonable because it goes through the largest cluster of points. The line of best fit is not reasonable because it does not go through any points. The line of best fit is not reasonable because it has more points below it than above it.
Answer:
The correct option is;
The line of best fit is not reasonable because it has more points below it than above it.
Step-by-step explanation:
Here we note that there are a total of seven points in the scatter plot and there are five of the points below the line of best fit and just two above the line.
Of the five points below the line of best fit, four are just about touching the underside of the line while one of the two points above the line is just about touching the line.
The proper positioning of the line can be reviewed, therefore, with a line drawn through the four points presently touching the underside of the line of best fit.
The feedback the other lab group should give is (d) the line of best fit is not reasonable because it has more points below it than above it.
From the scattered plot (see attachment), we can see that the scattered plot has a total of 7 dots,
5 of which are below the line of best fit, 1 is on the line of best fit, and the other one is above it.There are more points below the scattered plot, than above it.
This means that the scattered plot is not reasonable because of (d)
Read more about scattered plots at:
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A movie theater make a profit Of $15 on adult ticket and $7.50 for children ticket.Write an inequality in slope intercept from that expresses when the profit is less than $15000
Answer:
y < 2,000 - 2x
Step-by-step explanation:
Solution:-
- The number of adult ticket sold = x
- The number of children ticket sold = y
- The profit from a adult ticket, L1 = $15
- The profit from a adult ticket, L2 = $7.5
- The amount of profit ( P )made by selling "x" adult tickets and "y" children tickets:
P = L1*x + L2*y
P = 15x + 7.5y
Where, We are to set up an equality for Profit P < $15000:
15x + 7.5y < 150,000
- Re-arrange to develop the slope-intercept form of an inequality:
7.5y < 150,000 - 15x
y < 2,000 - 2x
Where, the slope = -2 and intercept = $2,000
Answer: y<-2x+2,000
Step-by-step explanation:
What's 0.11 as a fraction?
Answer:
11/100
Step-by-step explanation:
Write a recursive rule for the sequence.
X, X, 2x, 3x, 5x, 8x, ...
Answer:
The recursive rule for the sequence is
[tex]a_{1}[/tex] = x
[tex]a_{2}[/tex] = x
[tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + [tex]a_{n-2}[/tex]
Step-by-step explanation:
∵ The first term is x
∴ [tex]a_{1}[/tex] = x
∵ The second term is x
∴ [tex]a_{2}[/tex] = x
∵ The third term is 2x
- That means the third term is the sum of the 1st and 2nd terms
∴ [tex]a_{3}[/tex] = [tex]a_{1}[/tex] + [tex]a_{2}[/tex]
∵ The fourth term is 3x
∴ [tex]a_{4}[/tex] = [tex]a_{2}[/tex] + [tex]a_{3}[/tex]
∵ The fifth term is 5x
∴ [tex]a_{5}[/tex] = [tex]a_{3}[/tex] + [tex]a_{4}[/tex]
∵ The sixth term is 8x
∴ [tex]a_{6}[/tex] = [tex]a_{4}[/tex] + [tex]a_{5}[/tex]
From all above the sequence is Fibonacci sequence where its recursive rule is
[tex]a_{1}[/tex] = first term
[tex]a_{2}[/tex] = second term
[tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + [tex]a_{n-2}[/tex]
The recursive rule for the sequence is
[tex]a_{1}[/tex] = x
[tex]a_{2}[/tex] = x
[tex]a_{n}[/tex] = [tex]a_{n-1}[/tex] + [tex]a_{n-2}[/tex]
Final answer:
The sequence provided follows a pattern similar to the Fibonacci sequence but starts with two identical terms. The recursive rule is F(1) = X, F(2) = X, and for n > 2, F(n) = F(n-1) + F(n-2).
Explanation:
The given sequence is reminiscent of the Fibonacci sequence, where each term after the first two is the sum of the two preceding ones. However, since the first two terms here are identical, both being X, it alters the common Fibonacci sequence slightly. To write a recursive rule for the sequence X, X, 2X, 3X, 5X, 8X, ... we would use the standard recursion formula with a modification accounting for the sequence's unique start.
Recursive Rule for the Modified Fibonacci Sequence:
F(1) = X (First term)
F(2) = X (Second term)
For n > 2, F(n) = F(n-1) + F(n-2) (Subsequent terms)
Here, F(n) represents the nth term in the sequence. By following this recursive rule, you can generate any term in the sequence by summing up the two previous terms.
The owner of the car dealership decides to treat the value 22 as an outlier.which measure of center or spread is affected the most if the owner removes this outlier ?
Answer:
See explanation
Step-by-step explanation:
Solution:-
- The effect of an outlier on the mean, median and range is to be investigated.
- Mean: It is the average of all the values. If the outlier "22" is lies on the upper spectrum of the center value. If the outlier is removed the value of center or mean will decrease.
- Median: The median value is mostly defined as the value around which their is a cluster of data. The value of the outlier "22" if close to that cluster of data points is omitted there will be small deviation in the value of median. If the value of the outlier "22" if far away to that cluster of data points is omitted there will be significant deviation in the value of median.
- Range: Is defined by the uppermost and lowermost value from a set of data points that is considered. The value of outlier will equally effect either of these limits depending where the outlier lies close to upper limit or lower limit of the range.
The mean is the measure of center most affected by an outlier, especially in a skewed distribution. The range and standard deviation, measures of spread, will also be notably impacted by the removal of an outlier.
The removal of the value 22 as an outlier from a set of data will affect measures of center and spread, with the mean being the measure of central tendency most susceptible to change due to an outlier. The standard deviation and range are measures of spread that would also be influenced when removing an outlier, especially if the dataset is not symmetrical. However, the median and mode are more robust to outliers and would likely be less affected by the removal of a single value.
In the context of a car dealership data analysis where the sales are right-skewed, indicating that higher values are more variable, the inclusion of a high outlier can raise the mean significantly. By removing the outlier, the mean will decrease, potentially providing a more accurate representation of the central tendency of the data. The range will also decrease, as the outlier might be the highest value in the set. The standard deviation might decrease as well, as it measures the average distance from the mean, which would be less skewed without the outlier.
***EASY***** HELP!!!!!! i have been crying over this problem for an hour. please help
Answer:
UT = 15
Step-by-step explanation:
Its okay this was a hard one to know. since its a 90 degrees angle you can use the Pythagorean theorem.
A^2 + B^2 =C^2
but in this case they give you c and as for b
so it be C^2 - A^2 = B^2
17^2 - 8^2 = B^2
289 - 64 = B^2
225 = B^2
now you square both sides getting B = [tex]\sqrt{225}[/tex]
The steps to convert 37 over 4 to a decimal are shown below:
Division is shown with divisor 4, quotient 9.21, and dividend 37 labeled as Step 1. The numbers below 37 in sequence are subtract 36 lebeled as Step 2, 10 labeled as Step 3, subtract 6 labeled as step 4, 4, subtract 4, and 0.
In what step is the first error?
Answer:
Step 4
Step-by-step explanation:
We are given that
[tex]\frac{37}{4}[/tex]
We have to convert [tex]\frac{37}{4}[/tex] into decimal.
Divisor:The number which divides the dividend.
Dividend:The number which is divided by divisor.
Quotient:The number obtained when the divisor divides the dividend.
[tex]\frac{37}{4}=9.25[/tex]
Step:1
Dividend 37
Step 2:
37
-36
=1
Step 3:
10
Step 4:
10
-8
=2
Step 5:
20
-20
=0
Hence, the first error in step 4.
The slide is 4 meters due west of the tire swing and 3 meters due south of the monkey bars. What is the distance between the tire swing and the monkey bars?
Answer:
The distance between the tire swing and the monkey bars = 5 m
Step-by-step explanation:
Given:
Distance between tire swing and the slide = 4 m
Distance between monkey bars and slide = 3 m
We have to find the distance between tire swing and the monkey bars.
Let the distance be "h" meters.
From the diagram we can see that when they are arranged they form a right angled triangle.
Where the distance we have to find is the hypotenuse of the triangle.
Using Pythagoras formula:
⇒ (hypotenuse)^2 = (base)^2 + (perpendicular)^2
Plugging the values:
⇒ [tex](hypotenuse)^2 = (base)^2 + (perpendicular)^2[/tex]
⇒ [tex]h^2=b^2+p^2[/tex]
⇒ [tex]h=\sqrt{b^2+p^2}[/tex]
⇒ [tex]h=\sqrt{(4)^2+(3)^2}[/tex]
⇒ [tex]h=\sqrt{16+9}[/tex]
⇒ [tex]h=\sqrt{25}[/tex]
⇒ [tex]h=5[/tex] m
So,
The distance between the tire swing and the monkey bars = 5 m
Adam wants to compare the fractions 2 over 5, 1 over 6, and 1 over 3. He wants to order them from least to greatest and rewrite them so they all have the same denominator. Explain how Adam can rewrote the fractions. I'm in 4th grade to..
Answer:
Least to greatest should be 1 over 6, 1 over 3, and then 2 over 5. least to greatest with the same denominator should be 5 over 30, 10 over 30, and 12 over 30
Step-by-step explanation:
First step is to find the common denominator, which is 30. Then you want to figure out how you got to 30 and multiply that by nominator. after you have that all complete, then just sort from least to greatest. To listed from least to greatest is you take the smallest number, for example which would be 5 over 30 and then sorts continuously from the greater number as well, which would be 12 over 30
3. Find the angle between the given vectors to the nearest tenth of a degree.
u = <8,7>, V = <9,7>
a) -8.3
b) 1.7
c) 3.3
d) 13.3
Answer:
Step-by-step explanation:
[tex]u.v=|u||v| cos \theta\\(8)(9)+(7)(7)=\sqrt{8^2+7^2 } \sqrt{9^2+7^2} cos \theta\\72+49=\sqrt{64+49} \sqrt{81+49}~ cos ~\theta\\121=\sqrt{113} \sqrt{130} ~cos~\theta\\cos~\theta=\frac{121}{\sqrt{113} \sqrt{130} } \\\theta=cos^{-1} (\frac{121}{\sqrt{113}\sqrt{130} } )\approx 3.3^\circ[/tex]
In the figure below, BD and EC are diameters of circle P. What is the arc measure of ADE in degrees?
Answer:
Hey briannah its haley lol. I thought I'd help you out with this for Mrs F's class! The answer is 333.
Step-by-step explanation: Basically you do 180-63-90 and that equals 27. Then from there you subtract 360 from 27, then there is your answer 333!!
Basically you do 180-63-90 and that equals 27. Then from there you subtract 360 from 27, then there is your answer 333!!
Charlie , a frequent business traveler, is considering signing up for hotel reward program.Hotel Elegance is currently offering 50 points for signing up, plus 15 points for every night charlie books at the hotel. Alternatively , hotel Paradiso is offering a sign-up bonus of 250 points, plus 5 points per night. If charlie books a certain number of nights, points he earns the hotel will be the same. How many points will he have
Answer:
350 points.
Step-by-step explanation:
Given:
Charlie , a frequent business traveler, is considering signing up for hotel reward program.
Hotel Elegance is currently offering 50 points for signing up, plus 15 points for every night charlie books at the hotel.
Alternatively , hotel Paradiso is offering a sign-up bonus of 250 points, plus 5 points per night.
If charlie books a certain number of nights, points he earns the hotel will be the same.
Question asked:
How many points will he have ?
Solution:
Let Charlie books for number of nights = [tex]x[/tex]
Hotel Elegance is offering = 50 points for signing up + 15 points for every night
Total points, Charlie can earn from this hotel [tex]=50+15x[/tex]
Similarly, Charlie can earn from hotel Paradiso = [tex]250+5x[/tex]
As given, that for certain number of nights, points he earns the hotel will be the same. Hence, the equation will be:-
[tex]50+15x=250+5x\\ \\ By\ subtracting \ both\ sides\ by\ 5x\ and\ 50\\ \\ 50-50+15x-5x=250-50+5x-5x\\ \\ 10x=200\\ \\ By \ dividing\ both\ sides\ by\ 10\\ \\ x=20[/tex]
By substituting the value:-
Charlie books for number of nights = [tex]x[/tex] = 20
You can take any one equation as both are equal.
Number of points, he will get [tex]=50+15x[/tex]
[tex]=50+15\times20\\ \\= 50+300\\ \\ =350[/tex]
Thus, he will have 350 points.
Final answer:
By setting up an equation to equate points from Hotel Elegance and Hotel Paradiso, solving for 'n' shows Charlie must book 20 nights at either hotel to earn a total of 350 points.
Explanation:
To determine the number of nights Charlie would need to book for the points he earns at Hotel Elegance and Hotel Paradiso to be the same, we need to set up an equation. Hotel Elegance offers 50 points for signing up and 15 points per night, and Hotel Paradiso offers 250 points for signing up and 5 points per night. Let n represent the number of nights Charlie books. For Hotel Elegance, the points earned will be 50 + 15n, and for Hotel Paradiso it will be 250 + 5n. We set up the equation 50 + 15n = 250 + 5n.
Solving the equation, we subtract 5n from both sides to get 50 + 10n = 250. Then, we subtract 50 from both sides which gives us 10n = 200. Dividing both sides by 10, we get n = 20. This means Charlie needs to book 20 nights for the points to be equal. Substituting n = 20 back into either equation, we get 50 + 15(20) or 250 + 5(20), both resulting in 350 points.
20 points please help!!
Answer:
3/7+r = 13/15
r=46/105
Step-by-step explanation:
Given ribbon of her project= 13/15
Used 3/7
left=r
3/7+r = 13/15
r=13/15 - 3/7
r=46/105
Misty is buying a pair of jeans for $54 and a pair of boots for $95. She has a coupon for 15% off is she spends at least $75. If the sales tax is 8% how much will she pay in total?
Answer:
$136.78
Step-by-step explanation:
Step 1: add 54 + 95= 149
Step 2: Multiply 149 and 15% (You can convert to a decimal) = 22.35
Step 3: Minus 149- 22.35= 126.65
Step 4: Multiply 126.65 by 8% (or 0.08)= 10.1320
Step 5: Round 10.1320 to the nearest cent= 10.13
Step 6: Add 126.65 to 10.13= 136.78
Misty spent $136.78 total.
(make sure to include your units!)
Patti is using mental math to evaluate the expression (70 + 12.8 + 30) + 6.1. She recognizes that the expression will be easier to simplify if she can combine 70 and 30 before she works with the other numbers. How do the properties of operations allow Patti to take that approach?
Answer:54
Step-by-step explanation:
Answer:
D. She can use the commutative property to rewrite the expression as (70 + 30 + 12.8) + 6.1
Step-by-step explanation:
what happens to the frequency as the period of a sinusoidal function increases
answer choices:
a. increases
b. decreases
c. depends on whether it is sin or cosine
It should be c because the relationship between the frequency and the function matters.
One teacher wants to give each student 35 of a slice of pizza. If the teacher has 6 slices of pizza, then how many students will she be able to hand out pizza to?
Answer:
Total ten students can feed on 6 slices of pizza with each student getting [tex]\frac{3}{5}[/tex] of a pizza slice
Step-by-step explanation:
Given-
Teacher wished to give [tex]\frac{3}{5}[/tex] of a pizza slice to each student
This means that each student will get only [tex]\frac{3}{5} *[/tex] one slice of pizza
Now the total number of slices of pizza [tex]= 6[/tex]
Thus,
[tex]\frac{3}{5}[/tex] of a pizza slice to each student [tex]*[/tex] total number of students [tex]= 6[/tex]
Rearranging the above equation, we get -
[tex]\frac{3}{5} * 1 * X = 6[/tex]
Where X is the number of students
[tex]X = \frac{6}{\frac{3}{5} } \\X = \frac{5}{3} * 6\\X = 10[/tex]
Total ten students can feed on 6 slices of pizza with each student getting [tex]\frac{3}{5}[/tex] of a pizza slice
What is the solution to the equation?
6x-3=3x+12
Answer:
x=5
Step-by-step explanation:
Step 1: Subtract 3x from both sides.
6x−3−3x=3x+12−3x
3x−3=12
Step 2: Add 3 to both sides.
3x−3+3=12+3
3x=15
Step 3: Divide both sides by 3.
3x/3=15
x=5
The solution to the equation 6x - 3 = 3x + 12 is x = 5.
We have,
To solve the equation 6x - 3 = 3x + 12 for x, we can follow these steps:
Begin by isolating the x-terms on one side of the equation.
To do this, subtract 3x from both sides:
6x - 3 - 3x = 3x + 12 - 3x
Simplifying
3x - 3 = 12
Next, move the constant term (-3) to the other side by adding 3 to both sides:
3x - 3 + 3 = 12 + 3
Simplifying:
3x = 15
Finally, divide both sides of the equation by 3 to solve for x:
3x / 3 = 15 / 3
Simplifying:
x = 5
Therefore,
The solution to the equation 6x - 3 = 3x + 12 is x = 5.
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Every day a group of nine workers pick peas pods in a field each worker picks 2400 P pods and each Peapod contain six peas how many peas do the workers pick every day day
If angle a measures 67 degrees ,how much does angle b measure ?
Answer:
∠ b = 23°
Step-by-step explanation:
Complementary angles sum to 90°, thus
∠ a + ∠ b = 90°, that is
67° + ∠ b = 90° ( subtract 67° from both sides )
∠ b = 23°
In the equation y=-3x - 4, what is
the slope and y-intercept?
A. slope = -4 y-intercept = -3
B. slope = -3 y-intercept = 4
C. slope = 3 y-intercept = -4
D. slope = -3 y-intercept = -4
Answer:
The slope is -3 and the y-intercept is -4. So D.
Step-by-step explanation:
The height of a door is 2.5 feet longer than its width, and its front area is 1610.5 square feet. Find the width and height of the door.
Question A) Work out the distance travelled on a cycle in the first 14 seconds.
Answer:
96m
Step-by-step explanation:
distance traveled =
area of the graph( up to 14 sec)
= (1/2*4*8)+(8*10)
= 16 + 80 = 96
The distance traveled by a cycle in a certain time is calculated by the formula 'distance = velocity x time'. But without the velocity (speed of the cycle), we cannot calculate the covered distance.
Explanation:To find the distance covered by a moving object, we use the formula distance = velocity x time. However, from your query, it seems we don't have the velocity (speed of the cycle) given. Once we know the velocity (say in km/hr or m/s), we can simply multiply it with the time (in this case 14 seconds) to get the distance covered. For example, if the velocity was 1 m/s, the distance covered in 14 seconds would be just 1m/s x 14s = 14 meters.
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What is the answer to this question
Determine the quadrant when the terminal side of the angle lies according to the following conditions: cos (t) < 0, csc (t) > 0.
Answer:
The angle is in the second quadrant.
Step-by-step explanation:
The cosecant of an angle is the same as the reciprocal of the sine of that angle. In other words, as long as [tex]\sin (t) \ne 0[/tex],
[tex]\displaystyle \csc t = \frac{1}{\sin t}[/tex].
Therefore, [tex]\csc(t) > 0[/tex] is equivalent to [tex]\sin (t) > 0[/tex].
Consider a unit circle centered at the origin. If the terminal side of angle [tex]t[/tex] intersects the unit circle at point [tex](x,\, y)[/tex], then
[tex]\cos (t) = x[/tex], and[tex]\sin(t) = y[/tex].For angle [tex]t[/tex],
[tex]x = \cos(t) < 0[/tex], meaning that the intersection is to the left of the [tex]y[/tex]-axis.[tex]y = \sin(t) > 0[/tex], meaning that the intersection is above the [tex]x[/tex]-axis.In other words, this intersection is above and to the left of the origin. That corresponds to second quadrant of the cartesian plane.
Is the point (1, 13) a solution to y> 2x + 5
Answer:
(1,13) is a solution
Step-by-step explanation:
y> 2x + 5
Substitute the point in and see if the inequality is true
13> 2(1) + 5
13 > 2+5
13>7
This is true so the point is a solution
Answer:
Yes
Step-by-step explanation:
Substitute the point into the inequality to see if it is true.
(1,13) --> (x,y)
So, x is 1, and y is 13, and we can substitute them in to the inequality
y> 2x + 5
13 > 2(1) + 5
13> 2 + 5
13 > 7
Since 13 is greater than 7, it is true, and (1,13) is a solution
Given a linear function y =23x+2, which two points make the line?
(0, -2) and (3, 4)
(0, 2) and (3, 4)
(0, 0) and (2, 3)
(2, 3) and (0, 0)
Answer:
(0, 2) and (3, 4)
Step-by-step explanation:
The equation is in slope-intercept form:
y = mx +b . . . . . a line with slope m and y-intercept b
So, you know immediately that the y-intercept is (0, 2). This matches only one answer choice.
(0, 2) and (3, 4)
Seawater has density 1025 kg/m3 and flows in a velocity field v = y i + x j, where x, y, and z are measured in meters and the components of v in meters per second. find the rate of flow outward through the hemisphere x2 + y2 + z2 = 81, z ≥ 0.
Answer:
Flow (Q) = 0 kg/s
Step-by-step explanation:
Given:-
- The density of seawater, ρ = 1025 kg/m^3
- The velocity field of flow, v ( x,y,z ) = y i + x j
Find:-
The rate of flow outward through the hemisphere x2 + y2 + z2 = 81, z ≥ 0.
Solution:-
For some flow, if we know the point density at (x,y,z) is ρ ( x , y ,z ).
And the velocity of a unit element at point (x,y,z) is v ( x , y ,z ).
Then the rate of outflow of substance through surface "S" is given by:
[tex]Flow (Q) = \int\int\limits_S {F} \, .ds[/tex]
Where, F = ρ ( x , y ,z )*v ( x , y ,z )
- Therefore for the given data:
F = ( 1025 kg/m^3)*(y i + x j)
- We will now parametrize the surface "S" given as:
x2 + y2 + z2 = 81, z ≥ 0
S: r(u) = 3 sin (v) cos (u) i + 3 sin (v) sin (u) j + 3 cos (v) k
Where, u ε [ 0 , 2π) and since z ≥ 0, v ε [ 0 , π/2 ]
Therefore,
F [ r(u) ] = ( 1025 kg/m^3)*( 3 sin (v) sin (u) i + 3 sin (v) cos (u) j).
Note:
[tex]Flow (Q) = \int\int\limits_S {F} \, .ds = \int\int\limits_D {F} \, .n.ds[/tex]
Where, n: Unit vector normal to surface "S"
- Such that, [tex]dS = magnitude(r_u x r_v).dA[/tex]
The Flux through a surface is defined only if the surface is orientable.
Note that if the substance is orientable, there will be two normal unit vectors at every point (x,y,z):
- We can write:
[tex]dS = n.dA = +/- \frac{r_u x r_v}{magnitude (r_u x r_v)}*magnitude (r_u x r_v).dA \\\\dS = +/- (r_u x r_v).dA[/tex]
- For finding the flux we always use the positive orientation of unit normal vector (n).
- The parameterisation for sphere "S":
r ( u , v ) = 3 sin (v) cos (u) i + 3 sin (v) sin (u) j + 3 cos (v) k
Where, u ε ( 0 , 2π) and since z ≥ 0, v ε ( 0 , π/2 ),
[tex]r_u = -3 sin(v)*sin(u) i + 3 sin(v)*cos (u) j \\\\r_v = 3 cos (v) *cos (u) i + 3 cos (v)* sin (u) j - 3 sin (v) k[/tex]
- Compute ( r_u x r_v ) :
[tex]r_u xr_v = \left[\begin{array}{ccc}i&j&k\\-3sin(v)*sin(u)&3sin(v)*cos(u)&0\\3cos(v)*cos(u)&3cos(v)*sin(u)&3sin(v)\end{array}\right] \\\\\\r_u xr_v = -9 sin^2(v)*cos(u) i -9 sin^2(v)*sin(u)j -9 sin(v)*cos(v) k[/tex]
- The vector ( r_u x r_v) points towards the origin, therefore the positive unit normal vector ( r_u x r_v) would be:
[tex]dS = - (r_u xr_v).dA = [ 9 sin^2(v)*cos(u) i +9 sin^2(v)*sin(u)j +9 sin(v)*cos(v) k] .dA[/tex]
Therefore,
[tex]Flow (Q) = \int\int\limits_S {F} \, .ds = 3075\int\int\limits_D { [2sin^3(v)*sin(u)*cos(u) ]} . dA\\\\Flow (Q) = 3075\int\int\limits_D { [sin^3(v)*sin(2u) ]} . dA\\\\\\u = e ( a1 = 0 , b1 = 2\pi ), v = e ( a2 = 0 , b2 = \pi/2)\\\\\\Flow (Q) = 3075\int\limits^b_0 {sin(2u)} \, du*\int\limits^b_0 {sin^3(v)} \, dv\\\\Flow (Q) = 3075[ - \frac{cos(2u)}{2}]\limits^2^\pi _0 * \int\limits^b_0 {sin^3(v)} \, dv\\\\Flow (Q) = 3075[ 0 ] * \int\limits^b_0 {sin^3(v)} \, dv = 0\\\\[/tex]
Answer: So the flow through the surface "S" is Flow (Q) = 0 kg/s
The given flow field has zero divergence which leads to zero net outward flux, implying that the rate of flow outward through the hemisphere x² + y² + z² = 81, z ≥ 0 is 0 m³/s.
Explanation:This problem is best approached through Gauss' Theorem which states that the total outward flux through a surface is equal to the volume integral of the divergence of the vector field over the volume enclosed by the surface. The divergence of the velocity vector, v = y i + x j , is obtained by adding the derivatives of the vector components with respect to their corresponding spatial coordinates.
I.e. ∇.v = ∇.(y i + x j) = ∂y/∂x + ∂x/∂y = 0 (since we can see that y is not dependent on x, and x is not dependent on y).
The total flux through the given hemisphere can now be calculated by substituting the divergence of velocity in the volume integral of Gauss’ theorem. However, since divergence is zero, the volume integral will be zero, hence implying that the net flux through the hemisphere will also be zero.
Therefore, the rate of the flow outward through the hemisphere x² + y² + z² = 81, z ≥ 0, is 0 m³/s.
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