Answer:
The flux of F across S is 8.627.
Step-by-step explanation:
Given,
F(x, y, z) = xy i + yz j + zx k
or F=(xy, yz, zx)
S is the part of the paraboloid [tex]z=8-x^{2} -y^{2}[/tex] above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
By differentiating z with respect to x we get:
fx=-2x
By differentiating z with respect to y we get:
fy=-2y
So, Surface integral is given by:
[tex]=\int\limits^1_0\int\limits^1_0( {(xy)*(2x)+y(8-x^{2} -y^{2} *(2y)+x(8-x^{2} -y^{2})} \, )dx \, dy[/tex][tex]=\int\limits^1_0\int\limits^1_0 (2x^{2} y+16y^{2} -2x^{2} y^{2} -2y^{4}+8x-x^{3}-xy^{2} } \,) dx \, dy[/tex]
Integrating with respect y:
[tex]=\int\limits^1_0(x^{2} y^{2} +\frac{16}{3} y^{3} -\frac{2}{3} x^{2} y^{3} -\frac{2}{5} y^{5}+8xy-x^{3}y-\frac{1}{3} xy^{3} } \, )dx \,[/tex]
After Substituting limits of y, we get:
[tex]=\int\limits^1_0(x^{2} +\frac{16}{3} -\frac{2}{3} x^{2} -\frac{2}{5}+8x-x^{3}-\frac{1}{3} x } \,) dx \,[/tex]
Integrating with respect x:
[tex]=(\frac{1}{3} x^{3} +\frac{16}{3}x -\frac{2}{9} x^{3} -\frac{2}{5}x+4 x^{2} -\frac{1}{4} x^{4}-\frac{1}{6} x^{2} } \,)[/tex]
After Substituting limits of x, we get:
[tex]=(\frac{1}{3} +\frac{16}{3} -\frac{2}{9} -\frac{2}{5}+4 -\frac{1}{4} -\frac{1}{6} } \,)\\\\=\frac{1553}{180}[/tex]
[tex]= 8.627[/tex]
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To evaluate the surface integral of the given vector field F across the given surface S, we need to use the formula: Φ = ∫∫S F · dS. In this case, the vector field F(x, y, z) = xy i + yz j + zx k and the surface S is the part of the paraboloid z = 8 - x^2 - y^2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, with upward orientation. We can parametrize the surface S and calculate the normal vector to evaluate the surface integral using the given formula.
Explanation:To evaluate the surface integral of the given vector field F across the given surface S, we need to use the formula:
Φ = ∫∫S F · dS
In this case, the vector field F(x, y, z) = xy i + yz j + zx k and the surface S is the part of the paraboloid z = 8 - x2 - y2 that lies above the square 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, with upward orientation.
We can parametrize the surface S as r(u, v) = (u, v, 8 - u2 - v2), where 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.
Next, we calculate the normal vector to S by taking the cross product of the partial derivatives of r(u, v) with respect to u and v: N = (-∂r/∂u) x (-∂r/∂v) = (2u, 2v, 1).
Now, we can evaluate the surface integral using the formula:
Φ = ∫∫S F · dS = ∫∫R F(r(u, v)) · (N · (∂r/∂u) x (∂r/∂v)) du dv
Substituting the values for F and N, we get:
Φ = ∫∫R (u2v + 4uv + 4uv) (2u, 2v, 1) · (2u, 2v, 1) du dv
Calculating this integral over the region R: 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1, we find the flux of F across S.
Arrange the circles (represented by their equations in general form) in ascending order of their radius lengths.
please help
a)
[tex]x^2+y^2-2x+2y-1=0[/tex]
It could be expressed as:
[tex](x-1)^2-1+(y+1)^2-1-1=0\\\\\\i.e.\\\\\\(x-1)^2+(y+1)^2=3\\\\\\i.e.\\\\\\(x-1)^2+(y+1)^2=(\sqrt{3})^2[/tex]
Hence, the radius of circle is: √3≈1.732 units
b)
[tex]x^2+y^2-4x+4y-10=0[/tex]
It is represented as:
[tex](x-2)^2-4+(y+2)^2-4-10=0\\\\\\i.e.\\\\\\(x-2)^2+(y+2)^2=18\\\\\\(x-2)^2+(y+2)^2=(3\sqrt{2})^2[/tex]
Hence, the radius of circle is: 3√2≈4.242 units
c)
[tex]x^2+y^2-8x-6y-20=0[/tex]
on converting to standard form
[tex](x-4)^2+(y-3)^2=(3\sqrt{5})^2[/tex]
Hence, the radius of circle is: 3√5≈6.708 units
d)
[tex]4x^2+4y^2+16x+24y-40=0[/tex]
on dividing both side by 4 we obtain:
[tex]x^2+y^2+4x+6y-10=0\\\\\\(x+2)^2+(y+3)^2=(\sqrt{23})^2[/tex]
Hence, radius of circle is: √23=4.796 units
e)
[tex]5x^2+5y^2-20x+30y+40=0[/tex]
on dividing both side by 5 we obtain:
[tex]x^2+y^2-4x+6y+8=0[/tex]
[tex](x-2)^2+(y+3)^2=(\sqrt{5})^2[/tex]
Hence, radius of circle is: √5=2.236 units
f)
[tex]2x^2+2y^2-28x-32y-8=0[/tex]
which could also be represented as follows:
[tex]x^2+y^2-14x-16y-4=0\\\\\\(x-7)^2+(y-8)^2=(\sqrt{117})^2[/tex]
Hence, the radius of circle is: [tex]\sqrt{117}[/tex]≈ 10.817 units
g)
[tex]x^2+y^2+12x-2y-9=0[/tex]
It could also be written as:
[tex](x+6)^2+(y-1)^2=(\sqrt{46})^2[/tex]
Hence, the radius of circle is: [tex]\sqrt{46}[/tex]≈ 6.782 units
The ascending order is:
a → e → b → d → c → g → f
The radius of a circle can be found from its equation in general form, which can be rearranged into the format (x-h)² + (y-k)² = r². From there, the radii of all the circles can be determined, and the circles arranged in ascending order by these lengths.
Explanation:To arrange the circles in ascending order of their radii, we need to understand the general equation of a circle which is in the format "(x-h)² + (y-k)² = r²". Here, (h,k) are the coordinates of the center of the circle, and 'r' is the radius of the circle. The given equations of the circles usually can be rewritten into this format.
To ascertain the radius of a circle from its equation, identify the constant term on the right hand side of the equation, which is the square of the radius (r²). The square root of this term will give you the 'r' - radius of the circle.
Once you know the radii of all the circles, arrange the equations in ascending order of these radii value. Remember, the smaller the r, the smaller the circle's circumference and area.
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The volleyball reached its maximum height at 3 seconds.
The function f(x) = -2(t - 3) +23 represents the height, in feet, t seconds after a volleyball is served. Which of the following statements are true? Select all that apply.
A. The maximum height of the volleyball was 23 feet.
B. If the volleyball is not returned by the opposing team, it will hit the ground in 5.5 seconds.
C. The graph that models the volleyball’s height over time is exponential.
D. The volleyball was served from a height of 5 feet
Answer:
A. The maximum height of the volleyball was 23 feet.
Step-by-step explanation:
Given the equation for the height as
f(x) = -2(t - 3) +23
The time taken to attain maximum height = 3 seconds
Finding the maximum height; we substitute value of t with 3 sec in the function;
f(x) = -2(t - 3) +23
f(3) = -2(3-3) +23
maximum height= 23
What are the solution(s) to the quadratic equation 50 – x2 = 0?
x = ±2
x = ±6
x = ±5
no real solution
Answer:
No real solution
Step-by-step explanation:
Work is on another question for this. You have asked it multiple times
For this case we must find the solution of the following quadratic equation:
[tex]50-x ^ 2 = 0[/tex]
Subtracting 50 from both sides of the equation:
[tex]-x ^ 2 = -50[/tex]
Multiplying by -1 on both sides of the equation:
[tex]x ^ 2 = 50[/tex]
We apply square root on both sides to eliminate the exponent:
[tex]x = \sqrt {50}\\x = \pm \sqrt {25 * 2}\\x = \pm \sqrt {5 ^ 2 * 2}\\x = \pm5 \sqrt {2}[/tex]
ANswer:
[tex]x = \pm5 \sqrt {2}[/tex]
Jared bought one share of stock for $225.
A. He sold the stock for a profit of $55. what was the selling price of the stock?
B. The price of the stock dropped $40 the day after Jared sold it. At what price would Jared had sold it if he had waited until then?
Write equate and solve! Please Help!
Answer:
A. 280
B. 185
Step-by-step explanation:
A. Buying price of share stock= $ 225
Profit = $55
Selling price= Buying price + profit
[tex]= 225+ 55= 280[/tex]
Selling price= $280
B.
Decrease in price was by $40
Buying price= $225
New selling price if he had waited till then will be=
[tex]= 225-40 = 185[/tex]
=$185
Officials project that between 2010 and 2050, the population of a certain city will drastically change. The model below gives the projection of the city's population , P, in thousands, with respect to time, t, in years, where 2010 corresponds to t = 0.P=175 + (11/2)tWhat does the 175 mean in the equationin 2010 the population of the city was 175 thousand
in 2050 the population of the city will be 175 thousand
Between 2010 and 2050, the population of the city will increase by 175 thousand
Between 2010 and 2050 the population of the city will decrease by 175 thousand
Answer:
In the year 2010, the population of the city was 175,000
Step-by-step explanation:
If we rewrote this as a linear expression in standard form (it is linear, btw), it would look like this:
[tex]P(t)=\frac{11}{2}t+175[/tex]
The rate of change, the slope of this line, is 11/2. If the year 2010 is our time zero (in other words, we start the clock at that year), then 0 time has gone by in the year 2010. In the year 2011, t = 1 (one year goes by from 2010 to 2011); in the year 2012, t = 2 (two years have gone by from 2010 to 2012), etc. If we plug in a 0 for t we get that y = 175,000. That is our y-intercept, which also serves to give us the starting amount of something time-related when NO time has gone by.
Solve (x + 1 < 4) ∩ (x - 8 > -7).
1 < x < 3 because
( x < 3 ) intersecting ( x > 1)
For this case we must find the intersection of the following inequations:
[tex]x + 1 <4\\x-8> -7[/tex]
So:
[tex]x + 1 <4\\x <4-1\\x <3[/tex]
All values of "x" less than 3.
[tex]x-8> -7\\x> -7 + 8\\x> 1[/tex]
All values of "x" greater than 1.
Thus, the intersection of the equations will be given by the values of "x" greater than 1 and less than 3.
[tex](1 <x <3)[/tex]
ANswer:[tex](1 <x <3)[/tex]
An arithmetic series contains 20 numbers. The first number is 102. The last number is 159. Which expression represents the sum of the series?
A.(20 (159-102/2)
B.(20(102+159/2)
C.20(102+159)
D.20(159-102)
Answer:
Option B
Step-by-step explanation:
we know that
The sum of an arithmetic series is equal to
S=n(a1+an)/2
where
a1 is the first term
an is the last term
n is the number of terms
In this problem we have
n=20
a1=102
an=159
substitute the values in the formula
S=20(102+159)/2
which of the following sets is closed under subtraction?
A. Integers
B. Whole Numbers
C. natural numbers
D. irrational numbers
Answer:
A. Integers
Step-by-step explanation:
Subtraction of whole or natural numbers can result in a negative number that is not in the set. Subtraction of irrational numbers can result in a rational number (√2 -√2 = 0, for example).
Final answer:
The Integers are closed under subtraction because the difference of any two integers is always an integer, while Whole Numbers, Natural Numbers, and Irrational Numbers are not, as their differences can result in numbers outside of their respective sets. Therefore, the correct answer is A.
Explanation:
When considering which of the following sets is closed under subtraction, we must understand what it means for a set to be 'closed' under an operation. A set is closed under subtraction if, when you subtract any two elements in the set, the result is also an element of the set.
Integers include whole numbers as well as their negative counterparts, such as -1, 0, and 1. When you subtract any two integers, the result is always another integer. Therefore, the set of integers is closed under subtraction.
Whole numbers, on the other hand, include 0 and all the positive integers. Subtracting a larger whole number from a smaller one would result in a negative integer, which is not included in the set of whole numbers. Thus, this set is not closed under subtraction.
Natural numbers are like whole numbers but don't include 0. Just like whole numbers, subtracting a larger natural number from a smaller one would result in a negative integer, which is not a natural number. Hence, natural numbers are not closed under subtraction.
Irrational numbers include quantities like π and √2. Subtracting two irrational numbers might result in a rational number, which is not an irrational number. Therefore, the set of irrational numbers is not closed under subtraction.
Therefore, the correct answer is A. Integers.
please help asap!!!!!!
The answer is:
The difference between the areas of the circles will be:
[tex]Difference=36\pi -9\pi =27\pi[/tex]
Why?To find the diffence in area between the two circles, we need to find both areas and then, subtract the smallest circle area to the largest circle area.
So,
For the small circle, we have:
[tex]Area_{SmallCircle}=\pi *radius^{2} \\\\Area_{SmallCircle}=\pi *(3)^{2}=9\pi[/tex]
For the large circle, we have:
[tex]Area_{LargeCircle}=\pi *radius^{2} \\\\Area_{LargeCircle}=\pi *(6)^{2}=36\pi[/tex]
Hence, we have that the difference between the areas of the circles will be:
[tex]Difference=36\pi -9\pi =27\pi[/tex]
Have a nice day!
Answer:
Difference = 27π square units
Step-by-step explanation:
Points to remember
Area of circle = πr²
Where r - Radius of circle
To find the area of large circle
Here r = 6 units
Area = πr² = π * 6²
= 36π square units
To find the area of small circle
Here r = 3 units
Area = πr² = π * 3²
= 9π square units
To find the difference
Difference = area of large circle - area of small circle
= 36π - 9π = 27π square units
Which could be the area of one face of the rectangular prism?Check all that apply
Answer: 24, 44, and 66 [tex]cm^{2}[/tex]
Step-by-step explanation:
Check all of the possible combinations of the faces:
6 × 4 = 24
11 × 4 = 44
11 × 6 = 66
So the answers are 24, 44, and 66 [tex]cm^{2}[/tex]
Answer:
option B 24 cm² option C 44 cm² and option E 66 cm² are the correct options
Step-by-step explanation:
A rectangular prism has 6 faces, in which opposite faces are always similar.
So a rectangular prism has 3 different faces
(1) From the figure given
for face (1) Area = 11 × 4
= 44 cm²
For face (2) Area = 4 × 6
= 24 cm²
For face (3) Area = 11 × 6
=66 cm²
Now we can say that option B 24 cm² option C 44 cm² and option E 66 cm² are the correct options
PLEASE HELP
I marked the ones I'm stuck on
must show work
Use Slide method for some of them, some are perfect squares:
Watch this video and try it a few times, pay attention to the signs when you are seeing which one will be negative when you multiply it out, in picture is me working out all problems requested.
Ask your brother for further help!
Select the correct answer from each drop-down menu.
The table shows the heights of the 10 tallest buildings in San Francisco and Los Angeles.
The average height of the 10 tallest buildings in Los Angeles is than the average height of the tallest buildings in San Francisco. The mean absolute deviation for the 10 tallest buildings in San Francisco is
The answer:
Answer with explanation:
[tex]\text{Average}=\frac{\text{Sum of all the observation}}{\text{Total number of Observation}}[/tex]
Average Height of tallest Building in San Francisco
[tex]=\frac{260+237+212+197+184+183+183+175+174+173}{10}\\\\=\frac{1978}{10}\\\\=197.8[/tex]
Average Height of tallest Building in Los Angeles
[tex]=\frac{310+262+229+228+224+221+220+219+213+213}{10}\\\\=\frac{2339}{10}\\\\=233.9[/tex]
→→Difference between Height of tallest Building in Los Angeles and Height of tallest Building in San Francisco
=233.9-197.8
=36.1
⇒The average height of the 10 tallest buildings in Los Angeles is 36.1 more than the average height of the tallest buildings in San Francisco.
⇒Part B
Mean absolute deviation for the 10 tallest buildings in San Francisco
|260-197.8|=62.2
|237-197.8|=39.2
|212-197.8|=14.2
|197 -197.8|= 0.8
|184 -197.8|=13.8
|183-197.8|=14.8
|183-197.8|= 14.8
|175-197.8|=22.8
|174-197.8|=23.8
|173 -197.8|=24.8
Average of these numbers
[tex]=\frac{62.2+39.2+14.2+0.8+13.8+14.8+14.8+22.8+23.8+24.8}{10}\\\\=\frac{231.2}{10}\\\\=23.12[/tex]
Mean absolute deviation=23.12
Answer:
1st -36.1 meters or more
2nd -23.12
Step-by-step explanation:
In triangle ABC, how long is side c if A = 103°, C = 17°, and a = 21?
Answer:
c=6.30 units
Step-by-step explanation:
we know that
Applying the law of sines
a/sin(A)=c/sin(C)
Solve for c
c=a*sin(C)/sin(A)
substitute the values
c=21*sin(17°)/sin(103°)=6.30 units
Reposting with higher point count AND the actual, like, chart.
My daughter needs help and I can't find this solution anywhere online to help her. Photo of tri is in photo. I spent 70 points on this, so I hope this gets answered.
For Triangle TOE, the following facts are given:
TE = 6 cm
OT = 2 cm
OE = 5.8 cm
OG = 4.35 cm
AU = 0.45 cm
Use this information for the following answers:
a. Why is triangle OBG ~ to triangle OTE?
Now find the following missing lengths. Show all work or reasoning. Round non-integral lengths to the nearest hundredth.
b. GE
c. TS
d. OA
e. BT (Use side-splitting theorem.)
f. SE
g. OU
Her answers were:
a. AA similarity postulate (Unsure if correct but probably is, teacher didn't see answer.)
b. GE = 1.45 ("Fine?")
c. TS = 3 cm (INCORRECT)
d. OA = 1.35 cm (CORRECT)
e. BT = .5 cm (CORRECT)
f. SE = 3 cm (INCORRECT)
g. OU = 1.8 cm (CORRECT)
Answer:
The answers to a, b, d, e, g are correct (as noted in your problem statement).
Step-by-step explanation:
a) The triangles are similar because their apex angle is the same angle, and their base angles are corresponding angles where transversals cross parallel lines, hence congruent. The triangles are similar by AA (or AAA, if you like) since all corresponding angles are congruent.
__
b) GE = OE -OG = 5.8 -435 = 1.45 . . . cm
__
c) Technically speaking, there is not enough information in your posted question to allow TS to be found. You can find the length TU using the Pythagorean theorem. (First you need OU (see g below).) By that theorem, ...
TU^2 + OU^2 = OT^2
TU = √(OT^2 -OU^2) = √(2^2 -1.8^2) = √0.76 ≈ 0.87
By all appearances, US = TU. If you make that assumption, then ...
TS = 2·TU = 2·0.87 = 1.74 . . . cm
__
d) We have seen that OG = 3·GE, so OA will be 3·AU.
OA = 3·AU = 3·0.45 = 1.35 . . . cm
__
e) Using the same proportions we have observed elsewhere,
BT/OT = 1/4
BT = (2 cm)/4 = 0.5 cm
__
f) SE = TE - TS = 6 cm - 1.74 cm = 4.26 cm
(see part (c) above for the assumption we must make regarding this)
__
g) OU = OA + AU = 1.35 cm + 0.45 cm = 1.8 cm
Will give brainliest if answered please answer quick
Given: circle k(O)
m∠OFQ = 52°
measure of arc FQ = (5x+1°)
Find: x
Answer:
x = 15°
Step-by-step explanation:
m∠DFQ = m∠OFQ = 52° (given), so arc DQ = 2·52° = 104°. Then arc FQ is the supplement of that, 180° -104° = 76°. The given relation to x is then ...
76° = 5x +1°
75° = 5x . . . . . . . subtract 1°
15° = x . . . . . . . . .divide by 5
You invested $4,200 into an account that pays an interest rate of 3.5% compounded continuously. What is the balance of your account in 5 years?. Use formula
A=Pert
Answer:
[tex]A=\$5003.2[/tex]
Step-by-step explanation:
Use the exponential growth formula
[tex]A = Pe ^ {rt}[/tex]
Where A is the final amount in the account, P is the initial amount, r is the growth rate and t is the time in years
In this problem
We know that
[tex]P=4,200\\\\r=\frac{3.5\%}{100\%}= 0.035\\\\ t=5\ years[/tex]
So
[tex]A = 4,200e^{0.035t}[/tex]
Finally after 5 years the balance of the account is:
[tex]A=\$5003.2[/tex]
A cube-shaped tank that measures 4.64.6 meters on each side is full of water. If some water is drained to fill a cone with a height of 66 meters and a radius of 2.52.5 meters, approximately how much water is left in the tank? Use 3.143.14 for ππ and round your answer to the nearest tenth. The amount of water left in the tank is approximately __________ cubic meters.
Answer:
The amount of water left in the tank is approximately 58.1 cubic meters
Step-by-step explanation:
step 1
Find the volume of the a cube shape tank
The volume is equal to
[tex]V=b^{3}[/tex]
we have
[tex]b=4.6\ m[/tex]
substitute
[tex]V=4.6^{3}=97.336\ m^{3}[/tex]
step 2
Find the volume of cone
The volume is equal to
[tex]V=\frac{1}{3}\pi r^{2}h[/tex]
we have
[tex]r=2.5\ m[/tex]
[tex]h=6\ m[/tex]
[tex]\pi =3.14[/tex]
substitute
[tex]V=\frac{1}{3}(3.14)(2.5)^{2}(6)[/tex]
[tex]V=39.25\ m^{3}[/tex]
step 3
Find the difference of the volumes
[tex]97.336\ m^{3}-39.25\ m^{3}=58.1\ m^{3}[/tex]
Type the correct answer in the box. If you cut a 12-inch loaf of bread into 1/2-inch slices, you would have slices of bread.
Answer:
you would have 24 slices of bread
Step-by-step explanation:
(12 in)/(1/2 in/slice) = 12·2/1 slices = 24 slices
You have $50 in your bank account.
You make $8 per hour mowing lawns.
How many hours must you mow lawns to have a total of $130 in your account?
Use the drop-down menus to complete the sentences below.
Answer:
10 hours
Step-by-step explanation:
If I have $50 in my bank account, and I want to have a total of $130 in my account. It means that I need to work enough hours to make $130 - $50 = $80.
If I make $8 per hour, and I need to make $80, then I just have to work 10 hours. ($80/8 = 10)
A culture started with 1,500 bacteria. After 5 hours it grew to 2,300 bacteria. How many bacteria with be present after 12 hours? PLSSS
Answer:
4185
Step-by-step explanation:
A culture of bacteria grows exponentially according to the following general exponential growth function;
[tex]P_{t}=P_{0}e^{kt}[/tex]
where;
p(t) is the population at any given time t.
p(0) is the initial population
k is the growth constant
From the information given we have;
p(0) = 1500
at t = 5, p(t) = 2300; p(5) = 2300
We shall use this information to determine the value of k;
[tex]2300=1500e^{5k}[/tex]
Divide both sides by 1500;
[tex]\frac{23}{15}=e^{5k}\\\\ln(\frac{23}{15})=5k\\\\k=0.08549[/tex]
Therefore, the population of the bacteria at any time t is given by;
[tex]P_{t}=1500e^{0.08549t}\\\\P(12)=1500e^{0.08549(12)}=4184.3[/tex]
Membership to a national running club is shown in the table. Which answer describes the average rate of change from Year 3 to Year 5? Year: 1, 2, 3, 4, 5 Members(thousands): 87.5, 88.3, 96.8, 87.2, 83.6
answers:
Membership decreased by an average of 6,600 people per year from Year 3 to Year 5.
Membership increased by an average of 13,200 people per year from Year 3 to Year 5.
Membership decreased by an average of 13,200 people per year from Year 3 to Year 5.
Membership increased by an average of 6,600 people per year from Year 3 to Year 5.
Answer:
Membership decreased by an average of 6,600 people per year from Year 3 to Year 5
Step-by-step explanation:
The average rate of change from Year 3 to Year 5 will be given by the slope of the line joining the points;
(3, 96.8) and (5, 83.6)
The slope of a line given two points is calculated as;
( change in y)/( change in x)
In this case y is the number of members for a given year x.
average rate of change = (83.6-96.8)/(5-3)
= -6.6
Since the number of members is given in thousands, we have;
-6,600
The negative sign implies a decrease in the number of members. Therefore, membership decreased by an average of 6,600 people per year from Year 3 to Year 5
1. Draw the image of ABC under the dilation with scale factor 2 and center of dilation (-4,-3). Label the
image A'B'C'.
Answer:
-4
-
21
2
3
4
5
Answer:
see below
Step-by-step explanation:
Each point moves to twice its original distance from (-4, -3). The point (-4, -3) remains unmoved.
Answer: (-4,1) ; (2,-3) ; (-4,-3)
Verify that the line integral and the surface integral of Stokes' Theorem are equal for the following vector field, surface S, and closed curve C. Assume that C has counterclockwise orientation and S has a consistent orientation. Fequals=left angle y comma font size decreased by 6 minus x comma font size decreased by 6 14 right angley, −x, 14; S is the upper half of the sphere x squaredx2plus+y squaredy2plus+z squaredz2equals=44 and C is the circle x squaredx2plus+y squaredy2equals=44 in the xy-plane.
As near as I can tell, you're given the vector field
[tex]\vec F(x,y,z)=\langle y,-x,14\rangle[/tex]
and that [tex]S[/tex] is the part of the upper half of the sphere with equation
[tex]x^2+y^2+z^2=4[/tex]
with boundary [tex]C[/tex] the circle in the plane [tex]z=0[/tex].
Line integral:Parameterize [tex]C[/tex] by
[tex]\vec r(t)=\langle2\cos t,2\sin t,0\rangle[/tex]
with [tex]0\le t\le2\pi[/tex]. Then the line integral of [tex]\vec F(x,y,z)[/tex] along [tex]C[/tex] is
[tex]\displaystyle\int_C\vec F(x,y,z)\cdot\mathrm d\vec r=\int_0^{2\pi}\langle2\sin t,-2\cos t,14\rangle\cdot\langle-2\sin t,2\cos t,0\rangle\,\mathrm dt[/tex]
[tex]=\displaystyle-4\int_0^{2\pi}(\sin^2t+\cos^2t)\,\mathrm dt=\boxed{-8\pi}[/tex]
Surface integral:Parameterize [tex]S[/tex] by
[tex]\vec s(u,v)=\langle2\cos u\sin v,2\sin u\sin v,2\cos v\rangle[/tex]
with [tex]0\le u\le2\pi[/tex] and [tex]0\le v\le\dfrac\pi2[/tex]. We have
[tex]\nabla\times\vec F(x,y,z)=\langle0,0,-2\rangle[/tex]
Take the normal vector to [tex]S[/tex] to be
[tex]\vec s_v\times\vec s_u=\langle4\cos u\sin^2v,4\sin u\sin^2v,2\sin2v\rangle[/tex]
Then the surface integral of the curl of [tex]\vec F(x,y,z)[/tex] across [tex]S[/tex] is
[tex]\displaystyle\iint_S(\nabla\times\vec F(x,y,z))\cdot\mathrm d\vec S=\iint_S(\nabla\times\vec F(x(u,v),y(u,v),z(u,v)))\cdot(\vec s_v\times\vec s_u)\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle\int_0^{\pi/2}\int_0^{2\pi}\langle0,0,-2\rangle\cdot\langle4\cos u\sin^2v,4\sin u\sin^2v,2\sin2v\rangle\,\mathrm du\,\mathrm dv[/tex]
[tex]=\displaystyle-4\int_0^{\pi/2}\int_0^{2\pi}\sin2v\,\mathrm du\,\mathrm dv=\boxed{-8\pi}[/tex]
Max sneezes every 5 minutes, Lina coughs every 6 minutes, and their dog barks every 3 minutes. If there was sneezing, barking, and coughing at 3:15 PM, when is the next time that these three sounds will happen simultaneously?
Answer:
3:45 PM
Step-by-step explanation:
The least common multiple of 3, 5, and 6 is 30, so the next occurrence will be 30 minutes after 3:15 PM, at 3:45 PM.
Answer: The next time at which the three sounds will happen simultaneously at 3 : 45 PM.
Step-by-step explanation: Given that Max sneezes every 5 minutes, Lina coughs every 6 minutes and their dog barks every 3 minutes.
We are to find the time at which these three sounds will happen simultaneously if there was sneezing, barking, and coughing at 3:15 PM.
We have
the sneezing, barking and coughing happen simultaneously at an interval that is equal to the L.C.M. of 5, 6 and 3 minutes.
Now,
L.C.M. (5, 6, 3) = 30.
Therefore, the sneezing, barking and coughing happen simultaneously at an interval of 30 minutes.
Since there was sneezing, barking, and coughing at 3:15 PM, so the next time at which the three sounds will happen simultaneously is
3 : 15 PM + 30 min = 3 : 45 PM.
Thus, the next time at which the three sounds will happen simultaneously is 3 : 45 PM.
Look at the sequence given below.
4, 0, -4, -8, ...
If n = 1 represents the first term in the sequence above, which of the following describes the nth term of the sequence?
A. -3n + 4
B. n + 3
C. -4n + 8
D. -4n + 3
Answer:
C. -4n + 8
Step-by-step explanation:
Try the formulas and see which works.
__
The common difference is -4, so the coefficient of n in the explicit formula is -4. Every term is divisible by 4, so there won't be 3 anywhere in the formula.
__
-4·1 +8 = 4
-4·2 +8 = 0
-4·3 +8 = -4
-4·4 +8 = -8
The formula -4n+8 reproduces the sequence exactly.
a person invest $1,700 in an account that earns 3.97% annual interest. find when the value of the investment reaches $3,200
Answer:
16 1/4 years
Step-by-step explanation:
Assuming interest is compounded annually, the account balance (A) after t years will be ...
A = P(1 +r)^t
3200 = 1700·1.0397^t
log(3200) = log(1700) +t·log(1.0397)
t = (log(3200) -log(1700))/log(1.0397) ≈ 16.247
The account will reach a balance of $3200 after about 16 1/4 years.
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You may be asked to round your answer to the nearest integer or tenth. We leave that exercise to the student.
What is the amplitude of the function graphed?
The function crosses the x-axis at points (-4,0), (-1,0), (2,0), and (5,0).
The amplitude is a measure of how far the function oscillates from its equilibrium position (usually the x-axis). Here are the steps to find the amplitude:
1. Identify the Peaks and Troughs:
- Observe the graph and locate the highest point (peak) and the lowest point (trough) of the waveform.
- In our case:
- Peak (Maximum Point) = 5
- Trough (Minimum Point) = -3
2. Calculate the Amplitude:
- The amplitude can be found using the formula:
[tex]\[ \text{Amplitude} = \frac{\text{Peak} - \text{Trough}}{2} \][/tex]
- Substituting the values:
[tex]\[ \text{Amplitude} = \frac{5 - (-3)}{2} = \frac{8}{2} = 4 \][/tex]
Therefore, based on visual estimation, the amplitude of this wave is approximately 4.
3. Graphical Representation:
- The graph represents a sinusoidal function with two complete cycles visible.
- Peaks occur at approximately (y = 5), and troughs occur at approximately (y = -3).
- The function crosses the x-axis at points (-4,0), (-1,0), (2,0), and (5,0).
A roulette wheel has 38 spaces: 18 red, 18 black, and 2 green. Suppose that in each spin of the wheel, the ball is equally likely to land on any of the 38 spaces, and that spins are independent. The wheel will be spun five times.
The chance that the ball lands on black in all five spins is closest to
a) 0.688%
b) 6.366%
c) 12.044%
d) 17.722%
e) 23.401%
Answer:
2.29%
Step-by-step explanation:
1. Chance of landing on black for one spin:
There are 38 spaces, and 18 lead to the wanted result. That means the chance is ¹⁸/₃₈, or about 0.47.
2. Chance for 5 spins.
We need to find (0.47)⁵, which is about 0.0229, which is 2.29%
That is none of the choices, but from every way I did this problem, that is the only solution I got.
Solve the following system: y = x + 3
3x + y = 19
(7, 4)
(−4, 7)
(4, 7)
(4, −7)
Answer:
(4,7)
Step-by-step explanation:
3x+x+3=19
4x=16
x=4
so y=7
c. (4,7)
Sally got the following solutions when three different systems of linear equations. Tell which indicates one solution, no solution, and infinitely many solutions.
System #1
x=13
System #2
0=13
System #3
0=0
answer honestly and right il give you brainleist
Try this:
if x=13 it means 'one solution'; the only point;
if 0=13 it means 'no solution'; wrong equation = no points;
if 0=0 it means 'many solutions'; no variable in the equation = much points.
Finally:
System #1 - one solution;
System #2 - no solution;
System #3 - many solutions.