Find the arclength of x = 1/3 (√y)(y - 3) on the interval 1 ≤y ≤ 9.
a. Set up the integral and then evaluate the integral. Show all of your work. b. Then find the value of the definite integral. Show all of your work. Write an exact answer (NOT A DECIMAL).
Answers
The arc length of the curve x = 1/3 (√y)(y - 3) on the interval 1 ≤y ≤ 9 is exactly √2.
To find the arc length of the curve x = 1/3 (√y)(y - 3) on the interval 1 ≤y ≤ 9, we use the formula for arc length:
L = ∫[a,b] √(1 + (dy/dx)^2) dx
where a and b are the endpoints of the interval and dy/dx can be found by differentiating x with respect to y:
dx/dy = (√y - (3/2)√y)/(2√y) = (1/2)√y - (3/4)
Squaring and simplifying, we get:
1 + (dx/dy)^2 = 1 + y/4 - (3/2)√y + 9/16
Substituting into the arc length formula, we get:
L = ∫[1,9] √(1 + y/4 - (3/2)√y + 9/16) dy
To evaluate this integral, we make the substitution u = √y:
L = ∫[1,3] 2u√(1 + u^2/4 - (3/2)u + 9/16) du
We can simplify the expression under the square root by completing the square:
1 + u^2/4 - (3/2)u + 9/16 = (u/2 - 3/4)^2 + 1/16
Substituting back into the integral and simplifying, we get:
L = ∫[1,3] 2u√[(u/2 - 3/4)^2 + 1/16] du
= ∫[1,3] √(4u^2 - 12u + 13/4) du
= ∫[1,3] √[(2u - 3)^2 + 1] du
We can evaluate this integral using the trigonometric substitution u = (3/2) + (1/2)tanθ, which gives:
du = (1/2)sec^2(θ)/2 dθ
Substituting into the integral and simplifying, we get:
L = ∫[α,β] secθ dθ
where α and β are the values of θ corresponding to u = 1 and u = 3, respectively. We can find these values by solving the equations:
(3/2) + (1/2)tanα = 1
(3/2) + (1/2)tanβ = 3
which give α = π/4 and β = 3π/4.
Substituting these values into the integral, we get:
L = ∫[π/4,3π/4] secθ dθ
To evaluate this integral, we use the substitution u = secθ, which gives:
du = secθ tanθ dθ
Substituting into the integral and simplifying, we get:
L = ∫[sec(π/4),sec(3π/4)] du
= sec(3π/4) - sec(π/4)
= -√2 + 2√2
= √2
Therefore, the arc length of the curve x = 1/3 (√y)(y - 3) on the interval 1 ≤y ≤ 9 is exactly √2.
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Related Questions
Measurements taken as a ball falls a distance D in a time T are: - D=1.215±0.004 m - T=0.495,0.498,0.503,0.496,0.501 s. The mean average value of T is 0.499 s and the acceleration due to gravity g is 9.77 m s
−2
(calculated using the formula g=
T
2
2D
). Calculate the: a percentage uncertainty in D b range in the measurements of T c absolute uncertainty in the average value of T d percentage uncertainty in the average value of T e percentage uncertainty in g (remember, as g=
T×T
2D
you add the percentage uncertainty in T twice to the percentage uncertainty in D ) f absolute uncertainty in g.
Answers
The calculated values are:
a) Percentage uncertainty in D ≈ 0.329%
b) Range in the measurements of T = 0.008 s
c) Absolute uncertainty in the average value of T ≈ 0.003 s
d) Percentage uncertainty in the average value of T ≈ 0.601%
e) Percentage uncertainty in g ≈ 1.531%
f) Absolute uncertainty in g ≈ 0.150 m/s^2
To calculate the requested values, we'll follow these steps:
a) Percentage uncertainty in D:
Percentage uncertainty = (Absolute uncertainty / Measured value) * 100
Absolute uncertainty = ±0.004 m
Measured value = 1.215 m
Percentage uncertainty in D = (0.004 / 1.215) * 100 ≈ 0.329%
b) Range in the measurements of T:
Range = Maximum value - Minimum value
Range = 0.503 s - 0.495 s = 0.008 s
c) Absolute uncertainty in the average value of T:
Absolute uncertainty = Range / √(Number of measurements)
Number of measurements = 5
Absolute uncertainty in the average value of T = 0.008 s / √5 ≈ 0.003 s
d) Percentage uncertainty in the average value of T:
Percentage uncertainty = (Absolute uncertainty / Average value) * 100
Average value of T = 0.499 s
Percentage uncertainty in the average value of T = (0.003 / 0.499) * 100 ≈ 0.601%
e) Percentage uncertainty in g:
Percentage uncertainty in g = 2 * (Percentage uncertainty in T) + (Percentage uncertainty in D)
Since we have already calculated the percentage uncertainty in T and D, we can substitute the values:
Percentage uncertainty in g = 2 * 0.601% + 0.329% ≈ 1.531%
f) Absolute uncertainty in g:
Absolute uncertainty in g = Percentage uncertainty in g * g
g = 9.77 m/s^2
Absolute uncertainty in g = 1.531% * 9.77 m/s^2 ≈ 0.150 m/s^2
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find the value of X also show the process please
Answers
Answer: 36
Step-by-step explanation:
can someone help with my homework? Thank you !
Answers
Answer:
1000-0=1/2(2000-0)
y= 1/2x
-1x + 2y = 0
In your reservoir, you have a production well which flows for 48 hours at 200 STB/day, and then shut-in for 24 hours. The following additional data are given : Pi = 3100 psi Ct = 15x10^-6 psi^-1 Bo = 1.3 bbl/STB ϕ = 15% μ=1.2 cp K = 45 md and h = 60 ft
a-) Calculate the pressure in this production well at 12 hours of shut in
b-) Explain how can you use superposition in time to analyze a pressure build-up test.
Answers
a) To calculate the pressure at 12 hours of shut-in:
substitute the given values into the pressure buildup equation and solve for P(t=12).
b) Superposition in time is used in pressure buildup analysis by adding or summing the responses of multiple transient tests to analyze and interpret reservoir behavior and properties.
We have,
a) To calculate the pressure in the production well at 12 hours of a shut-in, we can use the equation for pressure transient analysis during shut-in periods, known as the pressure buildup equation:
P(t) = Pi + (Q / (4πKh)) * log((0.14ϕμCt(t + Δt)) / (Bo(ΔP + Δt)))
Where:
P(t) = Pressure at time t
Pi = Initial reservoir pressure
Q = Flow rate
K = Permeability
h = Reservoir thickness
ϕ = Porosity
μ = Viscosity
Ct = Total compressibility
t = Shut-in time (12 hours)
Δt = Time since the start of the flow period
Bo = Oil formation volume factor
ΔP = Pressure drop during the flow period
Given:
Pi = 3100 psi
Q = 200 STB/day
K = 45 md
h = 60 ft
ϕ = 15%
μ = 1.2 cp
Ct = 15x10^-6 psi^-1
Bo = 1.3 bbl/STB
t = 12 hours
Δt = 48 hours
ΔP = Pi - P(t=Δt) = Pi - (Q / (4πKh)) * log((0.14ϕμCt(Δt + Δt)) / (Bo(ΔP + Δt)))
Substituting the given values into the equation:
ΔP = 3100 - (200 / (4π * 45 * 60)) * log((0.14 * 0.15 * 1.2 * 15x\(10^{-6}\) * (48 + 48)) / (1.3 * (3100 - (200 / (4π * 45 * 60)) * log((0.14 * 0.15 * 1.2 * 15 x \(10^{-6}\) * (48 + 48)) / (1.3 * (0 + 48))))))
After evaluating the equation, we can find the pressure in the production well at 12 hours of shut-in.
b) Superposition in time is a principle used in pressure transient analysis to analyze and interpret pressure build-up tests.
It involves adding or superimposing the responses of multiple transient tests to simulate the pressure behavior of a reservoir.
The principle of superposition states that the response of a reservoir to a series of pressure changes is the sum of the individual responses to each change.
Superposition allows us to combine the information obtained from multiple tests and obtain a more comprehensive understanding of the reservoir's behavior and properties.
It is a powerful technique used in reservoir engineering to optimize production strategies and make informed decisions regarding reservoir management.
Thus,
a) To calculate the pressure at 12 hours of shut-in:
substitute the given values into the pressure buildup equation and solve for P(t=12).
b) Superposition in time is used in pressure buildup analysis by adding or summing the responses of multiple transient tests to analyze and interpret reservoir behavior and properties.
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path is made of white tiles and grey tiles.
1/4 of the tiles are white.
(a) Write down the ratio of white
Answers
Answer: 1 : 3
Step-by-step explanation:
¹/₄ of the tiles are white. To find the ratio, assume the tiles are 4 in number.
If there are 4 tiles;
White = ¹/₄ * 4 = 1 white tile
Grey tiles = 4 - 1 = 3 grey tiles
The ratio is therefore;
1 white tile : 3 grey tiles
1 : 3
just to say thanks that was the correct answer and i got all correct so thank you so much you are the best
Help I will give crown and brainiest
Answers
Answer:
quad then hex then oct then non
Step-by-step explanation:
write P(x)=6x³+3x²-24x-12 as a product of (2x+1)and a quadratic formula.
pls help
Answers
Answer:
3(2x+1)( \(x }\)-2)(x+2)
Step-by-step explanation:
3(2x+1)( \(x }\)-2)(x+2)
3(2x+1)( \(x^{2}\)-4)
(2x+1)(3\(x^{2}\)-12)
My'chael has read 120 pages, which is 45% of a book. How many
pages are in the book?
Answers
Answer:
267 pages
Step-by-step explanation:
120 ÷ 0.45 = 267
Answer:
The answer is 174 pages
Step-by-step explanation:
Find out how much is 45% = 0.45
multiply 0.45 by 120
you get 54
so then you add 54 to 120 and you get 175
54+120=175
Historical data on rates of return indicate that:
T-bills and T- bonds yield about the same rate of return on average.
On average T-bonds yield higher rates of return than T-bills.
On average T-bills yield higher rates of return than T-bonds.
T-bills have a higher standard deviation of returns compared to T-bonds.
Answers
On average, T-bonds yield higher rates of return than T-bills. This is supported by historical data on rates of return.
Historical data on rates of return indicates that T-bonds tend to yield higher rates of return compared to T-bills. T-bonds are long-term government bonds with longer maturities, while T-bills are short-term government securities with shorter maturities.
The longer duration of T-bonds exposes investors to a higher level of risk, but it also offers the potential for higher returns. T-bills, on the other hand, have lower risk due to their shorter durations but provide lower returns.
When comparing the rates of return between T-bills and T-bonds, it is important to consider the overall average over a significant period. While there may be individual instances where T-bills outperform T-bonds, the historical data supports the notion that T-bonds generally yield higher rates of return on average.
It is worth noting that rates of return can vary from year to year and are influenced by various factors such as economic conditions, interest rates, and market trends. Therefore, it is important for investors to carefully assess their risk tolerance and investment objectives when choosing between T-bills and T-bonds.
In summary, historical data suggests that, on average, T-bonds tend to yield higher rates of return compared to T-bills. However, individual investment decisions should consider personal circumstances and goals.
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A theater is tracking the number of adults and children in the audience at
various shows. The table shows how the number of adults and children
are related
Select the equation that matches the relationship.
X+y = 1000
X-y1000
y-x= 1000
Y= 1000xc
Adults
Children
500
500
450
550
400
600
350
650 I Need help i dont understand!!!!!!!!
Answers
Answer:
The correct option is;
x + y = 1000
Step-by-step explanation:
The data from the table can be presented as follows
Adults (x) \({}\) Children (y)
500 \({}\) 500
450 \({}\) 550
400 \({}\) 600
350 \({}\) 650
Therefore, we have the rate of change of the children, y to the adults, x, which is also the slope of the graph of the data given as follows;
The slope, m = dy/dx = (650 - 500)/(350 - 500) = 150/(-150) = -1
m = -1
The function of the graph in point and slope form, using the first point, becomes;
y - 500 = -1 × (x - 500) = -x + 500
∴ y - 500 = -x + 500
y + x = 500 + 500 = 1,000
Which gives
y + x = 1,000
y + x = x + y = 1,000
In the figure below, m||n. Match the angle pairs with the
correct label for the pairs.
Answers
On solving the provided question, by properties of parallel lines we got to know that - 1 - A 2 - C 3 - B 4 - D
what is alternate exterior angles?When two or more lines cross the intersection line, alternate exterior angles result. These angles are created on a number of sides outside the transverse lines. Any two parallel lines that cross one another create two angles with the horizontal line. In the area between the parallel lines, interior angles are formed, while alternate exterior angles are formed in the area outside the parallel lines.
Matches are -
1 - A
2 - C
3 - B
4 - D
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Which term refers to a process that is deployed to ensure the confidentiality and integrity of data while being stored or when it is transmitted Brainly?
Answers
The term refers to a process that is deployed to ensure the confidentiality and integrity of data while being stored or when it is transmitted is called "Encryption".
What is encryption?Data encryption is a computational procedure that converts plaintext/cleartext (unencrypted, readable data) in ciphertext (encrypted data), which is only available to authorised users with the correct cryptographic key.
Some key features regarding the encryption are-
Data is converted into cipher text through encryption that used a cipher (an encrypted technique) and an encryption key. A key (the same key for symmetric key encryption; a distinct, related value for asymmetric cryptography) is employed to decode the cipher - text into the original value once it has been communicated to the receiving party. Since encryption keys function similarly to physical keys, only users who have the appropriate key can "unlock" or decode the encrypted information.Encryption is frequently important to uphold compliance laws imposed by numerous organisations or standards bodies, in addition to the strong data privacy protection it offers.To know more about encryption, here
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Two boats A and B left port C at the same time on different routes B travelled on a bearing of 150° and A travelled on the north side of B. When A had travelled 8km and B had travelled 10km, the distance between the two boats was found to be 12km. Calculate the bearing of A's route from C
Answers
Using sine rule, the bearing of A's route from C is 109.1°
What is the bearing of A's route from C?To calculate the bearing of A's route from port C, we can use trigonometry and the given information. Let's denote the bearing of A's route from C as θ.
Since we have the value of three sides and only one angle, we can use sine rule to find the missing side.
a / sin A = b / sin B
10/ sin 40 = 8 / sin B
sin B = 8sin 40/ 10
sin B = 0.51423
B = sin⁻¹ (0.51423)
B = 30.94
Using the sum of angles in a triangle;
30.94 + 40 + x = 180
x = 109.1°
The bearing of A to C is 109.1°
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help me please i do not understand
Answers
Answer:
the equation of the line is:
y=-2x+-3
if the objective function is q=x^2y and you know that x y=10 write the objective function first in terms of x and then in terms of y.
Answers
If the objective function is q=\(x^2y\) and you know that x y=10, the objective function first in terms of x is q = 10x and then in terms of y is q = 100/y.
To write the objective function in terms of x, we can use the given value of xy = 10 and solve for y. Dividing both sides by x, we get:
y = 10/x
Now we can substitute this expression for y into the original objective function, q = \(x^2y\), to get:
q = x^2(10/x)
Simplifying this, we get:
q = 10x
So the objective function in terms of x is q = 10x.
To write the objective function in terms of y, we can use the same approach. Solving the given equation xy = 10 for x, we get:
x = 10/y
Substituting this into the original objective function, q = \(x^2y\), we get:
q = \((10/y)^2y\)
Simplifying this, we get:
q = 100/y
So the objective function in terms of y is q = 100/y.
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Two lines are intersecting. What is the value of x? Enter your answer in the box. Two straight lines intersecting each other where the consecutive angles formed are x plus twenty three degrees and two x plus four degrees.
Answers
Answer:
x = 51
Step-by-step explanation:
Supplementary angles total 180º
(2x + 4) + (x + 23) = 180
Combine like terms
3x + 27 = 180
Subtract 27 from both sides
3x = 153
Divide both sides by 3
x = 51
Answer:
Hello! x = 51
Step-by-step explanation:
OMG I CAN ANSWER FINALLY!
it is forming a supplementary angle meaning it will add up to 180 degrees
51 × 2 = 102 102 + 4 = 106
51 + 23 = 74
106 + 74 = 180 therefore x = 51 Hope that helps.
evaluate the scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2).
Answers
The scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2) is equal to 78√26/5
To evaluate the scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2), we first need to parameterize the curve c.
Let t be the parameter such that
x = -1 + 5t,
y = 3 - t,
for 0 ≤ t ≤ 1.
The length of the curve c is given by the integral ∫ c ds, which can be calculated using the formula ∫ a to b \(√(dx/dt)^2 + (dy/dt)^2\) dt. Plugging in the values from the parameterization, we get
\(∫ c ds = ∫ 0 to 1 √(5^2 + (-1)^2) dt = ∫ 0 to 1 √26 dt = √26.\)
Using the parameterization, we can now write the integral as
\(∫ c (3x y) ds = ∫ 0 to 1 (3(-1+5t)(3-t)) √(5^2 + (-1)^2) dt = 78√26/5.\)
Therefore, the scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2) is equal to 78√26/5.
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The scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2), is approximately equal to
22.229.
We can do this by letting x = t and y = 3 - t/2, where -1 ≤ t ≤ 4.
Then, we can find ds/dt using the formula \(ds/dt = \sqrt{(dx/dt^2 + dy/dt^2)}\), which simplifies to
\(ds/dt = \sqrt{(1 + 1/4) } = \sqrt{(5)/2} .\)
Next, we can substitute x and y in terms of t into the integrand and simplify to get:
\(3x y = 3t(3 - t/2) = 9t - (3/2)t^2\)
Now, we can evaluate the integral by integrating with respect to t from -1 to 4:
\(\int c (3x y) ds = ∫ from -1 to 4 (9t - (3/2)t^2) (\sqrt{(5)/2)} dt\)
\(= (\sqrt{(5)/2)} [ (9t^2/2) - (3/8)t^3 ] evaluated from -1 to 4\)
\(= (\sqrt{(5)/2)} [ (81/2) - (243/8) - (-27/8) + (3/8) ]\)
\(= \sqrt{(5)/2)} [ (189/8) ]\)
= 22.229
Therefore, the scalar line integral ∫ c (3x y) ds, where c is the line segment from (−1,3) to (4,2), is approximately equal to
22.229.
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can you guys help me please
Answers
Answer:
C
Step-by-step explanation:
A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.
C is the correct answer since it can be expressed as a fraction which equals to 7/9.
Hope it helps:)
Answer:
A
Step-by-step explanation:
Pi is an irrational number so it can't be B.
√(5) is an irrational number so it can't be D.
It could be either A or C... Hm, go with A. Let me know if it works.
Now that you’ve calculated all the missing values, complete the table.
Answers
Missing Blank for:
Raul and his friends: (3/4) / (1/20) = 15
Items: 15
Cars: (7/8) / 2 = 7/16
Weight per item: 7/16
Baby elephants: (3/4) / (3/8) = 2
Items: 2
Goats: (1/2) / 4 = 1/8
Weight per item: 1/8
Desks: (3/8) / (1/32) = 12
Items: 12
-5x-2y=
−5x−2y=
\,\,-40
−40
y=
y=
\,\,2x+2
2x+2
Answers
Answer: look at the ss :))
Step-by-step explanation:
Y more than 5
Translate the phrase into an algebraic expression
Answers
Answer:
y+5
Step-by-step explanation:
Answer:
y + 5
Step-by-step explanation:
Which could be a set of treatment and control groups for a retrospective study on the effects of endocrine disruptors on humans
Answers
A set of treatment and control groups for a retrospective study on the effects of endocrine disruptors on humans a group of children with mothers exposed to PCBs during pregnancy and a group of children who were exposed to DDT after they were born (option c).
In a retrospective study, researchers investigate the effects of endocrine disruptors on humans by analyzing past data and observing correlations. To ensure reliable findings, it is crucial to establish appropriate treatment and control groups. These groups are designed to compare individuals who have been exposed to different levels or types of endocrine disruptors, allowing researchers to assess the effects accurately. Let's explore the potential treatment and control groups for such a study.
A group of children with mothers exposed to PCBs during pregnancy and a group of children who were exposed to DDT after they were born:
In this case, the treatment group would consist of children whose mothers were exposed to PCBs during pregnancy. The control group, on the other hand, would include children who were exposed to DDT after birth. PCBs and DDT are both endocrine disruptors, but by comparing these two groups, researchers can examine the effects of prenatal exposure (PCBs) versus postnatal exposure (DDT).
Hence the correct option is (c).
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Complete Question
Which could be a set of treatment and control groups for a retrospective study on the effects of endocrine disruptors on humans?
(a) A group of farmers that are planning to spray crops with atrazine and a group of farmers that are planning to use organic pesticides,
(b) A group of construction workers that have been exposed to asbestos and a group of construction workers that have been exposed to PCBs,
(c) A group of children with mothers exposed to PCBs during pregnancy and a group of children who were exposed to DDT after they were born,
(d) Members of a town that was downwind of a commercial farm that sprayed pesticides and members of a town that was downwind of an organic farm,
(e) Humans raised when DDT was used and humans raised after DDT was no longer used.
a1 is fouled on an unsuccessful two-point shot attempt. a1 is injured on the play and cannot shoot the free throws. team a has seven eligible players on the bench. a1's free throws must be shot by:
Answers
When a player is fouled and injured on an unsuccessful two-point shot attempt, the opposing team's coach is responsible for choosing the replacement free throw shooter from the injured player's team bench. This ensures a fair and balanced game.
In basketball, when a player (A1) is fouled during an unsuccessful two-point shot attempt and is injured, the opposing team's coach selects the replacement free throw shooter from the seven eligible players on the bench. This rule ensures fairness in the game, as it prevents the injured player's team from gaining an advantage by choosing their best free throw shooter.
Since A1 is injured and cannot shoot the free throws, the opposing team's coach will pick a substitute from the seven available players on Team A's bench. This decision maintains a balance in the game, as it avoids giving Team A an unfair advantage by selecting their own substitute.
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For a pancake distribution of sin(a), where a = 0, determine the ratio of the average flux for e > 45 to the omnidirectional flux. What I need from here is:Directional Flux, Omnidirectional Flux,Directional Solid Angle, Omnidirectional Solid Angle. Then: Find the Flux per Solid Angle (For both the directional and omnidirectional cases) And find the ratio of those two
Answers
For the ratio of the average flux for e > 45 degrees to the omnidirectional flux, we divide the flux per solid angle for the directional case by the flux per solid angle for the omnidirectional case.
To find the ratio of the average flux for e > 45 degrees to the omnidirectional flux in a pancake distribution of sin(a) where a = 0, we need to calculate the directional flux, omnidirectional flux, directional solid angle, and omnidirectional solid angle.
Directional Flux:
The directional flux is the flux within a specific direction or range of angles. In this case, we are interested in e > 45 degrees.
Omnidirectional Flux:
The omnidirectional flux is the total flux in all directions or over the entire solid angle.
Directional Solid Angle:
The directional solid angle is the solid angle subtended by the specified direction or range of angles. In this case, it would be the solid angle corresponding to e > 45 degrees.
Omnidirectional Solid Angle:
The omnidirectional solid angle is the total solid angle subtended by all possible directions or over the entire sphere.
To find the flux per solid angle for both the directional and omnidirectional cases, we can use the formula:
Flux per Solid Angle = Total Flux / Solid Angle
Finally, to find the ratio of the average flux for e > 45 degrees to the omnidirectional flux, we divide the flux per solid angle for the directional case by the flux per solid angle for the omnidirectional case.
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n – 8 = 16 to a verbal sentence
Answers
Answer:
n is subtracted from 8 is equal to eight
Answer:
n = 24
Step-by-step explanation:
n - 8 = 16
You will add 8 on both sides.
The 8's would cancel out.
You'll then be left with n = 16 + 8 and 16 + 8 = 24
Therefore, n = 24
Hope this helps :)
Teresa’s restaurant bill comes to $29.99 before tax. If the sales tax is 6.25% and she tips the waiter 20%, what is the total cost of the meal
Answers
Suppose that the numbers of residents in zip codes are normally distributed with an unknown mean and standard deviation. The numbers of residents of 50 randomly sampled zip codes are used to estimate the mean of the population. What t-score should be used to find the 99% confidence interval for the population mean?
Answers
Answer:
Step-by-step explanation:
The required t score would be determined from the t distribution table. In order to use the t distribution, we would determine the degree of freedom, df for the sample.
df = n - 1 = 50 - 1 = 49
Since confidence level = 99% = 0.99, α = 1 - CL = 1 – 0.99 = 0.01
α/2 = 0.01/2 = 0.005
the area to the right of z0.005 is 0.005 and the area to the left of z0.005 is 1 - 0.005 = 0.995
Looking at the t distribution table,
t score = 2.678
3.17 scores on stats final: below are final exam scores of 20 introductory statistics students. 57 66 69 71 72 73 74 77 78 78 79 79 81 81 82 83 83 88 89 94 a) the mean score is 77.7 points and the standard deviation is 8.44 points. do the exam scores approximately follow the 68-95-99.7% rule? how can you decide?
Answers
Overall, the exam scores approximately follow the 68-95-99.7% rule, suggesting that the distribution of scores is approximately normal.
The 68-95-99.7% rule, also known as the empirical rule, states that for a normal distribution, approximately:
68% of the data falls within one standard deviation of the mean
95% of the data falls within two standard deviations of the mean
99.7% of the data falls within three standard deviations of the mean.
Exam scores approximately follow this rule, we can calculate the range that includes one, two, and three standard deviations from the mean and see if the corresponding percentages of scores fall within these ranges.
The mean score is 77.7 points and the standard deviation is 8.44 points.
One standard deviation from the mean would be 77.7 ± 8.44 = [69.26, 86.14].
We can see that 16 out of the 20 scores fall within this range, which is approximately 80%, close to the 68% we would expect for a normal distribution.
Two standard deviations from the mean would be 77.7 ± 2(8.44) = [60.82, 94.58].
We can see that 19 out of the 20 scores fall within this range, which is approximately 95%, in line with the 95% we would expect for a normal distribution.
Three standard deviations from the mean would be 77.7 ± 3(8.44) = [52.38, 103.02].
All 20 scores fall within this range, which is 100%, more than the 99.7% we would expect for a normal distribution.
Could be due to the small sample size.
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2. Find the present value of an ordinary annuity with regular quarterly opayments worth P1, 000 at 3% annual interest rate compouded quarterly at the end of 4 years.
Answers
The present value of an ordinary annuity with regular quarterly payments is ₹1126.99.
The interest charged on a loan or deposit is known as compound interest. It is the idea that we employ the most frequently on a daily basis. Compound interest is calculated for an amount based on both the principal and cumulative interest. The major distinction between compound and simple interest is this.
Given that,
P = ₹1000
R = 3% compounded quarterly
T = 4 years
Amount = P ( 1+ r / 4* 100 )^4t )
= ₹1000 ( 1+ 3/400)^16)
= 1000(1+0.0075)^16)
= 1000 * 1.126
= 1126.99
Thus, The present value of an ordinary annuity with regular quarterly payments is ₹1126.99.
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let f be the function given by f(x)=1(2 x). what is the coefficient of x3 in the taylor series for f about x = 0 ?
Answers
The coefficient of x^3 in the Taylor series for f(x) is 0, since there is no term involving x^3.
To find the Taylor series of the function f(x) = 1/(2x) about x = 0, we can use the formula:
\(f(x) = f(0) + f'(0)x + (1/2!)f''(0)x^2 + (1/3!)f'''(0)x^3 + ...\)
where f'(x), f''(x), f'''(x), etc. denote the derivatives of f(x).
First, we need to find the derivatives of f(x):
f'(x) = -1/(2x^2)
f''(x) = 2/(x^3)
f'''(x) = -6/(x^4)
f''''(x) = 24/(x^5)
Next, we evaluate these derivatives at x = 0 to get:
f(0) = 1/(2(0)) = undefined
f'(0) = -1/(2(0)^2) = undefined
f''(0) = 2/(0)^3 = undefined
f'''(0) = -6/(0)^4 = undefined
f''''(0) = 24/(0)^5 = undefined
Since the derivatives are undefined at x = 0, we need to use a different method to find the Taylor series. We can use the identity:
1/(1 - t) = 1 + t + t^2 + t^3 + ...
where |t| < 1.
Substituting t = -x^2/a^2, we get:
1/(1 + x^2/a^2) = 1 - x^2/a^2 + x^4/a^4 - x^6/a^6 + ...
This is the Taylor series for 1/(1 + x^2/a^2) about x = 0. To get the Taylor series for f(x) = 1/(2x), we need to replace x with ax^2:
f(x) = 1/(2(ax^2)) = 1/(2a) * 1/(1 + x^2/a^2)
Substituting the Taylor series for 1/(1 + x^2/a^2), we get:
f(x) = 1/(2a) - x^2/(2a^3) + x^4/(2a^5) - x^6/(2a^7) + ...
Therefore, the coefficient of x^3 in the Taylor series for f(x) is 0, since there is no term involving x^3.
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