George's dog is approximately 32.41 meters away from its crate, if it ran 22 meters, turned and ran 11 meters, and then turned 120° to face its crate.
To determine the distance from George's dog to its crate after the described movements, we can use the concept of a triangle and trigonometry.
The dog initially runs 22 meters, then turns and runs 11 meters, forming the two sides of a triangle. The third side of the triangle represents the distance from the dog's final position to the crate.
To find this distance, we can use the Law of Cosines, which states that in a triangle with sides a, b, and c and angle C opposite side c, the equation is c² = a² + b² - 2abcos(C).
In this case, a = 22 meters, b = 11 meters, and C = 120°. Plugging these values into the equation, we have
c² = 22² + 11² - 2(22)(11)cos(120°).
Evaluating the expression, we get
c ≈ 32.41 meters.
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A bird is flying 15 feet above sea level. A dolphin is swimming 3 feet below sea level. What is the difference in elevation? (I just need help checking again)
Answer:
18ft
Step-by-step explanation:
15+3=18
A function is defined by f(x) = x²+2, ≥0. A region R is enclosed by y = f(x), the y-axis line y = 4.
Find the exact volume generated when the region R is rotated through 27 radians about the y-axis.
Given that a function is defined by `f(x) = x² + 2`, and the region R is enclosed by `y = f(x)`, the `y-axis` line `y = 4`. We need to find the exact volume generated when the region R is rotated through `27 radians` about the `y-axis`.
Explanation: The formula for finding the volume generated by rotating the region R about the `y-axis` is given by: `V = ∫ [from a to b] 2πxf(x) dx`. Here, the value of `a` is `0` because it's given that `f(x) = x² + 2`, and `f(x)` is greater than or equal to `0`. Also, the line `y = 4` intersects `f(x)` at `x = 2`. So, the value of `b` is `2`.
Therefore, the volume generated is given by: V = ∫ [from 0 to 2] 2πx (x² + 2) dx`=`2π ∫ [from 0 to 2] (x³ + 2x) dx`=`2π [(x⁴/4) + x²] {from 0 to 2}`=`2π [(2⁴/4) + 2²] - 0`=`2π [4 + 4]`=`16π` cubic units.
So, the exact volume generated when the region R is rotated through `27 radians` about the `y-axis` is `16π` cubic units.
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PLEASE ASAP, PLEASEEEE
Answer: (1,2)
Step-by-step explanation:
-2+3=1 and 4-2=2
(1,2)
What is the length of the legs of the triangle if the hypotenuse of an isosceles right triangle is (sqrt)23 feet?
4 feet
8 feet
16 feet
16.2 feet
question in the pic
Select all expressions that are NOT written correctly in Scientific Notation.
The expressions that are not written correctly in scientific notation are: 48,200, 36.105, 8.7.10-1, and 0.78.10-3
Scientific notation is a way to express numbers in a concise form, using a number between 1 and 10 multiplied by a power of 10. Let's analyze the given expressions and identify the ones that are not written correctly in scientific notation:
48,200: This expression is not written in scientific notation. It should be expressed as 4.82 × 10^4 or 4.82e4.
0.00099: This expression is correctly written in scientific notation. It can be expressed as 9.9 × 10^-4 or 9.9e-4.
36.105: This expression is not written in scientific notation. It should be expressed as 3.6105 × 10^1 or 3.6105e1.
8.7.10-1: This expression is not written correctly in scientific notation. Scientific notation only allows one decimal point in the number. The correct representation would be 8.7 × 10^-1 or 8.7e-1.
0.78.10-3: Similar to the previous expression, this is not written correctly in scientific notation. The correct representation would be 7.8 × 10^-3 or 7.8e-3.
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Help ASAP!!!!!!!!!!!!!!!!!!!!!!
square
its the middle bit that looks darker than the rest.
A study was conducted to determine if husbands and wives like the same TV shows. Married couples ranked 15th TV shows, and the Spearman rank correlation coefficient was found to be rs = 0.47.
a. Specify the competing hypotheses to determine if there is a positive correlation between the two rankings.
Null Hypothesis (H0): There is no positive correlation between the rankings of TV shows by husbands and wives in married couples.
Alternative Hypothesis (HA): There is a positive correlation between the rankings of TV shows by husbands and wives in married couples.
In simpler terms, the null hypothesis suggests that there is no relationship or association between the rankings of TV shows by husbands and wives.
The alternative hypothesis, on the other hand, proposes that there is a positive correlation between the rankings, indicating that husbands and wives tend to have similar preferences when it comes to TV shows.
The competing hypotheses in this study aim to determine whether there is evidence to support the idea that husbands and wives tend to like the same TV shows.
The null hypothesis assumes that there is no correlation between the rankings, meaning that the preferences of husbands and wives are independent of each other.
The alternative hypothesis, in contrast, suggests that there is a positive correlation, indicating a tendency for spouses to have similar preferences for TV shows.
To test these hypotheses, the researchers used the Spearman rank correlation coefficient (rs) to quantify the strength and direction of the relationship between the rankings.
The Spearman rank correlation is a statistical measure that assesses the monotonic relationship between two ranked variables, in this case, the rankings of TV shows by husbands and wives. A value of rs = 0.47 indicates a moderate positive correlation between the rankings.
To evaluate the hypotheses, statistical tests can be conducted. The significance level (alpha) is typically set in advance (e.g., 0.05) to determine the threshold for accepting or rejecting the null hypothesis.
If the p-value associated with the test is less than the chosen significance level, the null hypothesis is rejected in favor of the alternative hypothesis, suggesting that there is evidence of a positive correlation between the rankings of TV shows by husbands and wives.
Conversely, if the p-value is greater than the significance level, there is insufficient evidence to reject the null hypothesis, and the data does not provide support for a positive correlation.
It is important to note that the interpretation of the Spearman rank correlation coefficient and the conclusions drawn from the study should consider other factors, such as the sample size, sampling method, and the specific characteristics of the TV shows ranked by the couples.
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Find the solution of the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0. = = - 2 NOTE: Enter an exact answer. y(t) =
The solution of the initial value problem is:[tex]Y(t) = e^(3t) + e^(-t) - 3tet[/tex]
To solve the initial value problem Y" – 2y – 3y = 15tet, y(0) = 2, y'(0) = 0, we can use the method of undetermined coefficients.
First, we find the general solution of the homogeneous equation Y" – 2y – 3y = 0.
The characteristic equation is:
[tex]r^2 - 2r - 3 = 0[/tex]
Factoring the quadratic equation, we have:
(r - 3)(r + 1) = 0
This gives us two distinct roots: r = 3 and r = -1.
Therefore, the general solution of the homogeneous equation is:
[tex]Yh(t) = C1e^(3t) + C2e^(-t)[/tex]
To find a particular solution Yp(t) for the non-homogeneous equation, we assume a solution of the form Yp(t) = Atet, where A is a constant to be determined.
Taking the first and second derivatives of Yp(t), we have:
[tex]Yp'(t) = Ate^t + Aet[/tex]
[tex]Yp"(t) = Ate^t + 2Aet[/tex]
Substituting these derivatives into the non-homogeneous equation, we get:
[tex](Ate^t + 2Aet) - 2(Atet) - 3(Atet) = 15tet[/tex]
Simplifying the equation, we have:
[tex]Ate^t + 2Aet - 2Ate^t - 3Ate^t = 15tet[/tex]
Combining like terms, we get:
[tex](-4A + 2A - 3A)te^t = 15tet[/tex]
Simplifying further, we have:
[tex]-5Ate^t = 15tet[/tex]
Cancelling out the common terms, we get:
-5A = 15
Solving for A, we find:
A = -3
Now, we have the particular solution Yp(t) = -3tet.
The general solution of the non-homogeneous equation is the sum of the general solution of the homogeneous equation and the particular solution:
Y(t) = Yh(t) + Yp(t)
[tex]Y(t) = C1e^(3t) + C2e^(-t) - 3tet[/tex]
Using the initial conditions y(0) = 2 and y'(0) = 0, we can solve for the values of C1 and C2.
When t = 0:
[tex]Y(0) = C1e^(3(0)) + C2e^(-0) - 3(0)e^(0)[/tex]
2 = C1 + C2
Taking the derivative of Y(t) with respect to t and evaluating it at t = 0:
[tex]Y'(t) = 3C1e^(3t) - C2e^(-t) - 3te^(3t)Y'(0) = 3C1e^(3(0)) - C2e^(-0) - 3(0)e^(3(0))[/tex]
0 = 3C1 - C2
Solving these equations simultaneously, we find C1 = 1 and C2 = 1.
Therefore, the solution of the initial value problem is:
[tex]Y(t) = e^(3t) + e^(-t) - 3tet[/tex]
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PLease help me thank you if you do have a good day
Answer:
acceleration
Step-by-step explanation:
use POE:
the object is moving because the graph is not a straight line
the object is not slowing down because the graph is going up
there is not a constant speed because the line is exponential not linear
the only one left is a. acceleration
help i need this to be answerd
Answer:
What is it, you need help with?
Step-by-step explanation:
:)
How would I go about solving number 3 in Math Practice? The question is Calculate the percent change from phytoplankton to smelt. Show your setup and answer.
To calculate the percent change from phytoplankton to smelt, you would first need to find the difference between the two concentrations. Then, you would divide the difference by the original concentration and multiply by 100%.
For example, if the concentration of phytoplankton is 100 parts per million (ppm) and the concentration of smelt is 150 ppm, then the difference between the two concentrations is 50 ppm.
Dividing 50 ppm by 100 ppm and multiplying by 100% gives us a percent change of 50%. This means that the concentration of PCBs in smelt is 50% higher than the concentration of PCBs in phytoplankton.
To calculate the percent change from phytoplankton to smelt, you can use the following formula:
Percent change = [tex]\frac{(new value - old value)}{old value } *100%[/tex]
In this case, the new value is the concentration of PCBs in smelt and the old value is the concentration of PCBs in phytoplankton.
Once you have calculated the percent change, you can interpret it by comparing it to a reference value. For example, a percent change of 50% is considered to be a large change. This means that the concentration of PCBs in smelt is much higher than the concentration of PCBs in phytoplankton.
It is important to note that the percent change is only a measure of the relative change between two values. It does not take into account the absolute values of the two values. For example, a percent change of 50% is the same whether the original value is 100 or 1000.
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Maya bought a board game that cost $24. She had to pay a 7 1/2% sales tax. How much change did she get from thirty dollars?
Answer: $4.20
Step-by-step explanation:
24(.075)=1.80
24+1.80=25.80
30-25.80=4.20
For the following nonlinear system, 73 2 = y + 3x2 + 3x 91(x, y) = 7 y2 + 2y – X – 2 92(x, y) = 2 y = use the initial approximation (po, qo) = (-0.3, -1.3), and compute the next three approximations to the fixed point using (a) Jacobi iteration (b) Seidel iteration.
To compute the next three approximations to the fixed point using Jacobi iteration and Seidel iteration, we will use the initial approximation (p0, q0) = (-0.3, -1.3) for the given nonlinear system.
(a) Jacobi Iteration:
In Jacobi iteration, we update the variables simultaneously using the previous values.
(b) Seidel Iteration:
In Seidel iteration, we update the variables using the most recently computed values.
To compute the next three approximations to the fixed point using Jacobi iteration and Seidel iteration, we will use the initial approximation (p0, q0) = (-0.3, -1.3) for the given nonlinear system.
(a) Jacobi Iteration:
In Jacobi iteration, we update the variables simultaneously using the previous values. The iteration formula for this system is as follows:
p(n+1) = (y(n) + 3x(n)^2 + 3x(n) - 73)/3
q(n+1) = (7y(n)^2 + 2y(n) - x(n) - 2)/92
Using the initial approximation (-0.3, -1.3), we can compute the next three approximations as follows:
Iteration 1:
p(1) = (1 + 3(-0.3)^2 + 3(-0.3) - 73)/3 ≈ -8.300
q(1) = (7(-1.3)^2 + 2(-1.3) - (-0.3) - 2)/92 ≈ -1.317
Iteration 2:
p(2) = (1 + 3(-8.300)^2 + 3(-8.300) - 73)/3 ≈ -209.034
q(2) = (7(-1.317)^2 + 2(-1.317) - (-8.300) - 2)/92 ≈ -2.924
Iteration 3:
p(3) = (1 + 3(-209.034)^2 + 3(-209.034) - 73)/3 ≈ -14314.328
q(3) = (7(-2.924)^2 + 2(-2.924) - (-209.034) - 2)/92 ≈ -6.344
(b) Seidel Iteration:
In Seidel iteration, we update the variables using the most recently computed values. The iteration formula for this system is as follows:
p(n+1) = (y(n) + 3x(n)^2 + 3x(n) - 73)/3
q(n+1) = (7y(n+1)^2 + 2y(n+1) - x(n) - 2)/92
Using the initial approximation (-0.3, -1.3), we can compute the next three approximations as follows:
Iteration 1:
p(1) = (1 + 3(-0.3)^2 + 3(-0.3) - 73)/3 ≈ -8.300
q(1) = (7(-1.3)^2 + 2(-1.3) - (-0.3) - 2)/92 ≈ -1.315
Iteration 2:
p(2) = (1 + 3(-1.315)^2 + 3(-1.315) - 73)/3 ≈ -8.264
q(2) = (7(-8.264)^2 + 2(-8.264) - (-1.315) - 2)/92 ≈ -3.471
Iteration 3:
p(3) = (1 + 3(-3.471)^2 + 3(-3.471) - 73)/3 ≈ -1.252
q(3) = (7(-1.252)^2 + 2(-1.252) - (-3.471) - 2)/92 ≈ -1.100
These are the next three approximations to the fixed point using Jacobi iteration and Seidel iteration with the given initial approximation.
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Branliest!!!! 100 points! solve for x in the image below:
also here's the equation: 3x+9= 90 degrees
Answer:
Hi! The answer to your question is x = 27
Step-by-step explanation:
☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆☆*: .｡..｡.:*☆
☁Brainliest is greatly appreciated!☁
Hope this helps!!
- Brooklynn Deka
Answer:
x=27
Step-by-step explanation:
3x+9=90
subtract 9 from both sides
3x=90-9
3x=81
divide by 3
x= 27
checking
3x27 +9= 90
81+9=90
I hope this makes sense
help your girl outtt !!!
n^2-8n=0
can i have the work shown?
Answer:
n=1/8N^2
Step-by-step explanation:
n^2-8n=0
-8n=-N^2
n=1/8N^2
What percentage of videos on the streaming site are between 264 and 489 seconds? 0.15% 49.85% 95% 99.7%
Answer:
49.85%
Step-by-step explanation:
Streaming is defined as sending and receiving an video or an audio content in continuous flow over any network. It allows one to begin the playback and while it sends the rest of the data.
In the context, the percentage of the videos on a streaming site that is between 264 and 489 seconds is 49.85% which is found by a graph where the distribution of the lengths of the videos in seconds on a popular video streaming sites.
i have more to come check out my account:( please
Answer:
Part A
The two figures, ΔABC and ΔPQR are congruent by the SSS rule of congruency
Part B
The rigid motion that maps ΔABC to ΔPQR are a 180° clockwise rotation about the origin, followed by a horizontal shift of 1 unit to the right
Step-by-step explanation:
Part A
The given coordinates of the vertices of triangle ABC are;
A(-8, -2), B(-3, -6), C(-2, -2)
The length of side AB = √((-6 - (-2))² + (-3 - (-8))²) = √41
The length of side AC = √((-2 - (-2))² + (-2 - (-8))²) = 6
The length of side BC = √((-6 - (-2))² + (-3 - (-2))²) = √17
The given coordinates of the vertices of triangle PQR are;
P(9, 2), Q(4, 6), R(3, 2)
The length of side PQ = √((6 - 2)² + (4 - 9)²) = √41
The length of side PR = √((2 - 2)² + (3 - 9)²) = 6
The length of side RQ = √((6 - 2)² + (4 - 3)²) = √17
Given that the length of the three sides of triangle ABC are equal to the lengths of the three sides of triangle PQR, we have;
ΔABC ≅ ΔPQR by Side Side Side rule of congruency
Part B
Whereby AB and PQ, and BC and RQ are pair of corresponding sides, the rigid motion that maps ΔABC to ΔPQR are;
1) A 180° clockwise (or counterclockwise) rotation about the origin followed by
2) A shift of 1 unit to the right.
ok so Im getting banned and these cant go to waist so have fun :D
Answer:
Thx u
Step-by-step explanation:
that's not good oof
:3
:)
:0
:>
what is 4 + 7 x 2 – 8
Answer:
The answer is 10.
Step-by-step explanation:
4 + 7 x 2 - 8.
7 x 2 = 14
4 + 14 = 18
18 - 8 = 10
So, the answer is 10
Hope this helps! :)
Using the BODMAS rule, the value of the expression (4+7x2 –8) is 10.
What is BODMAS?BODMAS is a system that decides the preference of mathematical operations to solve an expression.
B stands for Bracket
O stands for of
D stands for division
M stands for multiplication
A stands for addition
S stands for subtraction.
So, In the given expression (4 + 7 x 2 – 8) first, we will do multiplication
i.e. 4+14-8
Secondly, we will do the addition
i.e. 18-8
Lastly, we will subtract
10
Hence, using the BODMAS rule, the value of the expression (4+7x2 –8) is 10.
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Please help with the square roots
Step-by-step explanation:
-√81=-√(9)²=-9
A number inside a Square root cannot ne negative. So, the second one is not a real no.
Answer:
[tex] - \sqrt{81} = - 9 \\ \sqrt{ - 25 = - 5} [/tex]
Step-by-step explanation:
-81/9 9+9+9+9+9+9+9+9+9(81)
-9
-25/5 (5+5+5+5+5(25)
-5
Which of the following expressions are equivalent to 10 – 12? Choose all answers that apply:
А. 2.5 - 6
B.2(5 - 6)
C.None of the above
Answer:
B: 2(5−6)
Step-by-step explanation:
My Samsung charger broke so I went to Five Below to get a new one. They were on sale for $15,35 (not so five below). There
was a 6% tax too. How much did the new charger cost?
Answer:
It would be in thE 20.75
Step-by-step explanation:
Just keep adding change by 9 6 times
A Russian fighter jet is carrying a ready atomic missile over to Ukraine. The pilot shoots the missile so that it travels in a parabolic motion from a height of 22,500ft above the ground, Assuming that 22,500ft is the maximum point and the equation is - x² + 100x + 20000. a.) Assuming the fighter jet started from the ground and followed the path of the equation, calculate the horizontal distance between the fighter jet at ground level and the point of impact of the missile. b.) Graph the equation, showing the highest point and the two ends from which the equation represents. [c.) Determine the instantaneous rate of change when the missile has half of the horizontal journey left.
a) The horizontal distance between the fighter jet and the point of impact of the missile is 100 units.
b) The graph of the equation is a downward-opening parabola passing through the points (50, 22500), (100, 0), and (0, 0).
c) The instantaneous rate of change when the missile has half of the horizontal journey left is 0, indicating no change in height with respect to the horizontal distance at that point.
To find the horizontal distance between the fighter jet and the point of impact of the missile, we need to determine the x-coordinate when the missile hits the ground. This can be done by finding the x-intercepts of the equation -x² + 100x + 20000, which represents the path of the missile.
To find the x-intercepts, we set the equation equal to zero:
-x² + 100x + 20000 = 0
Using the quadratic formula, where a = -1, b = 100, and c = 20000, we can calculate the x-coordinate:
x = (-b ± √(b² - 4ac)) / (2a)
Plugging in the values, we get:
x = (-100 ± √(100² - 4(-1)(20000))) / (2(-1))
Simplifying further:
x = (-100 ± √(10000 + 80000)) / (-2)
x = (-100 ± √90000) / (-2)
x = (-100 ± 300) / (-2)
x = (200 or -100) / 2
Since negative values are not meaningful in this context, we take the positive value, which is x = 100. Therefore, the horizontal distance between the fighter jet and the point of impact of the missile is 100 units.
To graph the equation, we plot the points on a coordinate system. The equation -x² + 100x + 20000 represents a downward-opening parabola. The highest point of the parabola is at (50, 22500) because the x-coordinate represents the midpoint of the parabolic path, and the maximum height is reached when x = 50. The two ends of the parabolic path are located at the x-intercepts we calculated earlier, which are (100, 0) and (0, 0).
The graph of the equation would show a downward-opening parabola passing through the points (50, 22500), (100, 0), and (0, 0).
The instantaneous rate of change represents the derivative of the equation with respect to x at a given point. To find the instantaneous rate of change when the missile has half of the horizontal journey left, we need to find the derivative of the equation and evaluate it at that point.
Taking the derivative of -x² + 100x + 20000 with respect to x, we get -2x + 100. Evaluating this derivative at x = 50 (when the missile has half of the horizontal journey left), we have:
-2(50) + 100 = -100 + 100 = 0
Therefore, the instantaneous rate of change when the missile has half of the horizontal journey left is 0. This indicates that at that point, the height of the missile is not changing with respect to the horizontal distance.
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Is the set of all real-valued functions f(x) such that f(2)= 0, with the usual addition and scalar multiplication of functions, ((F+g)(x) = f(x) + g(x),(kf)(x) = kf (x)), a subspace of the vector space consisting of all real-valued functions? Answer yes or no and justify your answer. Is the solution set of a nonhomogeneous linear system Ax = b, of m equations in n unknowns, with b#0, a subspace of R" ? Answer yes or no and justify your answer.
No, the set of all real-valued functions f(x) such that f(2) = 0 is not a subspace of the vector space consisting of all real-valued functions. The solution set of a nonhomogeneous linear system Ax = b, with b ≠ 0, is also not a subspace of R.
To determine if a set is a subspace, it must satisfy three conditions: closure under addition, closure under scalar multiplication, and contain the zero vector. In the case of the set of real-valued functions f(x) such that f(2) = 0, it fails to satisfy closure under scalar multiplication. If we take a scalar k and multiply it with a function f(x) in the set, the resulting function kf(x) will not necessarily have f(2) = 0. Therefore, the set does not form a subspace.
For the solution set of a nonhomogeneous linear system Ax = b, where b ≠ 0, it also fails to be a subspace of R. A subspace must contain the zero vector, which corresponds to the homogeneous solution of the linear system. However, in a nonhomogeneous system, the zero vector is not a valid solution since Ax ≠ b. Therefore, the set of solutions does not contain the zero vector and cannot be considered a subspace.
In conclusion, neither the set of real-valued functions with f(2) = 0 nor the solution set of a nonhomogeneous linear system with b ≠ 0 form subspaces in their respective vector spaces.
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Find the value of x and the value of y.
A. x= 2 squared root of 3, y= 4 squared root of 3
B. x= 3,y= 6 squared root of 3
C. x= 6 squared root of 3, y=12
D. 2 squared root of 3, y= 6
Answer:
Option A
Step-by-step explanation:
To find all missing sides of a right triangle we use the sine, cosine or tangent ratio as below,
sinθ = [tex]\frac{\text{Opposite side}}{\text{Hypotenuse}}[/tex]
cosθ = [tex]\frac{\text{Adjacent side}}{\text{Hypotenuse}}[/tex]
tanθ = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
Now we take an angle measuring 60°
sin(60°) = [tex]\frac{6}{y}[/tex]
[tex]\frac{\sqrt{3} }{2}=\frac{6}{y}[/tex]
y = 4√3
tan(60°) = [tex]\frac{6}{x}[/tex]
√3 = [tex]\frac{6}{x}[/tex]
x = 2√3
Therefore, Option A will be the correct option.
In the data set #2 {75,80,85,75,85}, what is the range?
33. Box plots have been used successfully to describe
a. center and spread of a data set
b. the extent and nature of any departure from symmetry
c. identification of "outliers"
d. All of the choices.
e. none of the choices
34. A civil engineer is analyzing the compressive strength of concrete. Compressive strength is normally distributed with variance 1000(psi)². A random sample of 10 specimens has a mean compressive strength of 3250 psi. With what degree of confidence could we say that the mean compressive strength between 3235 and 3265?
a. 90%
b. 87%
c. 95%
d. 85%
e. 99%
33. The box plots have been used successfully to describe the center and spread of a data set, the extent and nature of any departure from symmetry and the identification of "outliers".
Hence, the correct option is (d) All of the choices.
34. We can say with a 95% degree of confidence that the mean compressive strength between 3235 and 3265, the correct option is (c) 95%.
Box plots are an excellent way of representing data, which has a statistical measure like variance, median, mean, mode, etc.
It presents the central tendency, variability, skewness, and even show the outliers.
A box plot, also called a box and whisker plot, shows the five-number summary of a set of data (minimum value, lower quartile, median, upper quartile, maximum value).
34. The given information is
Sample size, n = 10
Mean = 3250
Variance = 1000(psi)²
Standard Deviation = √1000(psi)²
= 31.62 psi
The degree of freedom is calculated as follows:
d. f . = n - 1
= 10 - 1
= 9
At 95% confidence level, the area in each tail is given by
α/2 = 0.05/2
= 0.025
Using the t-table, we can find that the t-value for 9 degrees of freedom and 0.025 area in each tail is 2.262.
Therefore, the critical values of t are
t₁ = -2.262 and
t₂ = 2.262.
We can calculate the confidence interval as follows:
Confidence Interval, CI = x± (t × σ/√n)
Plugging in the values, we get
CI = 3250 ± (2.262 × 31.62/√10)
= (3235, 3265)
Hence, we can say with a 95% degree of confidence that the mean compressive strength between 3235 and 3265.
The correct option is (c) 95%.
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Let v(x, y) = (2). (a) Show that v(x, y) is an ideal flow. (b) Find the complex potential (2) for v. (Aralytic methods. (c) Find the stagnation point (s) of v. (d) Find the streamlines (trajectories) of v, and hence show that v(x,y) is a tangent vector to the streamline at z = x+iy (excluding the stagnation point(s)). w= Sux-iny dz
(a) To show that v(x, y) is an ideal flow, we need to verify that it satisfies the conditions of being both irrotational and incompressible.
For irrotationality, we compute the curl of v(x, y):
curl(v) = ∂v_y/∂x - ∂v_x/∂y = 0 - 0 = 0
Since the curl is zero, v(x, y) is irrotational.
For incompressibility, we compute the divergence of v(x, y):
div(v) = ∂v_x/∂x + ∂v_y/∂y = 2 - 0 = 2
Since the divergence is not zero, v(x, y) is not incompressible. Therefore, v(x, y) is an irrotational flow but not an ideal flow.
(b) For the complex potential Φ for v(x, y), we can integrate the components of v(x, y) with respect to z = x + iy.
Φ = ∫ (2) dz = 2z = 2(x + iy) = 2x + 2iy
The complex potential Φ is given by Φ = 2x + 2iy.
(c) we need to solve for the points where both components of v are zero simultaneously:
v_x = 2x = 0
v_y = 0
From the first equation, x = 0. Substituting x = 0 into the second equation, we get v_y = 0, which holds for all values of y. Therefore, the stagnation point(s) of v(x, y) is at x = 0, y = y.
(d) For the streamlines (trajectories) of v, we can solve the differential equation given by dw/dz = Su_x - iu_y, where w is the complex potential Φ.
dw/dz = ∂Φ/∂x - i∂Φ/∂y = 2 - 2i
Integrating the above expression with respect to z, we get:
w = 2z - 2iz = 2(x + iy) - 2i(x + iy) = 2x + 2iy - 2ix - 2y = 2(x - y) + 2i(y - x)
The streamlines are given by the equation w = 2(x - y) + 2i(y - x), which shows that v(x, y) is a tangent vector to the streamline at z = x + iy (excluding the stagnation point(s)).
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