The maximum value of the objective function Z is 1800. The optimal values for the decision variables are X1 = 10, X2 = 0, and X3 = 0. The constraints are satisfied, and the optimal solution has been reached using the Simplex Method.
To compute the given problem using the Simplex Method, we need to convert it into a standard form.
The standard form of a linear programming problem consists of maximizing or minimizing a linear objective function subject to linear inequality constraints and non-negativity constraints.
Let's rewrite the problem in standard form:
Maximize:
Z = 50X1 + 20X2 + 10X3
Subject to the constraints:
2X1 + 4X2 + 5X3 <= 200
X1 + X3 <= 90
X1 + 2X2 <= 30
X1, X2, X3 >= 0
To convert the problem into standard form, we introduce slack variables (S1, S2, S3) for each constraint and rewrite the constraints as equalities:
2X1 + 4X2 + 5X3 + S1 = 200
X1 + X3 + S2 = 90
X1 + 2X2 + S3 = 30
Now, we have the following equations:
Objective function:
Z = 50X1 + 20X2 + 10X3 + 0S1 + 0S2 + 0S3
Constraints:
2X1 + 4X2 + 5X3 + S1 = 200
X1 + X3 + S2 = 90
X1 + 2X2 + S3 = 30
X1, X2, X3, S1, S2, S3 >= 0
Next, we will create a table representing the initial simplex tableau:
| X1 | X2 | X3 | S1 | S2 | S3 | RHS |
---------------------------------------
Z | 50 | 20 | 10 | 0 | 0 | 0 | 0 |
---------------------------------------
S1 | 2 | 4 | 5 | 1 | 0 | 0 | 200 |
---------------------------------------
S2 | 1 | 0 | 1 | 0 | 1 | 0 | 90 |
---------------------------------------
S3 | 1 | 2 | 0 | 0 | 0 | 1 | 30 |
---------------------------------------
To compute the optimal solution using the Simplex Method, we'll perform iterations by applying the simplex pivot operations until we reach an optimal solution.
Iterating through the simplex method steps, we can find the following tableau:
| X1 | X2 | X3 | S1 | S2 | S3 | RHS |
---------------------------------------
Z | 0 | 40 | 10 | 0 | 0 | -500| 1800|
---------------------------------------
S1 | 0 | 3 | 5 | 1 | 0 | -40 | 120 |
---------------------------------------
S2 | 1 | 0 | 1 | 0 | 1 | 0 | 90 |
---------------------------------------
X1 | 0 | 2 | 0 | 0 | 0 | -1 | 10 |
---------------------------------------
The optimal solution is Z = 1800, X1 = 10, X2 = 0, X3 = 0, S1 = 120, S2 = 90, S3 = 0.
Therefore, the maximum value of Z is 1800, and the values of X1, X2, and X3 that maximize Z are 10, 0, and 0, respectively.
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3 Write down the gradient of each line segment.
Answer:
One on the left = -1
One on the right = 1/3
Step-by-step explanation:
[tex]slope=\frac{rise}{run}[/tex]
One on the left = [tex]\frac{-3}{3} =-1[/tex]
One on the right = [tex]\frac{1}{3} =\frac{1}{3}[/tex]
The gradient of the first line and the second line will be 1 and 1/3, respectively.
What is the slope?The slope is the ratio of rising or falling and running. The difference between the ordinate is called rise or fall and the difference between the abscissa is called run.
The slope of the line is given as,
m = (y₂ - y₁) / (x₂ - x₁)
The gradient of the first line is given as,
m = 3 / 3
m = 1
The gradient of the second line is given as,
m = 1 / 3
The gradient of the first line and the second line will be 1 and 1/3, respectively.
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help i need this to be answerd
Answer:
What is it, you need help with?
Step-by-step explanation:
:)
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
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
Help ASAP!!!!!!!!!!!!!!!!!!!!!!
square
its the middle bit that looks darker than the rest.
A student was asked to find a 98% confidence interval for widget width using data from a random sample of size n = 21. Which of the following is a correct interpretation of the interval 12.3 < p < 31.2?
The interval 12.3 < p < 31.2 is a 98% confidence interval for the widget width, indicating that we can be 98% confident that the true population mean falls within this range based on the student's sample data of size n = 21.
The interval 12.3 < p < 31.2 is a 98% confidence interval for the widget width based on the student's sample of size n = 21.
Interpreting this confidence interval means that we can be 98% confident that the true population parameter, the mean widget width (p), falls between 12.3 and 31.2.
This confidence level suggests that if we were to take multiple random samples and calculate confidence intervals using the same method, approximately 98% of those intervals would capture the true population mean.
In other words, the student's sample data suggests that the true widget width has a high likelihood of falling within the range of 12.3 to 31.2 units.
However, it's important to note that this interpretation does not guarantee that the true value of the widget width is within this interval.
It simply provides a range of plausible values based on the sample data and the chosen confidence level.
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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
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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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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which of the following are least likely to be primary means of investigating normality in a distribution X?
a. graphically you data in a histogram and use eyeball to test to see if there in any asymmetry or skew
b. observe a computerized output for the Q-Q plot of the distribution
c. calculate the observed and expected z score values and determine any major deviations
d. start with a normal approximation as most variables are normal
d. "Start with a normal approximation as most variables are normal" is least likely to be primary means of investigating normality in a distribution X.
How to analyze all the options?
a. Graphically plot the data in a histogram and use the eyeball test to check for asymmetry or skew. This method involves visually examining the shape of the distribution by creating a histogram. Any noticeable asymmetry or skewness can indicate non-normality.
b. Observe a computerized output for the Q-Q plot of the distribution. A Q-Q plot compares the quantiles of the observed data with the quantiles of a theoretical distribution, such as the normal distribution. If the points on the Q-Q plot closely follow a straight line, it suggests the data is normally distributed.
c. Calculate the observed and expected z-score values and determine any major deviations. By transforming the data into z-scores and comparing them to the expected values under a normal distribution, deviations from normality can be identified. Significant deviations indicate departures from normality.
d. Start with a normal approximation as most variables are normal. This option suggests assuming normality without conducting specific tests or employing appropriate techniques to assess normality. While this approach may be reasonable in certain cases based on prior knowledge or theoretical considerations, it lacks a direct means of investigating normality.
d. "Start with a normal approximation as most variables are normal" is least likely to be primary means of investigating normality in a distribution X.
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Which of the following is the Laplace transformation L {f(t)} if f(t)=(sin(-4t) 2 ? 32 33 +645 O None of them s2 $4 + 3252 +256 $+32 S + 64s ) 16 54 + 3252 +256
The Laplace transform of f(t) = (sin(-4t))^2 is not among the given options.
The Laplace transform of f(t) = (sin(-4t))^2 is none of the given options. Let's find the correct Laplace transform for the given function.
The Laplace transform of a function f(t) is denoted as L{f(t)} and is defined as:
L{f(t)} = ∫[0 to ∞] f(t) * e^(-st) dt,
where s is the complex variable.
In this case, f(t) = (sin(-4t))^2. To find its Laplace transform, we need to apply the definition of the Laplace transform and evaluate the integral:
L{f(t)} = ∫[0 to ∞] (sin(-4t))^2 * e^(-st) dt.
However, before proceeding with the integration, we can simplify the function using trigonometric identities:
(sin(-4t))^2 = (-sin(4t))^2 = sin^2(4t).
Now, we can rewrite the Laplace transform as:
L{f(t)} = ∫[0 to ∞] sin^2(4t) * e^(-st) dt.
At this point, we can utilize a well-known trigonometric identity that relates the square of the sine function to a combination of 1 and cosine functions:
sin^2(θ) = (1 - cos(2θ))/2.
Applying this identity to our expression:
L{f(t)} = ∫[0 to ∞] (1 - cos(8t))/2 * e^(-st) dt.
Now, we can split this integral into two parts and simplify further:
L{f(t)} = (1/2) ∫[0 to ∞] e^(-st) dt - (1/2) ∫[0 to ∞] cos(8t) * e^(-st) dt.
The first integral represents the Laplace transform of 1, which is 1/s:
L{f(t)} = (1/2) * (1/s) - (1/2) ∫[0 to ∞] cos(8t) * e^(-st) dt.
The second integral can be evaluated using standard Laplace transform formulas. However, without additional information or constraints on the Laplace transform variable 's', we cannot simplify it further.
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31. Which equation can be used to describe the relationship between x and y shown in the graph below?
Study the graph below...
A. y = 45x − 1
B. y = −45x − 1
C. y = 54x − 1
D. y = −54x − 1
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.
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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the sixth term is 22 and the common difference is 6 what is the 15th term?
Answer:
76
Step-by-step explanation:
I made a chart to simplify it
1 | -8
2| -2
3| 4
4| 10
5| 16
6| 22
7| 28
8| 34
9| 40
10| 46
11| 52
12| 58
13| 64
14| 70
15| 76
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?
The sequence an = 5, an+1 = an + 8, a > 1 is an example of which of the following? a. a recursively-defined geometric sequence. b. none of these. c. a variation of the recursively-defined Factorial sequence. d. a recursively-defined arithmetic sequence. e. a recursively-defined Fibonacci-like sequence.
The sequence an = 5, an+1 = an + 8, a > 1 is an example of a recursively-defined arithmetic sequence. It does not fit the definitions of a geometric sequence, factorial sequence, or Fibonacci-like sequence. The correct option is (d) a recursively-defined arithmetic sequence.
An arithmetic sequence is characterized by a common difference between consecutive terms. In this case, each term (an+1) is obtained by adding a constant value of 8 to the previous term (an).
This satisfies the definition of an arithmetic sequence.
While the sequence does not fit the definition of a geometric sequence, factorial sequence, or Fibonacci-like sequence, it does follow the pattern of an arithmetic sequence.
The terms increase by a constant value of 8 with each step, making it a recursively-defined arithmetic sequence.
Therefore, the correct option is (d) a recursively-defined arithmetic sequence.
Note: A geometric sequence would have a common ratio between consecutive terms, which is not the case here. The factorial sequence involves multiplying terms by consecutive positive integers, and a Fibonacci-like sequence follows the pattern of adding the previous two terms to obtain the next term.
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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
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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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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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 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
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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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.
PLEASE ASAP, PLEASEEEE
Answer: (1,2)
Step-by-step explanation:
-2+3=1 and 4-2=2
(1,2)
question in the pic
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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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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