Introduction
In mathematics, a function is a relation between two sets of values, usually denoted as a set of input values and a set of output values. One of the important aspects of a function is its vertex, which is the highest or lowest point in a graph, depending on the specific type of function. The size and position of a graph’s vertex can be important when studying the properties of a function. In this paper, we will discuss three statements about a function and determine whether or not each statement is true.
Statement 1: The vertex of the function is at (–4,–15).
The first statement being discussed is that the vertex of the function is at (–4,–15). This statement is true. By looking at the graph of the function, it can be seen that the vertex of the function is indeed located at the point (–4,–15). At this point, the graph reaches its highest or lowest point.
Statement 2: The vertex of the function is at (–3,–16).
The second statement being discussed is that the vertex of the function is at (–3,–16). Unfortunately, this statement is false. By looking at the graph of the function, it can be seen that the vertex of the function is actually located at (–4,–15). The vertex is not located at (–3,–16).
Statement 3: The graph is increasing on the interval x > –3.
The third statement being discussed is that the graph is increasing on the interval x > –3. This statement is true. By looking at the graph, it can be seen that the graph is indeed increasing on the interval x > –3. On this interval, the y-values increase as the x-values increase.
Statement 4: The graph is positive only on the intervals where x < –7 and where x > 1.
The fourth statement being discussed is that the graph is positive only on the intervals where x < –7 and where x > 1. This statement is true. By looking at the graph, it can be seen that the graph is positive only on the intervals where x < –7 and where x > 1. On these intervals, the y-values are greater than 0.
Statement 5: The graph is negative on the interval x < –4.
The fifth statement being discussed is that the graph is negative on the interval x < –4. This statement is also true. By looking at the graph, it can be seen that the graph is indeed negative on the interval x < –4. On this interval, the y-values are less than 0.
Conclusion
In this paper, we discussed three statements about a function and determined whether or not each statement was true. We found that the first statement, that the vertex of the function is at (–4,–15), is true. We also found that the second statement, that the vertex of the function is at (–3,–16), is false. Furthermore, we found that the third, fourth, and fifth statements, that the graph is increasing on the interval x > –3, that the graph is positive only on the intervals where x < –7 and where x > 1, and that the graph is negative on the interval x < –4, respectively, are all true.
help i need this to be answerd
Answer:
What is it, you need help with?
Step-by-step explanation:
:)
Suppose you are interested in learning about how much time seventh grade students at your school spend outdoors on a typical school day. Select all the samples that are a part of the population you are interested in. a. The 20 students in a seventh grade math class. b. The first 20 students to arrive at school on a particular day. c. The seventh grade students participating in a science fair put on by the four middle schools in a school district. d. The 10 seventh graders on the school soccer team. e. The students on the school debate team.
Answer:
1. The 20 students in a seventh grade math class.
4. The 10 seventh graders on the school soccer team.
Step-by-step explanation:
I just submitted the form and these are the right answers :)
It might be not on time but still lol.
I AM SURE THOSE ARE RIGHT ANSWERS !!! :)))
The 20 students in a seventh grade math class, the seventh grade students participating in a science fair put on by the four middle schools in a school district and the 10 seventh graders on the school soccer team are the samples.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
The samples that are a part of the population of interest, which is seventh grade students at your school, are:
The 20 students in a seventh grade math class.
The seventh grade students participating in a science fair put on by the four middle schools in a school district (if the students are from your school).
The 10 seventh graders on the school soccer team (if they are from your school).
The samples that are not a part of the population of interest are:
The first 20 students to arrive at school on a particular day (as this may include students from different grades or schools).
The students on the school debate team (as this may include students from different grades or schools).
Hence, the 20 students in a seventh grade math class, the seventh grade students participating in a science fair put on by the four middle schools in a school district and the 10 seventh graders on the school soccer team are the samples
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Help ASAP!!!!!!!!!!!!!!!!!!!!!!
square
its the middle bit that looks darker than the rest.
If a hotel has 33 king-size beds, 24 queen-size beds, 25 double beds, and 24 twin beds, what is the probability that you will be given a queen-size or a twin-bed when you register, if the beds are chosen randomly?
The probability of being given a queen-size or a twin bed when registering is approximately 0.453 or 45.3%.
To calculate the probability of being given a queen-size or a twin bed, we need to determine the total number of queen-size and twin beds available, as well as the total number of beds overall.
Total number of queen-size beds = 24
Total number of twin beds = 24
Total number of beds = 33 (king-size) + 24 (queen-size) + 25 (double) + 24 (twin) = 106
To calculate the probability, we divide the number of favorable outcomes (queen-size or twin bed) by the number of possible outcomes (total number of beds).
Number of favorable outcomes = Number of queen-size beds + Number of twin beds = 24 + 24 = 48
Probability = Number of favorable outcomes / Total number of beds
Probability = 48 / 106 ≈ 0.453
Therefore, the probability of being given a queen-size or a twin bed when registering is approximately 0.453 or 45.3%.
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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
1. A psychologist was interested in the effect of Vitamin A deficiency on maze learning rats. A group of nine rats learned a simple maze. Next, these same rats were deprived of Vitamin A for six weeks and were then tested again on their ability to learn a second maze equal in difficulty to the first. The scores represent the number of maze errors made under the two treatment conditions. Perform an appropriate two-tailed test of the significance of the mean difference.
Answer: hello your question has some missing information attached below is the missing information
we fail to reject H0 hence μ1 = μ2mean difference ( d ) = -2.444Step-by-step explanation:
H0 : μ1 = μ2
Ha : μ1 ≠ μ2
where : μ1 = mean number of errors made in pretest
μ2 = mean number of errors made in post test
n = 9
attached below is the detailed solution
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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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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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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In the data set #2 {75,80,85,75,85}, what is the range?
PLEASE ASAP, PLEASEEEE
Answer: (1,2)
Step-by-step explanation:
-2+3=1 and 4-2=2
(1,2)
question in the pic
Hybrid and electric cars have gained in popularity in the last decade as a consequence of high gas prices. But their great gas mileages often come with higher car prices. There may be savings, but how much and how long before those savings are realized? Suppose you are considering buying a Honda Accord Hybrid, which starts around $31,665 and gets 48 mpg. A similarly equipped Honda Accord will run closer to $26,100 but will get 31 mpg. How long would it take for the Prius to recoup the price difference with its lower fuel costs, assuming you drive 800 miles per month? First, use the following formula for gas savings, where GM stands for gas mileage, to determine how far you will need to drive to recoup the cost difference in the vehicles. Use the known values and the average price of gas in your area to write a specific equation. $Gas is $4.35 Determine the type of equation that results, and then solve it algebraically. $Saved = $Gas x (distance driven) x ( GM now GM improved) Choose a Tesla (electric car) that has NO gas cost and compare it in a similar way to a gas-powered cari, the Honda Accord. How long will it take to recoup the price difference for the miles you drive per month? Assume you still drive 800 miles a month. Be sure to consider TOTAL COST of each car. Explain what you thought TOTAL COST meant in the previous question.
The Honda Accord Hybrid would recoup the price difference with its lower fuel costs in approximately 6.8 years when driving 800 miles per month.
How long does it take for the Honda Accord Hybrid to recover the price difference with its lower fuel costs, assuming a monthly mileage of 800 miles?The Honda Accord Hybrid, priced at $31,665, has a fuel efficiency of 48 mpg, while the gas-powered Honda Accord, priced at $26,100, has a fuel efficiency of 31 mpg. To determine the distance that needs to be driven to recoup the cost difference, we can use the formula: $Saved = $Gas x (distance driven) x (GM now / GM improved). Considering the average gas price of $4.35, we can substitute the values into the formula and solve for the distance driven.
Using algebraic calculations, we find that the distance needed to recoup the price difference is approximately 50,472 miles. With a monthly mileage of 800 miles, it would take approximately 63 months or 6.8 years to recover the cost difference between the two vehicles.
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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.
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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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
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
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
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
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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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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help your girl outtt !!!
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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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
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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Place each function below in the appropriate cell to show the transformation from f to g. (Desmos)
Answer:
First row: 2,1
Second row: 3,4
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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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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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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