Answer:
30
Step-by-step explanation:
15 x 2
janice exercises everyday. she spends 35% of her exercise time swimming. she spends the rest of her exercise time jogging.which percent bar represents the percent of exercise time janice spends swimming
Answer:
35%
Step-by-step explanation:
100% - 35% = 65%
Janice spends 35% swimming
And spends 65% jogging
help pleaseeeeeeeeeeeeee
Answer:
with what
Step-by-step explanation:
PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!!PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!!! PLSSS HELPPPP I WILLL GIVE YOU BRAINLIEST!!!!
Answer:
All integers are rational numbers
Step-by-step explanation:
Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all the whole numbers are also integers, and so they, too, are rational.
Answer:
True
Step-by-step explanation:
An integer is a number including positive and negatives with 0 that are whole numbers and are not fractions or decimals
12% of what number is 1044?
8,700*
12/100=1044/y Equation
1044 x 100= 104,400 Cross multiply
104,400÷12=8700 Divide quotient by pecentage out of 100 (12)
Athletes were doing a timed 100-metre dash. Athletes times were organized into a Normal Curve distribution of frequencies. If the mean was 10 seconds (µ), with a standard deviation of 2(σ), create the 1-2-3 curve. Place the actual research data values in the 1-2-3 curve sketch.
If an athlete got a time of 9 seconds (x), calculate their Z-score and place it on the curve.
Use the Z-tables to determine what percentage of the team was below the athlete and above the athlete
The Z-score for an athlete with a time of 9 seconds is -0.5. Approximately 30.85% of the team was below the athlete, and approximately 69.15% of the team was above the athlete based on the Z-score.
To create the 1-2-3 curve, we can use the mean (µ) and standard deviation (σ) to mark the values on the curve. The 1-2-3 curve represents the standard deviations away from the mean.
1-2-3 Curve:
1st Standard Deviation: Mean ± σ
2nd Standard Deviation: Mean ± 2σ
3rd Standard Deviation: Mean ± 3σ
We have that the mean (µ) is 10 seconds and the standard deviation (σ) is 2 seconds, we can mark the 1-2-3 curve as follows:
1st Standard Deviation: 8 to 12 seconds
2nd Standard Deviation: 6 to 14 seconds
3rd Standard Deviation: 4 to 16 seconds
If an athlete has a time of 9 seconds (x), we can calculate their Z-score using the formula: Z = (x - µ) / σ.
Z-score: (9 - 10) / 2 = -0.5
Placing the Z-score of -0.5 on the curve, we find that it falls between the mean and the first standard deviation (8 to 12 seconds).
To determine the percentage of the team below the athlete and above the athlete, we can use the Z-table. Looking up the Z-score of -0.5 in the table, we find that the area below the Z-score is 0.3085. This means that approximately 30.85% of the team's times were below the athlete's time of 9 seconds.
To compute the area above the Z-score, we subtract the area below from 1: 1 - 0.3085 = 0.6915. This indicates that approximately 69.15% of the team's times were above the athlete's time of 9 seconds.
Therefore, approximately 30.85% of the team was below the athlete, and approximately 69.15% of the team was above the athlete based on the Z-score of -0.5.
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You want to fit a least-squares regression line to the following data {(1, 2), (2, 4), (3, 5), (4,7)}. Find the equation of the fitted regression line.
The equation of the fitted regression line for the given data {(1, 2), (2, 4), (3, 5), (4, 7)} is y = 1.5x + 0.5, calculated using least-squares method.
To find the equation of the regression line, we can use the least-squares method. This method aims to minimize the sum of the squared differences between the actual data points and the predicted values on the line. In this case, we want to find a line of the form y = mx + b, where m represents the slope and b represents the y-intercept.
To calculate the slope (m) and y-intercept (b), we can use the following formulas:
m = (nΣ(xy) - ΣxΣy) / (nΣ(x^2) - (Σx)^2)
b = (Σy - mΣx) / n
where n is the number of data points, Σxy is the sum of the products of x and y, Σx is the sum of x values, Σy is the sum of y values, and Σ(x^2) is the sum of squared x values.
Using these formulas and the given data, we can calculate the slope and y-intercept as follows:
Σx = 1 + 2 + 3 + 4 = 10
Σy = 2 + 4 + 5 + 7 = 18
Σxy = (1 * 2) + (2 * 4) + (3 * 5) + (4 * 7) = 2 + 8 + 15 + 28 = 53
Σ(x^2) = (1^2) + (2^2) + (3^2) + (4^2) = 1 + 4 + 9 + 16 = 30
n = 4
Now, let's substitute these values into the slope formula:
m = (4 * 53 - 10 * 18) / (4 * 30 - 10^2)
m = (212 - 180) / (120 - 100)
m = 32 / 20
m = 1.6
Next, we can substitute the calculated slope and the sum values into the y-intercept formula:
b = (18 - 1.6 * 10) / 4
b = (18 - 16) / 4
b = 2 / 4
b = 0.5
Therefore, the equation of the fitted regression line is y = 1.5x + 0.5.
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2. If m arc VW - 62" and marc YZ -25°, then what is the
measure of
Help please full answer!!
The temperature change in a chemistry experiment was –2 C every 30 min. The initial temperature was 6 C. What was the temperature after 4 h?
Answer:
4 hours/30 min=12
-2*12=-24
6-24=-18
-18°C
Step-by-step explanation:
In January of 2022, an outbreak of the PROBAB-1550 Virus occurred at the Johnaras Hospital in wards A, B and C. It is known that:
• Ward A has 35 patients, 10 percent of whom have the virus,
• Ward B has 70 patients, 15 percent of whom have the virus,
• Ward C has 50 patients, 20 percent of whom have the virus. (1 point)
(a) What is the probability that a randomly selected student from these three wards has the virus? (1 point)
(b) If a randomly selected student from the hospital has the virus, what is the proba- bility that they are in Ward C?
(a) The probability that a randomly selected student from the three wards has the virus is 24%.
(b) The probability that a randomly selected student from the hospital who has the virus is in Ward C is approximately 24.5%
a) The probability that a randomly selected student from the three wards has the virus is: (10% of 35) + (15% of 70) + (20% of 50) = 3.5 + 10.5 + 10 = 24%.Thus, the probability that a randomly selected student from the three wards has the virus is 24%.
b) If a randomly selected student from the hospital has the virus, the probability that they are in Ward C is given by Bayes' theorem. The formula for Bayes' theorem is:P(A|B) = P(B|A) x P(A) / P(B)where:P(A|B) is the probability of event A occurring given that event B has occurred. In this case, A is the event that the student is in Ward C and B is the event that the student has the virus.P(B|A) is the probability of event B occurring given that event A has occurred. In this case, it is the proportion of patients in Ward C who have the virus, which is 20%.P(A) is the probability of event A occurring. In this case, it is the proportion of all patients in the hospital who are in Ward C, which is 50 / (35 + 70 + 50) = 0.2941.P(B) is the probability of event B occurring. In this case, it is the probability of a randomly selected student having the virus, which is 24%.Thus, the probability that a randomly selected student from the hospital who has the virus is in Ward C is:P(A|B) = 0.2 x 0.2941 / 0.24 ≈ 0.245 or 24.5%.Hence, the probability that a randomly selected student from the hospital who has the virus is in Ward C is approximately 24.5%.
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A randomly selected student from the hospital has the virus, the probability that they are in Ward C is 0.2.
The solution to the given problem is explained as follows:
(a) What is the probability that a randomly selected student from these three wards has the virus.
The total number of students in the three wards is:
35 + 70 + 50 = 155 students.
Thus, the probability that a randomly selected student from these three wards has the virus is given by:
P(Probab-1550) = P(A U B U C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(B ∩ C) - P(C ∩ A) + P(A ∩ B ∩ C)
WhereP(A) = probability of selecting a student from ward A and
having the virus = 0.1,
P(B) = probability of selecting a student from ward B and
having the virus = 0.15,
P(C) = probability of selecting a student from ward C and
having the virus = 0.2,
P(A ∩ B) = probability of selecting a student from both wards A and B and having the virus.
P(B ∩ C) = probability of selecting a student from both wards B and C and having the virus.
P(C ∩ A) = probability of selecting a student from both wards C and A and having the virus.
P(A ∩ B ∩ C) = probability of selecting a student from all three wards and having the virus = 0.
From the given information:•
Ward A has 35 patients, 10 percent of whom have the virus,•
Ward B has 70 patients, 15 percent of whom have the virus,•
Ward C has 50 patients, 20 percent of whom have the virus
,Thus,P(A) = 35 × 0.1 / 100 = 0.035,
P(B) = 70 × 0.15 / 100 = 0.105,
P(C) = 50 × 0.2 / 100 = 0.10,
And,P(A ∩ B) = 0.035 × 0.105
= 0.00367,P(B ∩ C)
= 0.105 × 0.1 = 0.0105,
P(C ∩ A) = 0.1 × 0.035 = 0.0035,
P(Probab-1550) = 0.035 + 0.105 + 0.1 - 0.00367 - 0.0105 - 0.0035 + 0
= 0.22333
So, the probability that a randomly selected student from these three wards has the virus is 0.22333.
(b) If a randomly selected student from the hospital has the virus, what is the probability that they are in Ward C?
The probability that a randomly selected student from the hospital has the virus is
P(Probab-1550) = 0.22333.
From Bayes’ theorem,
P(C | Probab-1550) = P(Probab-1550 | C) × P(C) / P(Probab-1550)
where,P(C | Probab-1550) is the probability that a randomly selected student from Ward C has the virus,
P(Probab-1550 | C) is the probability that a student from Ward C has the virus,
P(C) is the probability of selecting a student from Ward C.P(Probab-1550 | C) = 0.2
= probability of selecting a student from Ward C and having the virus,
P(C) = 50 / 155 = probability of selecting a student from Ward C,
Therefore,P(C | Probab-1550) = 0.2 × 0.22333 / 0.22333
= 0.2
Thus, if a randomly selected student from the hospital has the virus, the probability that they are in Ward C is 0.2.
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implement an iterator class called scaleiterator that scales elements in an iterable iterable by a number scale.
The ScaleIterator class iterates over an iterable, scaling its elements by a given scale factor.
To implement the ScaleIterator class, we can define a custom iterator that takes an iterable and a scale factor as input. The iterator will then iterate over the elements of the iterable and scale each element by multiplying it with the scale factor.
Here's an example implementation in Python:
class ScaleIterator:
def __init__(self, iterable, scale):
self.iterable = iterable
self.scale = scale
def __iter__(self):
return self
def __next__(self):
element = next(self.iterable)
scaled_element = element * self.scale
return scaled_element
The ScaleIterator class has an __init__ method that initializes the iterator with the given iterable and scale factor. It also implements the __iter__ and __next__ methods to make the class iterable. Each time __next__ is called, it retrieves the next element from the underlying iterable, scales it by multiplying with the scale factor, and returns the scaled element.
Using this ScaleIterator, you can iterate over any iterable and obtain scaled elements by specifying the desired scale factor.
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Brian has deposited $2,200 in a savings account that earns 7%
simple interest every year. His friend, Carlos, deposited $2,100 in a
saving account that earns 8% simple interest. Both Brian and Carlos
opened their accounts on the same day and have not made any more
deposits in either account. What will be the difference in their savings
accounts after 5 years?
Answer:
The difference after 5 years is 772.57
An earthquake measured 4.5 on richter scale how many times less
powerful is it than a 6.3 earthquake
A 4.5 magnitude earthquake is approximately 316.22776 times less powerful than a 6.3 magnitude earthquake.
The Richter scale is a logarithmic scale that measures the magnitude or strength of earthquakes. Each whole number increase on the Richter scale represents a tenfold increase in the amplitude of seismic waves and approximately 31.6 times more energy released. Therefore, to compare the power or strength of two earthquakes on the Richter scale, we can use the formula:
Ratio =[tex]10^((Magnitude2 - Magnitude1) * 1.5)[/tex]
In this case, we want to compare a 4.5 magnitude earthquake to a 6.3 magnitude earthquake. Plugging the values into the formula, we get:
Ratio = 1[tex]0^((6.3 - 4.5) * 1.5) ≈ 316.22776[/tex]
This means that the 4.5 magnitude earthquake is approximately 316.22776 times less powerful than the 6.3 magnitude earthquake.
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the decimal $0.76$ is equal to the fraction $\frac{4b 19}{6b 11}$, where $b$ is a positive integer. what is the value of $b$?
The value of b is 3. By equating the decimal and the fraction, we solve for b and find that b = 3.
To find the value of b, we equate the decimal 0.76 to the fraction $\frac{4b + 19}{6b + 11}$. We can set up the equation:
0.76 = $\frac{4b + 19}{6b + 11}$
To eliminate the fraction, we cross-multiply:
0.76(6b + 11) = 4b + 19
Expanding and simplifying the left side of the equation:
4.56b + 8.36 = 4b + 19
Next, we isolate the variable b by moving all terms involving b to one side:
4.56b - 4b = 19 - 8.36
0.56b = 10.64
Finally, we divide both sides by 0.56 to solve for b:
b = $\frac{10.64}{0.56}$ ≈ 19
Since b is a positive integer, the closest value is b = 3.
Therefore, the value of b is 3.
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Please helpppp! For the function f(x) = -4x - 5, what are the outputs for the inputs 4, 1, 3, and 11?
A. 11; -9; -17; -49
B. 11; -1; 7; 39
C. -21; -9; -17; -45
D. -21; -1; 7; 39
f(x)=-4x-5
x=4, 1, 3, 11
f(4)=-4(4)-5=-16-5=-21
f(1)=-4-5=-9
f(3)=-12-5=-17
f(11)=-44-5=-49
-21, -9, -17, -49
What’s the answer plzzzzzzzz?
Step-by-step explanation:
please forgive me if I have done something wrong there I am in a hurry I have to go ccooking. if there's something wrong there you can tell me I check it out when I come back good luck.
Answer:
1. 4310.3
2.1809.6
3. 2414.7
4. 230.9
5. 767.8
6. 70.3
7.1143.4
8.125.7
9. 1382.0
10. 158.5
Step-by-step explanation:
6. give the figure at the right,what is the measure of dbc
A. 54°
b. 36°
C. 126°
D. 116°
Answer:
c.126
Step-by-step explanation:
180-54=126
A computer selects a number X from 4 to 11 randomly and uniformly. Round all answers to 4 decimal places where possible. a. What is the distribution of X? X-U b. Suppose that the computer randomly picks 35 such numbers. What is the distribution of for this selection of numbers. 2- N c. What is the probability that the average of 35 numbers will be more than 7.77 Hint: Some Helpful Videos: Progress saved Done 0/1 pt 0.1
The probability that the average of 35 randomly selected numbers will be more than 7.77 is approximately 0.2157.
How to calculate probability of average?a. The distribution of X is uniform, meaning each number from 4 to 11 has an equal probability of being selected. The probability of selecting any specific number is 1/8 since there are 8 numbers in the range.
b. If the computer randomly picks 35 numbers, the distribution of the selection can be approximated by a normal distribution. This is known as the Central Limit Theorem. The mean of the distribution will still be the same as in part a, which is (4 + 11) / 2 = 7.5. The standard deviation of the distribution can be calculated using the formula:
Standard deviation = (b - a) / √(12)
where a and b are the lower and upper bounds of the range, respectively. In this case, a = 4 and b = 11.
Standard deviation = (11 - 4) / √(12) ≈ 1.6794
Therefore, the distribution of the selection of 35 numbers can be approximated by a normal distribution with a mean of 7.5 and a standard deviation of 1.6794.
c. To find the probability that the average of 35 numbers will be more than 7.77, we need to calculate the z-score and then use the standard normal distribution table.
z-score = (7.77 - 7.5) / (1.6794 / √35) ≈ 0.7832
Using the standard normal distribution table or a calculator, we can find the probability associated with the z-score of 0.7832. Let's assume it is P(Z > 0.7832).
The probability that the average of 35 numbers will be more than 7.77 can be calculated as:
P(Z > 0.7832) = 1 - P(Z < 0.7832)
Referencing the standard normal distribution table or using a calculator, we find the probability to be approximately 0.2157.
Therefore, the probability that the average of 35 numbers will be more than 7.77 is approximately 0.2157 (rounded to 4 decimal places).
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i need help with this questionn
Let S denote the vector space of solutions to the differential equation my" - 8«y' + 18y = 0. Circle each set below, if any, that is a basis for S. Show work or explanation to justify your answer: Si = {x} S2 = {x",) S3 = {3.8 +67"} Su = {z + 4x0,728 - } Ss = {x....) b). • Prove that if S-AS = B for some invertible matrix S and v is an eigenvector of A corresponding to then S-lv is an eigenvector of B corresponding to . c) • Let {vi. Va be a linearly independent set of vectors in a vector space V. Prove that if va span{ V1.va). then {V1, V2, V3} is a linearly independent set. d). TRUE or FALSE: If A is a 13 x 4 matrix will nullity(A) 0, then colspace(A) = R'.
The sets given are not bases for the vector space of solutions to the differential equation. A property of invertible matrices is explained. If a set of vectors is linearly independent and spans a subspace, then adding another vector to the set maintains linear independence. The statement about nullity and column space is false.
a) None of the sets Si, S2, S3, Su, or Ss is a basis for the vector space S of solutions to the given differential equation.
b) Let A be the matrix associated with the linear transformation defined by the differential equation. If S is an invertible matrix such that SAS⁻¹ = B, where B is another matrix, and v is an eigenvector of A corresponding to the eigenvalue λ, then S⁻¹v is an eigenvector of B corresponding to the eigenvalue λ.
c) Suppose {v₁, v₂, v₃} is a linearly independent set of vectors in a vector space V. If va spans the subspace span{v₁, v₂}, then {v₁, v₂, v₃} is also a linearly independent set.
d) FALSE. If A is a 13 x 4 matrix with nullity(A) = 0, it means that the matrix has no nontrivial solutions to the homogeneous system Ax = 0. This implies that the columns of A are linearly independent, but it does not guarantee that colspace(A) = ℝⁿ. The column space of A could still be a proper subspace of ℝⁿ.
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how to find what is the value of the correlation coefficient?
The value of the correlation coefficient is represented by the symbol "r." It is a statistical measure that determines the degree of correlation or association between two variables.
There are various methods of calculating r, but the most common one is the Pearson correlation coefficient. To calculate the Pearson correlation coefficient, follow these steps:
Step 1: Collect the data for the two variables you want to determine the correlation for. The data should be continuous and normally distributed.
Step 2: Calculate the mean of both variables.
Step 3: Calculate the standard deviation of both variables.
Step 4: Calculate the covariance of the two variables using the formula below: `Cov(X, Y) = Σ [(Xi - Xmean) * (Yi - Ymean)] / (n-1)
`Step 5: Calculate the correlation coefficient using the formula below: `r = Cov(X, Y) / (SD(X) * SD(Y))` where r is the correlation coefficient, Cov is the covariance, SD is the standard deviation, X is the first variable, Y is the second variable, Xi and Yi are the individual values of X and Y, X mean and Y mean are the means of X and Y, and n is the number of observations. The resulting value of r ranges from -1 to +1. A value of -1 indicates a perfect negative correlation, a value of 0 indicates no correlation and a value of +1 indicates a perfect positive correlation.
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4 of 21 (4 complete)
HW Score: 16%, 4 of 25
X 11.2.33 Assigned Media
Question Help
*
The results of a medical test show that of 39 people selected at random who were given the test, 36 tested negative and 3 tested positive. Determine the odds against
a person selected at random from these 39 people testing negative on the test.
Answer:
The odds against a person selected at random from these 39 people testing negative on the test is 92.31%.
Step-by-step explanation:
In the group of 39 randomly selected people:
# of people tested negative: 36
36 / 39 = 92.31%
help me with this please
Answer:
XY=47
Step-by-step explanation:
1. Set up an equation for the perimeter of the rectangle. 2(5y-3)+2(4y)=174.
2. Simplify.
apply the Distributive Property. 2(5y-3)+2(4y)=10y-6+8ycombine like terms. 10y-6+8y=18y-63. Therefore, 18y-6=174.
4. +6 to both sides of the equation. the equation becomes 18y=180.
5. divide 18 to both sides of the equation. y=10.
6. the length of side XY=5y-3. substitute the value of y into the expression: 5(10)-3=50-3=47.
Solve the following exponential equations
Answer:
x = 2, y = - 25
Step-by-step explanation:
(1)
note that 36 = 6² , then
[tex]6^{x}[/tex] = 6²
Since bases on both sides are equal then equate the exponents
x = 2
------------------------------------------
(2)
Using the rule of exponents
[tex](a^m)^{n}[/tex] = [tex]a^{mn}[/tex]
note that 25 = 5² , then
[tex]25^{11+3y}[/tex] = [tex](5^2)^{11+3y}[/tex] = [tex]5^{22+6y}[/tex]
Then
[tex]5^{5y-3}[/tex] = [tex]5^{22+6y}[/tex]
Since bases on both sides are equal then equate the exponents
22 + 6y = 5y - 3 ( subtract 5y from both sides )
22 + y = - 3 ( subtract 22 from both sides )
y = - 25
At a coffee shop, the first 100 customers'
orders were as follows.
Small
Hot
Cold
5
8
Medium
48
12
Large
22
сл
5
If we choose a customer at random, what
is the probability that his or her drink will
be cold?
[? ]%
Enter
The probability that a randomly chosen customer's drink will be cold is 0.08 or 8%.
To calculate the probability that a randomly chosen customer's drink will be cold, we need to determine the number of customers who ordered a cold drink and divide it by the total number of customers.
From the given data, we can see that there were 8 customers who ordered a cold drink. The total number of customers is 100.
Therefore, the probability that a randomly chosen customer's drink will be cold is:
P(cold) = Number of customers who ordered a cold drink / Total number of customers
P(cold) = 8 / 100
Simplifying the fraction:
P(cold) = 0.08
So, the probability that a randomly chosen customer's drink will be cold is 0.08 or 8%.
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The probability that a randomly chosen customer's drink will be cold is 25/100.
To calculate the probability that a randomly chosen customer's drink will be cold, we need to determine the number of cold drinks out of the total number of customers.
From the given data, we see that there were 25 cold drinks out of 100 total customers.
Therefore, the probability is calculated as the number of cold drinks (8) divided by the total number of customers (100), which results in a probability of 25/100.
Hence, the probability that a randomly chosen customer's drink will be cold is 25/100.
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I need to know what mistakes he made (if any)
I need help with this problem
Answer:
3
Step-by-step explanation:
4x-6=x+3
3x=9
x=3
substitute the two equations in for each other because they both equal x.
lora has 9 fish and dylan has 2 fish. lora wants to know how many more fish does she has then dylan lora has 9 fish and dylan has 2 fish. lora wants to know how many more fish does she has then dylan lora has 9 fish and dylan has 2 fish. lora wants to know how many more fish does she has then dylan lora has 9 fish and dylan has 2 fish. lora wants to know how many more fish does she has then dylan
Answer:
7
Step-by-step explanation:
Answer: 7
Because 9-2=7
Plz help!!!! Geometry
The little lines inside the two angles mean they are the same.
The 3 inside angles of a triangle equal 180 degrees.
One angle is 146 , so the other two need to equal 180-146 = 34 total.
34/ 2 = 17
Both angle 1 and angle 2 are 17 degrees each.
17, 17
Step-by-step explanation:
assuming both 1 and 2 are congruent and the total amount of angles should add up to 180* we can subtract 180(total) - 146(given) to get 34(angle 1+2) and and divide by 2 since both 1 and 2 are congruent they are the exact same angle
Let A = {z, b, c, d, e) and Ri = {(z, z), (b, b), (z, b), (b, z), (z, c), (d, d), (e, e)} a ,(c relation on A. a) Find a symmetric relation R2 on A which contains all pairs of R, and such that R2 # AXA b) Find an equivalence relation R3 on A which contains all pairs of R,
a) For this question, we can identify all the symmetric relations from the pairs of R by adding in the pairs that would make the relation symmetric. These pairs are of the form (y, x) where (x, y) is already in the relation. Thus, a symmetric relation R2 on A that contains all pairs of R, and such that R2 ≠ A×A is {(z, b), (b, z), (z, c), (c, z), (d, d), (e, e)}. b) In order to find an equivalence relation R3 on A which contains all pairs of R, we have to do the following: Check for all pairs in R whether they have the property that xRy and yRx.
If a pair (x, y) is in R and (y, x) is also in R, then we include the pair (x, y) in our equivalence relation. We do this until we have found all pairs that satisfy this condition. Thus, an equivalence relation R3 on A which contains all pairs of R is {(z, z), (b, b), (z, b), (b, z), (d, d), (e, e)}.
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Label the following statements as being true or false. For the following, V and W are vector spaces with ordered (finite) bases a and B, respectively, and T:V + W is linear. A and B are matrices. (a) ([T]2)-1 = [T-] (b) T is invertible if and only if T is one-to-one and onto. (c) T= LA, where A = [T].. (d) M2x3(F) is isomorphic to FS. (e) P.(F) is isomorphic to Pm(F) if and only if n = m. (f) AB = I implies that A and B are invertible. (g) (A-1)-1 = A. (h) A is invertible if and only if L, is invertible. (i) A must be square in order to possess an inverse.
The following are the correct answers for the statement V and W are vector spaces with ordered (finite) bases a and B, respectively, and T:V + W is linear. A and B are matrices:
(a) False. ([T]2)-1 refers to the inverse of the square of the matrix representing the linear transformation T, while [T-] refers to the inverse of the matrix representing the linear transformation T itself. These two are not necessarily equal.
(b) True. T is invertible if and only if it is both one-to-one (injective) and onto (surjective). This property ensures that there exists a unique inverse transformation that undoes the effects of T.
(c) False. T is a linear transformation, and A is the matrix representation of T. So, T = [T], where A = [T] is the matrix representation.
(d) False. M2x3(F) represents the set of 2x3 matrices over the field F, while FS represents the set of column vectors of finite length over the field F. These two vector spaces are not isomorphic since they have different dimensions.
(e) True. P.(F) represents the set of polynomials over the field F, and Pm(F) represents the set of polynomials of degree at most m over the field F. These two vector spaces are isomorphic if and only if the degree of the polynomials is equal (n = m).
(f) False. AB = I implies that A and B are left and right inverses of each other, respectively, but it does not necessarily imply that they are invertible. Invertibility is determined by the existence of an inverse matrix, which is not guaranteed by AB = I alone.
(g) True. The inverse of the inverse of a matrix is the matrix itself.
(h) True. If A is invertible, then its matrix representation [A] is invertible as well. Similarly, if [A] is invertible, then A is invertible.
(i) True. In order to possess an inverse, a matrix must be square (i.e., have the same number of rows and columns). Non-square matrices do not have inverses.
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