In the case of discrete random variables, the expectation of a function is defined as the sum of the function's values multiplied by their probabilities:
E(aX + bY + c) = ∑(aX + bY + c)P(X,Y)
We can break down the sum using properties of summation:
= a∑XP(X,Y) + b∑YP(X,Y) + c∑P(X,Y)
Since the sum of probabilities over all events equals 1:
= aE(X) + bE(Y) + c
For the continuous case, the expectation of a function is defined as the integral of the function's values multiplied by the joint probability density function (PDF):
E(aX + bY + c) = ∫∫(aX + bY + c)f(X,Y)dXdY
We can break down the integral using properties of integration:
= a∫∫Xf(X,Y)dXdY + b∫∫Yf(X,Y)dXdY + c∫∫f(X,Y)dXdY
Again, since the integral of the joint PDF over all events equals 1:
= aE(X) + bE(Y) + c
Thus, we have shown that for both discrete and continuous cases, the linearity of expectation holds:
E(aX + bY + c) = aE(X) + bE(Y) + c
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consider the function 1/1-x^3 write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. for example, if the series were , you would write . also indicate the radius of convergence. partial sum:
The partial sum for the power series which represents the function 1/(1-x³) consisting of the first 5 nonzero terms is: 1 + x³ + x⁶ + x⁹ + x¹² and the radius of convergence is 1.
The formula for the partial sum of a power series is given by:
Sₙ(x) = a₀ + a₁x + a₂x² + ... + aₙxⁿ
where a₀, a₁, a₂, ..., aₙ are the coefficients of the power series.
In this case, we can use the formula for the geometric series to find the coefficients:
1/(1-x³) = 1 + x³ + x⁶ + x⁹ + x¹² + ...
a₀ = 1
a₁ = 1
a₂ = 1
a₃ = 0
a₄ = 0
and so on.
Therefore, the first 5 nonzero terms of the power series are 1, x³, x⁶, x⁹, and x¹².
The radius of convergence for this power series can be found using the ratio test:
lim┬(n → ∞)|aₙ₊₁/aₙ| = lim┬(n → ∞)|x³/(1-x³)| = 1
Since the limit equals 1, the radius of convergence is 1.
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I need help with this problem.
Answer:
1414 tickets, in explanation
Hope this helps!
Step-by-step explanation:
1 ticket = $9.50
? tickets = $13,433
13,433 ÷ 9.50 = 1414
9.50 × 1414 = 13,433
1 ticket × 1414 = ? tickets
? tickets = 1414 tickets
A factory
produces cylindrical metal bar. The production process can be
modeled by normal distribution with mean length of 11 cm and
standard deviation of 0.25 cm.
There is 14% chance that a randomly selected cylindrical metal bar has a length longer than K. What is the value of K?
To solve this problem, we need to find the z-score corresponding to the 14th percentile of the normal distribution. We can then use this z-score to find the corresponding value of K.
First, we find the z-score corresponding to the 14th percentile using a standard normal distribution table or calculator. The 14th percentile is equivalent to a cumulative probability of 0.14, which corresponds to a z-score of approximately -1.08.
Next, we use the formula z = (x - μ) / σ to find the corresponding value of K. Rearranging this formula, we get x = μ + z * σ. Plugging in the values we know, we get:
K = 11 + (-1.08) * 0.25
K = 10.73 cm
Therefore, there is a 14% chance that a randomly selected cylindrical metal bar has a length longer than 10.73 cm.
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A group of students was surveyed in a middle school class. They were asked how many hours they work on math homework each week. The results from the survey were recorded.
Number of hours Total number of students
0 1
1 3
2 2
3 5
4 9
5 7
6 3
Determine the probability that a student studied for 5 hours.
23.0
0.70
0.23
0.16
The probability that a student studied for 5 hours is given as follows:
0.23.
How to calculate a probability?A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.
The total number of students in this problem is given as follows:
1 + 3 + 2 + 5 + 9 + 7 + 3 = 30.
Out of those 30 students, 7 studied five hours, hence the probability is given as follows:
p = 7/30 = 0.23.
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3 < 3x + 9 < 24 solve the compound inequality
The answer of the compound inequality 3 < 3x + 9 < 24 is -2 < x < 5.
To solve the compound inequality 3 < 3x + 9 < 24, we need to isolate the variable x.
First, we will subtract 9 from all parts of the inequality:
3 - 9 < 3x + 9 - 9 < 24 - 9
-6 < 3x < 15
Next, we will divide all parts of the inequality by 3 (remembering to flip the direction of the inequality if we divide by a negative number):
-6/3 < 3x/3 < 15/3
-2 < x < 5
Therefore, the solution to the compound inequality 3 < 3x + 9 < 24 is -2 < x < 5.
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What is the value of R?
Given a ray passing through a line at an angle of 29 degrees, the angle opposite to it (angle R) can be found by subtracting 29 degrees from 180 degrees. Therefore, the value of angle R is 151 degrees.
We are given that a ray passes through a line, making an angle of 29 degrees with the line. Let us represent this situation as follows
The angle R represents the angle opposite to the angle of 29 degrees. Since the ray and the line form a straight line, their angles add up to 180 degrees. Therefore, we can write
angle R + 29 degrees = 180 degrees
To solve for angle R, we can subtract 29 degrees from both sides of the equation
angle R = 180 degrees - 29 degrees
Simplifying the expression, we get
angle R = 151 degrees
Therefore, the value of angle R is 151 degrees.
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what is the median for the data set 2, 3, 4, 5, 6, 7, 8, 8, 8, 9, 10, 11, 12, 12, 13, 14.
Answer:9.5
Step-by-step explanation:
Answer: 8
Step-by-step explanation:
The median of this data set is 8. If you cross one number from both sides at the same time, you will eventually come to the middle of the data set, which is 8.
In an English literature course, the professor asks students to read three books by selecting one memoire, one book of poetry, and one novel to read. The students can select these books from a list of 8 memoires, 9 poetry books, and 4 novels. How many different ways can a student select their reading assignment of three books?
In an English literature course, the professor asks students to read three books by selecting one memoire, one book of poetry, and one novel to read. The students can select these books from a list of 8 memoires, 9 poetry books, and 4 novels.
To determine how many different ways a student can select their reading assignment of three books, we will use the multiplication principle.
1. Choose one memoire: There are 8 memoires to choose from, so there are 8 ways to make this choice.
2. Choose one book of poetry: There are 9 poetry books to choose from, so there are 9 ways to make this choice.
3. Choose one novel: There are 4 novels to choose from, so there are 4 ways to make this choice.
Now, multiply the number of choices for each step together to find the total number of ways to select the reading assignment:
8 (memoires) x 9 (poetry books) x 4 (novels) = 288 different ways to select the reading assignment of three books.
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point in rabbits, brown fur (B) is dominant to white fur (b) and short fur (H) is dominant to long fur (h). A brown. long-furred rabbit (Bbhh) is crossed with a white. short-furred rabbit (bbhh). Both the Band H traits assort independently from one another. What probability of the offspring will be brown with long fur?
The probability of the offspring being brown with long fur is 25%.
To determine the probability of offspring being brown with long fur from a cross between a brown, long-furred rabbit (Bbhh) and a white, short-furred rabbit (bbHh), we will use the terms dominant, recessive, and independent assortment.
Step 1: Set up the Punnett squares for each trait separately.
For fur color (B and b alleles):
Bb (brown, long-furred rabbit)×bb (white, short-furred rabbit)
Resulting in offspring genotypes:
Bb (brown fur)
Bb (brown fur)
bb (white fur)
bb (white fur)
For fur length (H and h alleles):
hh (brown, long-furred rabbit)×Hh (white, short-furred rabbit)
Resulting in offspring genotypes:
Hh (short fur)
Hh (short fur)
hh (long fur)
hh (long fur)
Step 2: Calculate the probabilities for each trait.
For brown fur: 2 out of 4 (50%)
For long fur: 2 out of 4 (50%)
Step 3: Calculate the combined probability.
Since both the B and H traits assort independently, we can multiply the probabilities of each trait occurring:
0.5 (brown fur) x 0.5 (long fur) = 0.25 (25%).
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what does boxplot tell?
Data$Density 0 20 40 60 80 100 120 BARN Data$Species OYST o 8 o
Also, minimum observations for both data sets are same, however there is a difference in the maximum for the both data sets.
From the given boxplots, it is observed that the boxplot for the species BARN has more variation than the boxplot for the species OYST. The boxplot for the species OYST indicates that there is are some outliers present in the data, however the boxplot for the BARN species indicates that there are no any outliers present in the data. It is observed that the median for the species OYST is less than the median for the species BARN. First quartiles (Q1) for both data sets are approximately same, but medians and third quartiles are not same. Also, minimum observations for both data sets are same, however there is a difference in the maximum for the both data sets.
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Stevic delivers newspapers. He has already earned $36 delivering the Sunday paper and $12 delivering the Saturday paper. He earns $4 for each Sunday paper delivered and $2.50 for each Saturday paper delivered.
Part A
Enter numbers in the boxes to complete the rules for finding Stevic's earnings.
Sunday newspaper: Start at $ and add $
Saturday newspaper: Start at $ and add $
Part B
Stevic wants to compute his total earnings after delivering 15 papers on each day.
I'm actually in fifth grade
Answer:
Part A:
Sunday newspaper: Start at $36 and add $4 for each paper delivered.
Saturday newspaper: Start at $12 and add $2.50 for each paper delivered.
Part B:
To calculate Stevic's total earnings after delivering 15 papers on each day:
Earnings from Sunday papers = $36 + ($4 x 15) = $96
Earnings from Saturday papers = $12 + ($2.50 x 15) = $49.50
Total earnings = Earnings from Sunday papers + Earnings from Saturday papers
Total earnings = $96 + $49.50
Total earnings = $145.50
Therefore, Stevic's total earnings after delivering 15 papers on each day is $145.50.
Step-by-step explanation:
A multiple linear regression model is to be constructed to determine if there is a relationship between a dependent variable (y) and two independent variables (x1 and x2). A random sample of size n has been collected and the values of x1i, x2i and yi for i = 1, 2, ..., n have been recorded. The residuals (ei) in this analysis are defined as the difference between the observed values of y and the values of y predicted by the regression equation.Select the condition that is one of the assumptions of a valid multiple linear regression model:the relationship between the dependent and independent variables is linearthe residuals are constantthe independent variables are independent of the dependent variablethe relationship between the dependent and independent variables is quadratic
The condition that is one of the assumptions of a valid multiple linear regression model is: the relationship between the dependent and independent variables is linear.
Condition that is one of the assumptions of a valid multiple linear regression model is that the relationship between the dependent and independent variables is linear. This means that the change in the dependent variable is proportional to the change in each independent variable, and there is no curved or nonlinear relationship between them. The assumption of linear independence of the independent variables is also important, meaning that they are not highly correlated with each other.
The assumption of constant residuals means that the errors in the model are consistent across all values of the independent variables. The assumption of a quadratic relationship between the dependent and independent variables is not appropriate for a multiple linear regression model.
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Which table contains only values that satisfy the equation y = 0. 5x + 14?
The table which contains only values that satisfy the equation of line defined as y = 0. 5x + 14, ( linear equation) is present in option(c). So, option(c) is right one.
We have a equation of line, y = 0.5x + 14, --(1) which is a equation of line . We have to recognise the table which satisfy the above line equation. The values are called roots of the equation. A value that is a solution of an equation is said to satisfy the equation, and the solutions of an equation create its solution set. The above table consists values of x and y, so we check which set of values form solution set of equation (1). Let x = 0 => y = 14 so, ( 0, 14) is solution of equation (1). Similarly, when x = 5
=> y = 5× 0.5 + 14 = 16.5
Similarly, we can check other point values. The table present in option (c) is correct answer.
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Complete question:
The above figure complete the question.
Which situation involves descriptive statistics?
A) Ten percent of the girls on the cheerleading squad are also on the track team.
B)To determine how many outlets might need to be changed, an electrician inspected 20 of them and found 1 that didn’t work.
C) A study shows that the average student leaves a four-year college with a student loan debt of more than $30,000.
D) A survey indicates that about 25% of a restaurant’s customers want more dessert options
Option C, "A study shows that the average student leaves a four-year college with a student loan debt of more than $30,000", involves descriptive statistics.
The area of statistics known as descriptive statistics deals with the gathering, organizing, organizing, analyzing, interpreting, and presenting of data. It summarizes and describes the main features of a dataset, including measures of central tendency (such as mean, median, and mode) and measures of variability (such as range, standard deviation, and variance). Option C presents a descriptive statistic (the average student loan debt) that summarizes a larger dataset, making it the correct answer.
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E7.5. Given the variance-covariance matrix of three random variables X1, X2 and X3,∑=
4 1 2
1 9 -3
2 -3 25 a. Find the correlation matrix p. b. Compute the correlation between X1, and i/2X2 + 1/2X3.
a. The correlation matrix p = [tex]\left[\begin{array}{ccc}1&1/3&2/5\\1/3&1&-3/5\\2/5&-3/5&1\end{array}\right][/tex]. b. The correlation between X1, and i/2X2 + 1/2X3 is 0.3.
a. The correlation matrix p can be calculated by dividing the covariance matrix by the product of the standard deviations of the variables:
p = [tex]\left[\begin{array}{ccc}1&1/3&2/5\\1/3&1&-3/5\\2/5&-3/5&1\end{array}\right][/tex]
b. To compute the correlation between X1 and i/2X2 + 1/2X3, we first need to calculate the standard deviations of the variables:
σ1 = sqrt(4) = 2
σ2 = sqrt(9) = 3
σ3 = sqrt(25) = 5
Then, we can calculate the covariance between X1 and i/2X2 + 1/2X3:
cov(X1, i/2X2 + 1/2X3) = cov(X1, i/2X2) + cov(X1, 1/2X3)
= i/2 * cov(X1, X2) + 1/2 * cov(X1, X3)
= i/2 * 1 + 1/2 * 2
= 1.5
Finally, we can compute the correlation using the formula:
corr(X1, i/2X2 + 1/2X3) = cov(X1, i/2X2 + 1/2X3) / (σ1 * σ2/2 + σ3/2)
= 1.5 / (2 * 3/2 + 5/2)
= 0.3
Therefore, the correlation between X1 and i/2X2 + 1/2X3 is 0.3.
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Solve the differential equation by variation of parameters. 4y'' − y = ex/2 8
The solution of the differential equation 4y'' − y = [tex] {e}^{x/2} [/tex] + 8 by variation of parameter method is y(x) = (15C - 16)[tex] {e}^{x/2} [/tex] + 15C' [tex] {ex}^{-x/2} [/tex]
To solve the differential equation by variation of parameters, we assume that the solution is of the form,
y(x) = u₁(x)y₁(x) + u₂(x)y₂(x), linearly independent solutions of the homogeneous equation are y₂(x) and y₂(x), and functions to be determined u₁(x) and u₂(x). The homogeneous equation associated with the given differential equation is,
4y'' - y = 0
The characteristic equation is,
4r² - 1 = 0 which has solutions r = ±1/2. Therefore, the general solution of the homogeneous equation is,
y(x) = C[tex] {e}^{x/2} [/tex] + C'[tex] {e}^{-x/2} [/tex]
C and C' are arbitrary constants.
Now, we need to find particular solutions of the non-homogeneous equation. We can guess that a particular solution has the form,
[tex] y_{p(x)} = A(x) {e}^{(x/2)} [/tex]
where A(x) is a function to be determined. We can find A(x) by substituting y_p(x) into the differential equation and solving for A(x). We have,
[tex] 4y_{p(x)} - y_{p(x)} = {e}^{(x/2)} +8 [/tex]
Differentiating twice and substituting these into the differential equation gives:
[tex]4( A"(x) + A'(x)) {e}^{2/y} 2 + \frac{A(x)}{4} - A(x) {e}^{(x/2)} = {e}^{(x/2)} + 8[/tex]
Simplifying and solving for A(x), we obtain,
A(x) = -16/15
Therefore, a particular solution of the differential equation is:
[tex]y_{p(x)} = \frac{ - 16}{15} {e}^{(x \div 2)} [/tex]
The general solution of the non-homogeneous equation is then,
y(x) = C[tex] {e}^{x/2} [/tex] + C'[tex] {e}^{-x/2} [/tex] [tex]\frac{ - 16}{15} {e}^{(x/2)} [/tex]
Simplifying and collecting terms, we get,
y(x) = (15C - 16)[tex] {e}^{x/2} [/tex] + 15C' [tex] {ex}^{-x/2} [/tex] ,where C and C' are arbitrary constants.
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Complete question - Solve the differential equation by variation of parameters. 4y'' − y = e^x/2 + 8.
Find the dimensions of the rectangle with area 225 square inches that has minimum perimeter, and then find the minimum perimeter.
1. Dimensions: 2. Minimum perimeter: Enter your result for the dimensions as a comma separated list of two numbers. Do not include the units.
the dimensions of the rectangle are L = 15 inches and W = 15 inches, and the minimum perimeter is: P = 2L + 2W = 60 inches.
Let the length and width of the rectangle be L and W, respectively, so that the area of the rectangle is A = LW = 225. We want to find the dimensions of the rectangle with minimum perimeter P = 2L + 2W, and then find the minimum perimeter.
Using the given area, we can solve for one of the variables in terms of the other:
L = 225/W
Substituting this expression for L into the expression for the perimeter, we get:
P = 2(225/W) + 2W
Taking the derivative of P with respect to W and setting it equal to zero to find the minimum, we get:
[tex]dP/dW = -450/W^2 + 2 = 0[/tex]
Solving for W, we get:
W^2 = 225
Since W must be positive (it is a length), we take the positive square root:
W = 15
Substituting this value of W back into the expression for L, we get:
L = 225/W = 15
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Part A)
A buffer solution is made that is 0. 304 M in H2CO3 and 0. 304 M in NaHCO3.
If Ka1 for H2CO3 is 4. 20 x 10^-7 , what is the pH of the buffer solution?
pH =
Write the net ionic equation for the reaction that occurs when 0. 088 mol KOH is added to 1. 00 L of the buffer solution.
(Use the lowest possible coefficients. Omit states of matter. )
PART B)
A buffer solution is made that is 0. 311 M in H2CO3 and 0. 311 M in KHCO3.
If ka1 for H2CO3 is 4. 20 x 10^-7, what is the pH of the buffer solution?
pH =
Write the net ionic equation for the reaction that occurs when 0. 089 mol HI is added to 1. 00 L of the buffer solution.
(Use the lowest possible coefficients. Omit states of matter. Use H3O instead of H )
Part A - The pH of the buffer solution is 6.37.
Net ionic equation is [tex]H_2CO_3[/tex] + [tex]OH^-[/tex] → [tex]HCO^{3-}[/tex] + [tex]H_2O[/tex]
Part B - The pH of the buffer solution is 6.38.
Net ionic equation is [tex]H_2CO_3[/tex] + [tex]I^-[/tex] → [tex]HCO^{3-}[/tex] + [tex]H_3O^+[/tex]
Part A: To find the pH of the buffer solution, we first need to calculate the pKa of the weak acid. The pKa is -log(Ka1), so pKa1 = -log(4.20 x [tex]10^{-7}[/tex]) = 6.38.
Next, we can use the Henderson-Hasselbalch equation to find the pH: pH = pKa1 + log([[tex]A^-[/tex]]/[HA]).
Plugging in the values for the buffer solution, we get pH = 6.38 + log(0.304/0.304) = 6.38. Therefore, the pH of the buffer solution is 6.38.
The net ionic equation for the reaction when 0.088 mol KOH is added to 1.00 L of the buffer solution is:
[tex]H^+[/tex] + [tex]OH^-[/tex] → [tex]H_2O[/tex]
Part B: Similar to Part A, we first need to calculate the pKa of the weak acid. pKa1 = -log(4.20 x [tex]10^{-7}[/tex]) = 6.38.
Then, we can use the Henderson-Hasselbalch equation to find the pH: pH = pKa1 + log([A-]/[HA]).
Plugging in the values for the buffer solution, we get pH = 6.38 + log(0.311/0.311) = 6.38. Therefore, the pH of the buffer solution is 6.38.
The net ionic equation for the reaction when 0.089 mol HI is added to 1.00 L of the buffer solution is:
[tex]H_3O^+[/tex] + [tex]I^-[/tex] → HI + H2O
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Question 2: (5+5+ 7+ 3 marks)
Solve the following inequalities and write the solution in interval form
i) 2|2x+71 +2 ≤ 24
ii) 33x-2 >24
The solution to the inequality is:
x ∈ (26/33, ∞)
i) We can simplify the left-hand side of the inequality as follows:
2|2x + 71| + 2 ≤ 24
2|2x + 71| ≤ 22
|2x + 71| ≤ 11
Next, we can split this into two separate inequalities, depending on the sign of (2x + 71):
2x + 71 ≤ 11
2x ≤ -60
x ≤ -30
or
2x + 71 ≥ -11
2x ≥ -82
x ≥ -41
Therefore, the solution to the inequality is:
x ∈ (-∞, -30] ∪ [-41, ∞)
ii) We can solve for x as follows:
33x - 2 > 24
33x > 26
x > 26/33
Therefore, the solution to the inequality is:
x ∈ (26/33, ∞)
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When conducting a hypothesis test, a(an) ___ is more appropriate than a 2-score when you don't know the population variance or the population standard deviation. a. alpha value b. t-statistic c. Sample variance
d. Cohen's d
When conducting a hypothesis test, a(an) **b. t-statistic** is more appropriate than a z-score when you don't know the population variance or the population standard deviation. The t-statistic takes into account the sample size and is better suited for situations where population parameters are unknown.
When conducting a hypothesis test, a t-statistic is more appropriate than a 2-score when you don't know the population variance or the population standard deviation. The t-statistic is used to test hypotheses about population means when the sample size is small or when the population standard deviation is unknown.
The t-statistic is calculated by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the mean, which takes into account the variability of the sample.
The t-statistic is compared to a critical value from a t-distribution with n-1 degrees of freedom, where n is the sample size. The level of significance, or alpha value, is also used to determine the critical value. Sample variance and Cohen's d are other statistical measures used in hypothesis testing but are not specifically related to the use of t-statistics.
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2. Determine the supremum and infimum in R of each of the following sets. Is this value also the maximum/minimum? (a) {1/n: 0 € N} (b) {z E Q: 22 < 3}
To determine the supremum and infimum of the given sets.
(a) The set {1/n: n ∈ N} consists of the reciprocals of positive integers. The smallest element in the set is 1, as it corresponds to n=1. The set has no largest element since it has an infinite number of elements getting smaller as n increases. Therefore, the infimum (greatest lower bound) of the set is 1, and there is no maximum. The supremum (least upper bound) of the set is not in the set itself, but it exists and equals 1.
(b) The set {z ∈ Q: 22 < 3} is an empty set since there is no rational number z that satisfies the condition 22 < 3. In this case, there is no supremum or infimum since the set has no elements. Consequently, there is no maximum or minimum value.
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The hazard of exposure to radioactive chemicals is mitigated with 3 independent barriers. If only 1 barrier works, the exposure is prevented. The probability of each barrier to fail is 0.001 and the consequence of hazard exposure is 3000 cancer-deaths per year. Develop an event tree showing all branches and outcome. What is the probability of exposure. What is the risk (probability x consequence) due to the hazard?
The risk due to the hazard of exposure to radioactive chemicals is 6 cancer-deaths per year. The probability of exposure is approximately 0.002. The event tree exposure with 1, 2, or 3 barriers failing, and no exposure if all 3 barriers work.
To calculate the probability of exposure and risk due to the hazard, we need to develop an event tree showing all the branches and outcomes.
The event tree for this scenario would look like this:
Barrier 1 fails (0.001 probability) -> Exposure -> 3000 cancer-deaths per year
Barrier 2 fails (0.001 probability) -> Barrier 1 works -> Exposure -> 3000 cancer-deaths per year
Barrier 3 fails (0.001 probability) -> Barrier 2 works -> Barrier 1 works -> Exposure -> 3000 cancer-deaths per year
All 3 barriers work -> No exposure -> No consequence
Start
|
Barrier 1
/ | \
Fail (0.001) | Pass (0.999)
| |
Exposure Barrier 2
(3000 cancer-deaths) / | \
/ | \
Barrier 2 | Barrier 3
Fail (0.001)| Pass (0.999)
| |
Exposure No exposure
(3000 cancer-deaths) |
|
Barrier 3
Fail (0.001)
|
Exposure
(3000 cancer-deaths)
From this event tree, we can see that there are 4 possible outcomes: exposure with 1, 2, or 3 barriers failing, and no exposure if all 3 barriers work.
The probability of exposure can be calculated by adding up the probabilities of each branch that leads to exposure:
0.001 + (0.001 x 0.999) + (0.001 x 0.999 x 0.999) = 0.001997
Therefore, the probability of exposure is approximately 0.002 (or 0.2%).
To calculate the risk, we need to multiply the probability of exposure by the consequence:
0.002 x 3000 = 6
Therefore, the risk due to the hazard of exposure to radioactive chemicals is 6 cancer-deaths per year. However, it is important to continue to monitor and maintain these barriers to ensure their effectiveness and minimize the risk of exposure.
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The faces of a cube are painted with three colors so that opposite faces are the same color. Which of the following shows the development of the cube?
Answer:
The correct answer is the option 3
AOC and BOD are diameters of a circle, centre O. Prove that triangle ABD and triangle DCA are congruent by RHS. B D
Given:
[tex]\text{AOC}[/tex] and [tex]\text{BOD}[/tex] are diameters of a circle and has center [tex]\text{O}[/tex].
To Find:
[tex]\Delta\text{ABD}[/tex] and [tex]\Delta\text{DCA}[/tex] are congruent by [tex]\text{RHS}[/tex].
Solution:
It is given that [tex]\text{AOC}[/tex] and [tex]\text{BOD}[/tex] are diameters of a circle.
[tex]\rightarrow \text{BD} = \text{CA}[/tex] [diameters of the circle]
[tex]\rightarrow \angle\text{BAD} = \angle\text{CDA}[/tex] [angles in semicircle is 90°]
[tex]\rightarrow \text{AD} = \text{AD}[/tex] [common in both the triangles]
[tex]\rightarrow \Delta\text{ABD} \cong \Delta\text{DCA}[/tex] [using RHS congruence criteria]
Hence, proved [tex]\Delta\bold{ABD} \cong \Delta\bold{DCA}[/tex] by [tex]\bold{RHS}[/tex] congruency criteria.
PLEASE HELP ME SOLVE THIS ONE QUESTION , I HAVE SOLVED I) IT IS II) I NEED HELP WITH
5. A is the point (1,5) and B is the point (3,9).M is the midpoint of AB
i) M = (2,5)
ii)Find the equation of the line that is perpendicular to AB and passes through M.
Give your answer in the form : y=mx+c
I)
[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ A(\stackrel{x_1}{1}~,~\stackrel{y_1}{5})\qquad B(\stackrel{x_2}{3}~,~\stackrel{y_2}{9}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 3 +1}{2}~~~ ,~~~ \cfrac{ 9 +5}{2} \right) \implies \left(\cfrac{ 4 }{2}~~~ ,~~~ \cfrac{ 14 }{2} \right)\implies (2~~,~~7)[/tex]
II)
keeping in mind that perpendicular lines have negative reciprocal slopes, let's check for the slope of the line AB
[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{3}~,~\stackrel{y_2}{9}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{9}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{3}-\underset{x_1}{1}}} \implies \cfrac{ 4 }{ 2 } \implies 2 \\\\[-0.35em] ~\dotfill[/tex]
[tex]\stackrel{~\hspace{5em}\textit{perpendicular lines have \underline{negative reciprocal} slopes}~\hspace{5em}} {\stackrel{slope}{ 2 \implies \cfrac{2}{1}} ~\hfill \stackrel{reciprocal}{\cfrac{1}{2}} ~\hfill \stackrel{negative~reciprocal}{-\cfrac{1}{2} }}[/tex]
so we're really looking for the equation of a line whose slope is -1/2 and it passes through (2 , 7)
[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{7})\hspace{10em} \stackrel{slope}{m} ~=~ - \cfrac{1}{2} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{7}=\stackrel{m}{- \cfrac{1}{2}}(x-\stackrel{x_1}{2}) \\\\\\ y-7=- \cfrac{1}{2}x+1\implies {\Large \begin{array}{llll} y=- \cfrac{1}{2}x+8 \end{array}}[/tex]
Evaluate the following iterated integral.
∫85∫√x12ye−xdydx
The value of the iterated integral ∫85∫√x12ye−xdydx is
-[tex]4e^(-5) + 7e^(-8)[/tex] where the inner integral is first integrated with respect to y.
We are inquiring to assess the iterated integral:
[tex]∫85∫√x12ye−xdydx[/tex]
We are able to coordinate the internal integral, to begin with regard to y:
[tex]∫√x12ye−xdy = (-1/2)e^(-x) y√x1/2 | from y = to y = √x^1/2[/tex]
[tex]= (-1/2)e^(-x) (√x^1/2)^2 - (-1/2)e^(-x) (0)[/tex]
[tex]= (-1/2)x e^(-x)[/tex]
Substituting this into the first necessity, we get:
[tex]∫85∫√x12ye−xdydx = ∫85(-1/2)x e^(-x)dx[/tex]
To assess this necessarily, we utilize integration by parts with u = x and [tex]dv = e^(-x) dx, so that du/dx = 1 and v = -e^(-x):[/tex]
[tex]∫85(-1/2)x e^(-x)dx = (-1/2)xe^(-x) + ∫85(1/2)e^(-x)dx[/tex]
[tex]= (-1/2)xe^(-x) - (1/2)e^(-x) | from x = 8 to x = 5[/tex]
[tex]= (-1/2)(8e^(-8) - 5e^(-5)) - (1/2)(e^(-8) - e^(-5))[/tex]
[tex]= -4e^(-5) + 7e^(-8)[/tex]
therefore, the value of the iterated integral is [tex]-4e^(-5) + 7e^(-8).[/tex]
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Solve for x.
4x -9 = 2x +5
Answer:
x = 7
Step-by-step explanation:
Solve for x.
4x - 9 = 2x + 5
4x - 2x = 5 + 9
2x = 14
x = 14 : 2
x = 7
-----------------
check (replace "x" with "7")
4 * 7 - 9 = 2 * 7 + 5 (remember PEMDAS)
28 - 9 = 14 + 5
19 = 19
the answer is good
Answer:
hence the required value of x is 7.
Find the missing angle.
The value of the unknown angle is 68°
What is trigonometric ratio?Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.
Sin(tetha) = opp/hyp
cos(tetha) = adj/hyp
tan(tetha) = opp/adj
In the triangle, 51 is the opposite and 55 is the hypotenuse.
therefore;
sin(tetha) = 51/55
sin(tetha) = 0.927
tetha = sin^-1( 0.927)
tetha = 67.97
approximately to 68°
therefore the value of the unknown angle is 68°
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The state of Colorado has a population of about 5.77 million people. The state of Pennsylvania has a population density 5 times greater than the population density of Colorado. Find the population of Pennsylvania.
The population of Pennsylvania is: 1304503 people
How to calculate population density?Population density is calculated by taking the total area of a region in question and dividing it by the total number of people that live in that area. The result will give the average number of inhabitants per square kilometre, mile, acre, meter, etc.
The parameters given are:
Population of colorado = 5,770,000 people
Area of colorado = 280 * 380
= 106,400 mi²
Population density here = 5,770,000/106,400
54.23 people per mi²
Area of Pennsylvania = 283 * 170
= 48110 mi²
Thus:
Population of Pennsylvania/48110 mi² = 5 * 54.23 people per mi²
Population of Pennsylvania = 48110 * mi² * 5 * 54.23 people per mi²
= 1304503 people
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4. From historical data it is known that the probability is 0.25 that a randomly selected WST111
student will be late for the 7h30 lecture on a Tuesday. Suppose five WST111 students are
selected randomly. Assume that punctuality of students (whether they are late or not) are
independent. Calculate the probability that at least one student is in time for the 7h30 lecture on
a Tuesday morning.
The probability that at least one WST111 student is in time for the 7h30 lecture on a Tuesday morning is 0.9961.
1. First, let's find the probability that a randomly selected student is on time for the lecture. Since the probability that a student is late is 0.25, the probability that a student is on time is 1 - 0.25 = 0.75.
2. Now, we need to calculate the probability that all five randomly selected students are late for the lecture. Since punctuality is independent, we can simply multiply each student's probability of being late: 0.25×0.25×0.25×0.25× 0.25 = 0.0009765625.
3. Finally, we want to find the probability that at least one student is on time. To do this, we'll subtract the probability that all students are late from 1:
1 - 0.0009765625 = 0.9961.
So, the probability that at least one WST111 student is in time for the 7h30 lecture on a Tuesday morning is 0.9961.
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