(a) Letting X1, ..., Xy be independent random variables and Using the union bound, we have P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0.
(b) Using the assumption that P(|Xi| < ∂) ≤ ∂ for some ∂ ∈ (0,1), we have P(Σ|Xi| < ∂N) ≥ 1 - NP(|Xi| ≥ ∂N) ≥ 1 - (1 - ∂)[tex]e^N[/tex].
Setting t = 2N[tex]e^ε[/tex], we obtain
P(|X1| + ... + |XN| ≥ 2Ne**ε) ≤ e**(-ε)which is equivalent to
P(|X1| + ... + |XN| < 2Ne**ε) ≥ 1 - e**(-ε).By setting ∂ = 2Ne**ε/N, we get
P(Σ|Xi| < ∂) ≥ 1 - e**(-ε), and therefore,
NP(Σ|Xi| < ∂) ≥ N(1 - e**(-ε)) ≥ Nε for ε > 0.Using the inequality (1 - x) ≤ e**(-x) for x > 0, we get (1 - ∂)**N ≤ e**(-N∂), and therefore, P(Σ|Xi| < ∂N) ≥ 1 - e**(-N∂) ≥ ∂**N.
Thus, we have shown that NP(Σ|Xi| < ∂N) ≥ ∂**N for some ∂ ∈ (0,1) and P(|X1| + ... + |XN| ≥ t) ≤ P(|X1| ≥ t/N) + ... + P(|XN| ≥ t/N) ≤ 2N[tex]e^{(-tε/N)}[/tex] for all t > 0
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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
Lisa recorded her earnings for six weeks: $50, $50, $50, $45, $50, $50, $180, $50. Does the mean or the mode best describe Lisa's typical weekly earnings? Explain your answer.
Isabel is going to rent a truck for one day. There are two companies she can choose from, and they have the following prices. Company A charges $90 and allows unlimited mileage. Company B has an initial fee of $75 and charges an additional $0. 60 for every mile driven. For what mileages will Company A charge less than Company B? Use m for the number of miles driven, and solve your inequality for m
Therefore, if Isabel plans to drive inequality more than 25 miles, Company B will be more expensive than Company A. If she plans to drive 25 miles or less, Company A will be more expensive.
Let's start by setting up an inequality to represent the mileages for which Company A charges less than Company B.
For Company A, the cost is a flat fee of $90, regardless of the number of miles driven.
For Company B, the cost depends on the number of miles driven. The initial fee is $75, and then there is an additional charge of $0.60 for every mile driven. So, the total cost for Company B can be represented by the equation:
Cost(B) = 0.60m + 75
here m is the number of miles driven.
We want to find the mileages for which Company A charges less than Company B. In other words, we want to find the values of m for which:
Cost(A) < Cost(B)
Substituting in the expressions for the costs, we get:
90 < 0.60m + 75
Simplifying and solving for m, we get:
15 < 0.60m
25 < m
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Students at Mendel middle school are planning a fair for their school’s fundraiser. Liam makes a poster for the fair. Maureen looks at the poster and says that the price per ticket decreases the more tickets a customer buys. Liam disagrees. Is Liam or Maureen correct? What is the y-intercept of the graph? Explain what it means in the problem situation.
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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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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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.
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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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]
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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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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a pizza parlor offers four sizes of pizza and 12 different toppings. a customer may choose any number of toppings (or no tipping at all). how many different pizzas does this parlor offer?
The pizza parlor offers 2^48, or approximately 281 trillion, different pizzas.
To calculate the total number of different pizzas offered, we need to consider all possible combinations of pizza sizes and toppings.
For each pizza size, there are 12 options for toppings (including no toppings). Therefore, the total number of pizzas for a single size is 2^12 (2 options for each of the 12 toppings). Since there are four different sizes, the total number of different pizzas offered by the parlor is:
2^12 x 2^12 x 2^12 x 2^12 = 2^(12+12+12+12) = 2^48
Thus, the pizza parlor offers 2^48, or approximately 281 trillion, different pizzas.
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Line q passes through points (2, 5) and (8, 10). Line r is parallel to line q. What is the slope of line r?
Answer:
Step-by-step explanation:
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.
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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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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Which is different? A cylinder is shown. The radius of its base is 5 centimeters and height is 12 centimeters. Responses How much does it take to fill the cylinder? How much does it take to fill the cylinder? What is the capacity of the cylinder? What is the capacity of the cylinder? How much does it take to cover the cylinder? How much does it take to cover the cylinder? How much does the cylinder contain?
How much does it take to cover the cylinder? is different from the rest, as it refers to the surface area of the cylinder, not its volume or capacity.
The term "cover" usually means to place something over the top or on the surface of something else, such as a lid covering a container.
In the context of a cylinder, "covering" would typically refer to finding the surface area of the cylinder, which includes both the top and bottom circles as well as the curved lateral surface.
In contrast, the other statements are related to the volume or capacity of the cylinder, which refers to how much space is contained inside the cylinder.
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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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Which could be the dimensions of a rectangular prism whose surface area is greater than 140 square feet? Select
three options.
6 feet by 2 feet by 3 feet
6 feet by 5 feet by 4 feet
7 feet by 6 feet by 4 feet
8 feet by 3 feet by 7 feet
8 feet by 4 feet by 3 feet
Mark this and return
Save and Exit
Next
Submit
The three options with dimensions resulting in a surface area greater than 140 square feet are:
6 feet by 5 feet by 4 feet7 feet by 6 feet by 4 feet8 feet by 3 feet by 7 feetTo determine whether the dimensions of a rectangular prism result in a surface area greater than 140 square feet, we can use the formula for the surface area of a rectangular prism:
Surface Area = 2lw + 2lh + 2wh
where l, w, and h are the length, width, and height of the rectangular prism, respectively.
Option 1: 6 feet by 2 feet by 3 feet
Surface Area = 2(6)(2) + 2(6)(3) + 2(2)(3) = 24 + 36 + 12 = 72 square feet
This option does not have a surface area greater than 140 square feet.
Option 2: 6 feet by 5 feet by 4 feet
Surface Area = 2(6)(5) + 2(6)(4) + 2(5)(4) = 60 + 48 + 40 = 148 square feet
This option has a surface area greater than 140 square feet.
Option 3: 7 feet by 6 feet by 4 feet
Surface Area = 2(7)(6) + 2(7)(4) + 2(6)(4) = 84 + 56 + 48 = 188 square feet
This option has a surface area greater than 140 square feet.
Option 4: 8 feet by 3 feet by 7 feet
Surface Area = 2(8)(3) + 2(8)(7) + 2(3)(7) = 48 + 112 + 42 = 202 square feet
This option has a surface area greater than 140 square feet.
Option 5: 8 feet by 4 feet by 3 feet
Surface Area = 2(8)(4) + 2(8)(3) + 2(4)(3) = 64 + 48 + 24 = 136 square feet
This option does not have a surface area greater than 140 square feet.
Therefore, the three options with dimensions resulting in a surface area greater than 140 square feet are:
6 feet by 5 feet by 4 feet7 feet by 6 feet by 4 feet8 feet by 3 feet by 7 feetTo learn more about Dimensions of Rectangular Prism,
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Use the equation for the velocity of a free-falling object,
v =
2gh
,
where v is the velocity measured in feet per second,
g = 32
feet per second per second, and h is the distance (in feet) the object has fallen. A stone strikes the water with a velocity of 138 feet per second. Estimate to two decimal places the height from which the stone was dropped
As per the given equation, the stone was dropped from a height of approximately 1.08 feet or about 13 inches.
The velocity of a free-falling object is an important concept in physics, and it is defined by the equation:
v = 2gh
In this equation, v represents the velocity of the object in feet per second, g represents the acceleration due to gravity in feet per second per second, and h represents the distance that the object has fallen in feet.
Suppose a stone is dropped from a certain height and strikes the water with a velocity of 138 feet per second. Our task is to estimate the height from which the stone was dropped.
To solve this problem, we need to rearrange the equation to solve for h. We start by dividing both sides of the equation by 2g:
h = v/2g
Substituting the given values, we get:
h = 138/2(32) = 1.078125
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How to write an equation to describe a proportional relationship
A proportional relationship is a type of linear relationship where the ratio of two variables is constant.
The number of hours studied (x) and the corresponding grade on a test (y) for a group of students. We observe that the grades are directly proportional to the number of hours studied. To write an equation to describe this proportional relationship, we can use the form y = kx, where k is the constant of proportionality.
To find the value of k, we can use any data point in the dataset. Let's say that when a student studies for 5 hours, they get a grade of 80. We can substitute these values into the equation:
[tex]80 = k[/tex] × [tex]5[/tex]
To solve for k, we can divide both sides by 5:
[tex]k = 80 / 5[/tex]
[tex]= 16[/tex]
Therefore, the equation to describe this proportional relationship is:
[tex]y = 16x[/tex]
This means that for every additional hour studied, the grade increases by 16 points.
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Complete Question:
How to write an equation to describe a proportional relationship?
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.
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.
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.
thank you all for any help it realy means a lot
Answer: x +
Step-by-step explanation:
the correct expression is:
5 x (8 + 4)
when expanded, this would give you:
5x8 + 5x4
so the blanks should be filled with x and +
Assume C is the center of the circle.
A quadratic equation has zeros at -6 and 2. Find standard form
The quadratic equation with zeros at -6 and 2 is y² + 4y - 12 = 0. This is in standard form, which is ax² + bx + c = 0, with a = 1, b = 4, and c = -12.
To find the quadratic equation with zeros at -6 and 2, we can start by using the fact that if a quadratic equation has roots x₁ and x₂, then it can be written in the form
(y - x₁)(y - x₂) = 0
where y is the variable in the quadratic equation.
Substituting the given values of the zeros, we get
(y - (-6))(y - 2) = 0
Simplifying this expression, we get
(y + 6)(y - 2) = 0
Expanding this expression, we get
y² - 2y + 6y - 12 = 0
Simplifying this expression further, we get
y² + 4y - 12 = 0
So the quadratic equation with zeros at -6 and 2 is
y² + 4y - 12 = 0
This is the standard form of a quadratic equation, which is
ax² + bx + c = 0
where a, b, and c are constants. In this case, a = 1, b = 4, and c = -12.
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H с Homework: 6.2 Homework Question 5, 6.2.11-T Construct the indicated confidence interval for the population mean y using the t-distribution. Assume the population is normally distributed. C= 0.98, *= 12.1, =0.95, n=15 (Round to one decimal place as needed.)
We are 98% confident that the mean of the true population y lies between 10.8 and 13.4.
To construct the confidence interval, we first need to calculate the critical value of t using the given values of C and n. Since C = 0.98, we can find the level of significance as α = 1 - C = 0.02.
Using a t-table or calculator, the critical value of t for a two-tailed test with 14 degrees of freedom and
[tex]\frac{\alpha}{2} = 0.01[/tex] is approximately 2.977.
Next, we can calculate the sample standard deviation as s = σ/√n = [tex]\frac{0.95}{\sqrt{15}}= 0.245[/tex].
Then, we can use the formula for a confidence interval for the population mean using the t-distribution:
(y ± t)×[tex]\frac{s}{\sqrt{n}}[/tex]
Substituting the given values, we get:
(12.1 ± 2.977)×[tex]\frac{0.245}{\sqrt{15}}[/tex]
Simplifying and rounding to one decimal place, we get the confidence interval: (10.8, 13.4)
Therefore, we are 98% confident that the true population mean y lies between 10.8 and 13.4.
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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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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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