Answer: To solve the third-order differential equation y''' + y' = cot(x) by variation of parameters, we first need to find the solution to the associated homogeneous equation, which is:
y''' + y' = 0
The characteristic equation is r^3 + r = 0, which can be factored as r(r^2 + 1) = 0. This gives us the roots r = 0, r = i, and r = -i. Therefore, the general solution to the homogeneous equation is:
y_h = c1 + c2 cos(x) + c3 sin(x)
To find a particular solution to the non-homogeneous equation using variation of parameters, we assume that the solution has the form:
y_p = u1(x) + u2(x) cos(x) + u3(x) sin(x)
where u1, u2, and u3 are functions to be determined.
We can find the derivatives of y_p:
y'_p = u1'(x) + u2'(x) cos(x) - u2(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x)
y''_p = u1''(x) + u2''(x) cos(x) - 2u2'(x) sin(x) - u2(x) cos(x) + u3''(x) sin(x) + 2u3'(x) cos(x) - u3(x) sin(x)
y'''_p = u1'''(x) + u2'''(x) cos(x) - 3u2''(x) sin(x) - 3u2'(x) cos(x) - u2(x) sin(x) + u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x)
Substituting these derivatives into the non-homogeneous equation, we get:
u1'''(x) + u2'''(x) cos(x) - 3u2''(x) sin(x) - 3u2'(x) cos(x) - u2(x) sin(x) + u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x) + u1'(x) + u2'(x) cos(x) - u2(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x) = cot(x)
Grouping the terms with the same functions together, we get:
u1'''(x) + u1'(x) = 0
u2'''(x) cos(x) - 3u2''(x) sin(x) - u2(x) sin(x) + u2'(x) cos(x) + u2'(x) cos(x) = cot(x) cos(x)
u3'''(x) sin(x) + 3u3''(x) cos(x) - 3u3'(x) sin(x) + u3'(x) sin(x) + u3(x) cos(x) = cot(x) sin(x)
The first equation is a first-order differential equation, which can be solved by integrating both sides:
u1'(x) + u1(x) = c1
where c1 is a constant of integration. The solution to this equation is:
u1(x) = c1 + c2 e^(-x)
where c2 is another constant of integration.
Step-by-step explanation:
Select the MEAN, MEDIUM, MODE and RANGE for the data below and how you worked it out
Employment status of parents in couple families
Labour force, parents or partners aged 15 years and over in Warragul
Both employed, worked full-time
580
Both employed, worked part-time
134
One employed full-time, one part-time
853
One employed full-time, other not working
471
One employed part-time, other not working
217
Both not working
799
Other (includes away from work)
193
Labour force status not stated (by one or both parents in a couple family)
185
Answer:
Measures of Central Tendancy
Mean: 429
Median: 344
Mode: 134,185,193,217,471,580,799,853
Range: 719
Step-by-step explanation:
Mean:The mean of a data set is commonly known as the average. You find the mean by taking the sum of all the data values and dividing that sum by the total number of data values. The formula for the mean of a population is
[tex]\mu = \frac{{\sum}x}{N}[/tex]
The formula for the mean of a sample is
[tex]\bar{x} = \frac{{\sum}x}{n}[/tex]
Both of these formulas use the same mathematical process: find the sum of the data values and divide by the total. For the data values entered above, the solution is:
[tex]\frac{3432}{8} = 429[/tex]
Median:The median of a data set is found by putting the data set in ascending numerical order and identifying the middle number. If there are an odd number of data values in the data set, the median is a single number. If there are an even number of data values in the data set, the median is the average of the two middle numbers. Sorting the data set for the values entered above we have:
[tex]134, 185, 193, 217, 471, 580, 799, 853[/tex]
Since there is an even number of data values in this data set, there are two middle numbers. With 8 data values, the middle numbers are the data values at positions 4 and 5. These are 217 and 471. The median is the average of these numbers. We have
[tex]{\frac{ 217 + 471 }{2}}[/tex]
Therefore, the median is
[tex]344[/tex]
Mode:The mode is the number that appears most frequently. A data set may have multiple modes. If it has two modes, the data set is called bimodal. If all the data values have the same frequency, all the data values are modes. Here, the mode(s) is/are
[tex]134,185,193,217,471,580,799,853[/tex]
5.7. Suppose n = 2911 and e = 11. Encrypt the following messages as in
Example (5.3).
a) "OK"
b) "HELP" (Break this up into two blocks.)
Note that,
the encrypted message for "OK" is the pair of numbers (616, 2385).
and the final encrypted message for "HELP" is the sequence of numbers (738, 1277, 1479, 2252).
To encrypt a message using RSA, we need to first represent the message as numbers using a suitable encoding scheme. For simplicity, we can use the ASCII code for each character, which is a standard encoding scheme for text.
a) To encrypt "OK", we first convert each letter to its corresponding ASCII code:
"O" = 79
"K" = 75
Next, we use the RSA encryption formula:
C ≡ [tex]M^{e}[/tex] (mod n)
For "O", we have C ≡ 79¹¹ (mod 2911) ≡ 616 (mod 2911)
For "K", we have C ≡ 75¹¹ (mod 2911) ≡ 2385 (mod 2911)
Therefore, the encrypted message for "OK" is the pair of numbers (616, 2385).
b) To encrypt "HELP", we break it up into two blocks:
Block 1: "HE"
Block 2: "LP"
For block 1, we have:
"H" = 72
"E" = 69
Using the RSA encryption formula, we get:
C1 ≡ 72¹¹ (mod 2911) ≡ 738 (mod 2911)
C2 ≡ 69¹¹ (mod 2911) ≡ 1277 (mod 2911)
Therefore, the encrypted message for "HE" is the pair of numbers (738, 1277).
For block 2, we have:
"L" = 76
"P" = 80
Using the RSA encryption formula, we get:
C3 ≡ 76¹¹ (mod 2911) ≡ 1479 (mod 2911)
C4 ≡ 80¹¹ (mod 2911) ≡ 2252 (mod 2911)
Therefore, the encrypted message for "LP" is the pair of numbers (1479, 2252).
The final encrypted message for "HELP" is the sequence of numbers (738, 1277, 1479, 2252).
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Write an expression describing all the angles that are coterminal with 8°. (Please use the variable k in your answer. Give your answer in degrees, but do not include a degree symbol in your answer.)
the expression describing all the angles that are coterminal with 8° is: θ = 8° + 360°k, where k is an integer.
How to solve and what is angle?
An angle of 8° has an initial side on the positive x-axis and rotates counterclockwise by 8°.
Any angle coterminal with 8° can be expressed as:
θ = 8° + 360°k
where k is an integer.
Therefore, the expression describing all the angles that are coterminal with 8° is:
θ = 8° + 360°k, where k is an integer.
An angle is a geometric figure formed by two rays, called the sides of the angle, that share a common endpoint, called the vertex of the angle. Angles are typically measured in degrees or radians and are used to measure the amount of rotation or turn between two intersecting lines or planes.
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Solve 2^x=32, and rewrite this equation in a logarithmic form.
Answer:
To solve 2^x = 32, we need to find the value of x that satisfies the equation.
We can rewrite 32 as a power of 2 by noting that 2^5 = 32. Therefore, we have:
2^x = 2^5
Since the bases of the powers are equal, we can equate the exponents:
x = 5
So the solution to 2^x = 32 is x = 5.
To rewrite this equation in a logarithmic form, we can use the definition of logarithms:
log(base 2)32 = x
Here, the logarithm with base 2 of 32 is equal to x.
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The vertices of figure PQRS are translated to form figure P'Q'R'S'. Select all the statements that describe the two figures. Q S R P' S' Q' 'R
the anawer choices are : A. P Q R S is the preimage of PQRS, B. the two figures are congruent, C. the two figures are in different positions , but have the same orientation, D. the two figures are in different positions and have oppsoite orientation , E. corresponding angles and sides of the figures have the same measures.
The true statements are:
(B) Both figures are congurent.
(C) The two figures have the same orientation but different positions.
(E) Corresponding angles and sides have the same measures.
What is orientation?In geometry, how an item is positioned in the space it occupies—such as a line, plane, or rigid body—is described in terms of its orientation, angular position, attitude, bearing, and direction.
It refers more particularly to the fictitious rotation required to shift an object from a reference placement to its present location.
To get to the current positioning, a rotation might not be sufficient.
It could be required to include a fictitious translation known as the object's location (or position, or linear position).
Together, the position and orientation completely explain where the object is situated in space.
Therefore, the true statements are:
(B) Both figures are congurent.
(C) The two figures have the same orientation but different positions.
(E) Corresponding angles and sides have the same measures.
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Use the following results from a test for marijuana use, which is provided by a certain drug testing company. Among 145 subjects with positive test results, there are 21 false positive results; among 156 negative results, there are 4 false negative results. If one of the test subjects is randomly selected, find the probability that the subject tested negative or did not use marijuana. (Hint: Construct a table.)
Question content area bottom
Part 1
The probability that a randomly selected subject tested negative or did not use marijuana is enter your response here.
(Do not round until the final answer. Then round to three decimal places as needed.)
Answer:
The probability that a randomly selected subject tested negative or did not use marijuana is 0.589.
Step-by-step explanation:
Find the sum. -
3/4 + 1/2 =
Answer:
Is the problem -3/4 + 1/2 or 3/4 + 1/2?
I'll just do both then.
-3/4 + 1/2 = -1/4
3/4 + 1/2 = 5/4 or 1 1/4
Step-by-step explanation:
You're welcome.
Answer:
[tex] \frac{3}{4 } + \frac{1}{2} [/tex]
take lcm of denominators i.e. 4and 2
so, the lcm of 4 and 2 is 4.
[tex] \frac{3}{4} + \frac{2}{4} [/tex]
[tex] \frac{3 + 2}{4} [/tex]
[tex] \frac{5}{4} [/tex]
Step-by-step explanation:
hope this will be helpful:)
Find the value of each variable.
The values of the variables in the semicircle shown are:
x = 63 degrees; y = 90 degrees.
What is the Angle Inscribed in a Semicircle Theorem?A semi-circle is exactly half of a full circle and has a measurement of 180 degrees; the two endpoints of the diameter form the endpoints of the semi-circle. If an angle is enclosed inside a semi-circle, the angle formed measures 90 degrees.
Therefore, it means the value of the variable, y = 90 degrees.
Thus, using the triangle sum theorem, we have:
x = 180 - 90 - 27
x = 63 degrees.
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An isosceles triangle whose sides are 5cm, 5cm and 6cm is inscribed in a circle. Find the radius of the circle.
Answer:
To find the radius of the circle inscribed in an isosceles triangle, we can use the following formula:
r = (a/2) * cot(π/n)
where r is the radius of the inscribed circle, a is the length of one of the equal sides of the isosceles triangle, and n is the number of sides of the polygon inscribed in the circle.
In this case, we have an isosceles triangle with two sides of 5cm and one side of 6cm. Since the triangle is isosceles, the angle opposite the 6cm side is bisected by the altitude and therefore, the two smaller angles are congruent. Let x be the measure of one of these angles. Using the Law of Cosines, we can solve for x:
6^2 = 5^2 + 5^2 - 2(5)(5)cos(x)
36 = 50 - 50cos(x)
cos(x) = (50 - 36)/50
cos(x) = 0.28
x = cos^-1(0.28) ≈ 73.7°
Since the isosceles triangle has two equal sides of length 5cm, we can divide the triangle into two congruent right triangles by drawing an altitude from the vertex opposite the 6cm side to the midpoint of the 6cm side. The length of this altitude can be found using the Pythagorean theorem:
(5/2)^2 + h^2 = 5^2
25/4 + h^2 = 25
h^2 = 75/4
h = sqrt(75)/2 = (5/2)sqrt(3)
Now we can find the radius of the inscribed circle using the formula:
r = (a/2) * cot(π/n)
where a = 5cm and n = 3 (since the circle is inscribed in a triangle, which is a 3-sided polygon). We can also use the fact that the distance from the center of the circle to the midpoint of each side of the triangle is equal to the radius of the circle. Therefore, the radius of the circle is equal to the altitude of the triangle from the vertex opposite the 6cm side:
r = (5/2) * cot(π/3) = (5/2) * sqrt(3) ≈ 2.89 cm
Therefore, the radius of the circle inscribed in the isosceles triangle with sides 5cm, 5cm, and 6cm is approximately 2.89 cm.
a bakery has 17 pounds of flour they want to put into 6 containers. They put the same amount of flour in each container. If they want to use up all the flour, how much flour should they put in each container?
Answer:
2.83333 pounds of flour in each container
Are the fractions 2/2 and 8/8 equivalent fractions
Answer:
Step-by-step explanation:
yes since they are both divisible by their denominators and equal the same thing
Answer:
yes, they are equivalent
Step-by-step explanation:
2/2 = (2/2)x(4/4) = 8/8 = 1
3 Cassie wants to determine the length of the shadow that a 60-foot tall telephone pole casts without measuring it. If Cassie's mailbox, which is 42 inches in height, casts a shadow that is 31.5 inches in length, how long is the shadow that the telephone pole casts? A. 43 feet B. 45 feet C. 52 feet D. 55 feet
The answer is (B) 45 feet
Step-by-step explanation:
We can use proportions to solve this problem.
Let x be the length of the shadow cast by the telephone pole. Then we have:
(42 / 31.5) = (60 / x)
We can cross-multiply to get:
42x = 31.5 * 60
Simplifying this equation, we get:
x = (31.5 * 60) / 42
x = 45 feet
Therefore, the length of the shadow that the telephone pole casts is 45 feet.
PLS COMPLETE ALL OF IT!! 50 POINTS!
A. The length of the cord needed to reach corner C is 17.6 m
B. The distance between the electrical outlet and corner N is 14.3 m
A. How do i determine the length of cord needed?The length of the cord needed can be obtained as follow:
Length BC = Opposite = 8 mAngle (θ) = 27°Length of cord =?Sine θ = opposite / hypotenuse
Sine 27 = 8 / Length of cord
Cross multiply
Length of cord × sine 27 = 8
Divide both sides by sine 27
Length of cord = 8 / sine 27
Length of cord = 17.6 m
B. How do i determine the distance between electrical outlet and corner NFirst, we shall determine the length OB. Details below:
Angle (θ) = 27°Length BC = Opposite = 8 mLength OB =?Tan θ = Opposite / Adjacent
Tan 27 = 8 / Length OB
Cross multiply
Length OB × tan 27 = 8
Divide both sides by tan 27
Length OB = 8 / tan 27
Length OB = 15.7 m
Finally, we shall determine the distance between the electrical outlet and corner N. Details below:
Length OB = 15.7 mLength BN = 30 mLength ON = Distance =?Length BN = Length OB + Length ON
30 = 15.7 + Distance
Collect like terms
Distance = 30 - 15.7
Distance = 14.3 m
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Twelve friends share 4 cookies equally. What fraction of a cookie does each friend get? Write in simpliest form
Answer:
2/5 of the cookie
Step-by-step explanation:
12 friends need to split 4 cookies
4 cookies needs to divided by 10 people
[tex]\frac{4cookies}{10 people}[/tex] = [tex]\frac{4}{10}[/tex]
simplify: [tex]\frac{4}{10} = \frac{2}{5}[/tex]
Bug S Bug S and Bug F is fast. Both bugs start at 0 on a number line and move in the positive direction. The bugs leave 0 at the same time and move at constant speeds. Four seconds later, F is at 12 and S is at 8. When will F and S be 100 units apart?
Answer:
Let's call the speed of Bug F v_F and the speed of Bug S v_S. Since both bugs started at 0, we can express their positions at any time t as:
Position of Bug F = 12 + v_F * t
Position of Bug S = 8 + v_S * t
To find out when F and S will be 100 units apart, we need to find the time t at which their positions differ by 100 units. In other words, we need to solve the following equation:
|12 + v_F * t - (8 + v_S * t)| = 100
We can simplify this equation by expanding the absolute value and rearranging the terms:
|4 + (v_F - v_S) * t| = 100
Now we can split this equation into two cases:
Case 1: 4 + (v_F - v_S) * t = 100
In this case, we have:
v_F - v_S > 0 (since Bug F is faster)
t = (100 - 4) / (v_F - v_S)
Case 2: 4 + (v_F - v_S) * t = -100
In this case, we have:
v_F - v_S < 0 (since Bug S is faster)
t = (-100 - 4) / (v_F - v_S)
Since we're only interested in positive values of t, we can discard the second case. Therefore, the time at which F and S will be 100 units apart is:
t = (100 - 4) / (v_F - v_S)
t = 96 / (v_F - v_S)
We don't know the values of v_F and v_S, but we can use the fact that Bug F is at 12 and Bug S is at 8, four seconds after they started. This gives us two equations:
12 = 4v_F + 0v_S
8 = 4v_S + 0v_F
Solving these equations for v_F and v_S, we get:
v_F = 3
v_S = 2
Substituting these values into the equation for t, we get:
t = 96 / (3 - 2)
t = 96
Therefore, F and S will be 100 units apart 96 seconds after they start.
please help, thank you so much
a. The probability of rolling a number greater than 10 is 1/6.
b. The probability of rolling a number less than 5 is 1/3.
c. The expected number of times a 4, 6, or 9 will be rolled in 200 rolls is 50
How to calculate the probabilitya. The probability of rolling a number greater than 10 is equal to the number of faces with numbers greater than 10 (i.e., 11 and 12) divided by the total number of faces. Thus, P(number greater than 10) = 2/12 = 1/6.
b. The probability of rolling a number less than 5 is equal to the number of faces with numbers less than 5 (i.e., 1, 2, 3, and 4) divided by the total number of faces. Thus, P(number less than 5) = 4/12 = 1/3.
c. The probability of rolling a 4, 6, or 9 is equal to the number of faces with those numbers (i.e., 1 each) divided by the total number of faces. Thus, the probability of rolling a 4, 6, or 9 is 3/12 = 1/4.
Therefore, the expected number of times a 4, 6, or 9 will be rolled in 200 rolls is:
(expected number of times) = (probability of rolling a 4, 6, or 9) x (total number of rolls)
= (1/4) x (200)
= 50
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h (x) = (3x - 4) (x + 2)^2 (x - 5)
• (2, 0)
• (-3/4, 0)
• (4/3, 0)
• (5, 0)
The zeros of the function H(x) = (3x - 4)(x + 2)^2(x - 5) are (4/3, 0), (-2, 0), and (5, 0).
Calculating the zeros of the polynomial functionTo find the zeros of the function H(x), we need to find the values of x that make the function equal to zero.
H(x) = (3x - 4)(x + 2)^2(x - 5)
Setting H(x) equal to zero, we have:
(3x - 4)(x + 2)^2(x - 5) = 0
Using the zero product property, we can see that H(x) will be equal to zero when any of the factors are equal to zero.
So, the zeros of the function H(x) are:
3x - 4 = 0, which gives x = 4/3
x + 2 = 0, which gives x = -2
x - 5 = 0, which gives x = 5
Therefore, the zeros of the function H(x) are (4/3, 0), (-2, 0), and (5, 0).
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Evan is going to invest in an account paying an interest rate of 5.4% compounded annually. How much would Evan need to invest, to the nearest dollar, for the value of the account to reach $1,360 in 5 years
On solving the provided question, we can say that - the value of the simple interest will be $ 881.28.
What is interest ?Multiplying the principal by the interest rate, time, and other factors yields simple interest. Simple return equals principle times interest times hours is the marketed formula. It is easiest to compute interest using this formula. A percentage of the principle balance is how interest is most commonly computed. The interest rate on the loan is known as this percentage.
here,
we have
P = 1360;
R = 5.4 ;
T = 12
so, we get,
SI = 1360 X 5.4 X 12 /100
SI =88128/100
= 881.28
Hence, On solving the provided question, we can say that - the value of the simple interest will be $ 881.28.
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At Christmas, Ben, Sam and Tom received cards in the ration 2 : 3 : 12.
If Tom received 60 cards.
(a) What fraction of the cards did Ben receive?
(b) What fraction did Ben and Sam receive between them?
(c) How many cards did Sam receive?
(d) How many cards did they receive altogether?
The box plots show the weights, in pounds, of the dogs in two different animal shelters.
Weights of Dogs in Shelter A
2 box plots. The number line goes from 6 to 30. For the weights of dogs in shelter A, the whiskers range from 8 to 30, and the box ranges from 17 to 28. A line divides the box at 21. For shelter B, the whiskers range from 10 to 28, and the box ranges from 16 to 20. A line divides the box at 18.
Weights of Dogs in Shelter B
Which animal shelter has the dog that weighs the least?
shelter A
Step-by-step explanation:
The minimum weight for shelter A is not provided in the given information, but we can compare the minimum weight of shelter B with shelter A's box plot.
As per the given information, the whisker of shelter A ranges from 8 to 30, which means the minimum weight in shelter A is 8 pounds. On the other hand, the whisker of shelter B ranges from 10 to 28, which means the minimum weight in shelter B is 10 pounds. Therefore, shelter A has the dog that weighs the least.
Answer:
Your answer is correct, it's shelter A.
Step-by-step explanation:
Help with math problems
Answer:
13.) y=(x-4)^(2)+3
Step-by-step explanation:
How to get the unadjusted cost of sales in cost and management accounting
Answer:
Step-by-step explanation:
To calculate unadjusted cost of goods sold, sum the beginning inventory value and the cost of goods manufactured, then subtract the ending inventory value.
what is a prime number
Answer: A prime number is a positive integer greater than 1 that has no positive integer divisors other than 1 and itself. In other words, a prime number is a number that is only divisible by 1 and itself.
Step-by-step explanation:
For example, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, and so on, are prime numbers because they can only be divided by 1 and themselves without any remainder.
However, 4 is not a prime number because it can be divided by 1, 2, and 4, and 6 is not a prime number because it can be divided by 1, 2, 3, and 6.
Answer:
A prime number is a number that can be multiplied by one and itself~
eg- 2,3,5,7,11
Let me know if this helps
A normal population has a mean of $76 and standard deviation of $6. You select random samples of 40.
1. What is the probability that a sample mean is less than $75? (Round z-value to 2 decimal places and final answer to 4 decimal places.)
2. What is the probability that a sample mean is between $75 and $77? (Round z-value to 2 decimal places and final answer to 4 decimal places.)
3. What is the probability that a sample mean is between $77 and $78? (Round z-value to 2 decimal places and final answer to 4 decimal places.)
4. What is the probability that the sampling error ( x¯
− μ) would be $1.50 or less? (Round z-value to 2 decimal places and final answer to 4 decimal places.)
Using a z-table, the probability of a z-score less than 1.58 is 0.9429 (rounded to 4 decimal places).
What is probability?Probability is a measure of the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. It is used in various fields such as mathematics, statistics, science, and finance to make predictions and analyze data.
Here,
1. The z-score for a sample mean of $75 is calculated as:
z = (75 - 76) / (6 / √(40)) = -2.36
Using a z-table, the probability of a z-score less than -2.36 is 0.0099 (rounded to 4 decimal places).
2. The z-score for a sample mean of $75 is calculated as:
z1 = (75 - 76) / (6 / √(40))
= -2.36
The z-score for a sample mean of $77 is calculated as:
z2 = (77 - 76) / (6 / √(40))
= 0.79
Using a z-table, the probability of a z-score between -2.36 and 0.79 is 0.8669 (rounded to 4 decimal places).
3. The z-score for a sample mean of $77 is calculated as:
z1 = (77 - 76) / (6 / √(40))
= 0.79
The z-score for a sample mean of $78 is calculated as:
z2 = (78 - 76) / (6 / √(40))
= 1.57
Using a z-table, the probability of a z-score between 0.79 and 1.57 is 0.0823 (rounded to 4 decimal places).
4. The standard error of the mean (SEM) is calculated as:
SEM = standard deviation / sqrt(sample size)
SEM = 6 / √(40) = 0.9487
The z-score for a sampling error of $1.50 is calculated as:
z = 1.50 / 0.9487 = 1.58
Using a z-table, the probability of a z-score less than 1.58 is 0.9429 (rounded to 4 decimal places).
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Please help need to get a good score
A container contains 145.2 ounces of lemonade. If the lemonade is poured equally into 15 cups, how many ounces will be poured into each cup?
A. 8.78
B. 9.12
C. 9.64
D. 9.68
Show answer.
Answer: D. 9.68
Step-by-step explanation:
145.2 oz/ 15 cups = 9.68 oz per cup
Complete the ratio table to convert the units of time from hours to weeks or weeks to hours.
Hours:
168 1 week
1,008. ____week
_____. 5 weeks
Answer:
6 weeks and 840 hours
Step-by-step explanation:
There are 168 hours in one week.
24 hrs/day * 7 days = 168 hours
1008 hours ÷ 24 hours(1 day) = 42 days ÷ 7 days in a week = 6 weeks
168 hours/week * 5 weeks = 840 hours
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Simultaneous equations are a set of equations that are solved together to determine the values of the variables that satisfy both equations.
What are Simultaneous equations?1) 4x + 5y = 3 ---(1)
y - 3x = -7 ----(2)
Using the substitution method;
y = -7 + 3x ----(3)
Thus
4x + 5(-7 + 3x) = 3
4x -35 +15x = 3
19x - 35 = 3
19x = 3 + 35
x = 2
Then
4(2) + 5y = 3
8 + 5y = 5
y = 13/5
y = 2 3/5
2) 2x - 4y = 24 ----- x 3
-3x + 2y = -48 ----- x 2
6x - 12 y = 72 ---- (3)
-6x + 4y = -96 ---- (4)
Add 3 and 4
16y = -24
y = -24/16
Substitute y = -24/16 into (1)
2x - 4(-24/16) = 24
2x + 6 = 24
x = 15
3) -x + y = 13 --- 1
3x - 4y = 46 ---- 2
y = 13 + x ---- 3
Substitute 3 into 2
3x - 4(13 + x) = 46
3x - 52 - 4x = 46
-x - 52 = 46
x = 98
Substitute x = 98 into (1)
-98 + y = 13
y = 13 + 98
y = 111
Let C be x and D be y
x + y = 180
x = 33 + 6y
x - 6y = 33
x = 180 - y
Substitute and obtain;
180 - y - 6y = 33
180 - 7y = 33
y = 33 - 180/-7
y = 21
Then
x + 21 = 180
x = 180 - 21
x = 159
Lastly
Let small = x , medium = y
x + y = 150
4x + 6y = 764
x = 150 - y
4(150 - y) + 6y = 764
600 - 4y + 6y = 764
600 + 2y = 764
2y = 764 - 600
y = 82
Then;
x + 82 = 150
x = 68
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The solutions to the simultaneous equations are:
1) x = 2 and y = -1
2) x = 18 and y = 3
3) y = -7 and x = 6
How to Solve the System of simultaneous Linear Equations?There are three main methods in solving simultaneous equations and they are:
1) Elimination Method
2) Substitution Method
3) Graphical Method
1) 4x + 5y = 3
y = 3x - 7
Substitute 3x - 7 for y in the first equation to get:
4x + 5(3x - 7) = 3
4x + 15x - 35 = 3
19x - 35 = 3
19x = 35 + 3
19x = 38
x = 38/19
x = 2
Thus:
y = 3(2) - 7
y = -1
2) 2x - 4y = 24
-3x + 2y = -48
Multiply eq 2 by 2 and eq 1 by 1 to get:
2x - 4y = 24 -----(3)
-6x + 4y = -96 -----(4)
Add eq 3 to eq 4 to get:
-4x = -72
x = 18
2(18) - 4y = 24
36 - 4y = 24
36 - 24 = 4y
4y = 12
y = 3
3) -x + y = -13
3x - 4y = 46
From eq 1, x = y + 13
Thus:
3(y + 13) - 4y = 46
3y + 39 - 4y = 46
39 - y = 46
y = 39 - 46
y = -7
x = -7 + 13
x = 6
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A quality assurance check is 91% accurate for non-defective devices and 97% accurate for defective devices. Of the devices checked, 84% are not defective. What is the probability of an incorrect conclusion? Round your answer to the nearest tenth of a percent.
Answer: To solve the problem, we can use Bayes' theorem. Let D be the event that a device is defective, and let A be the event that the quality assurance check concludes that a device is defective.
We want to find P(A and not D) + P(not A and D), which represents the probability of an incorrect conclusion.
We know that P(D) = 1 - P(not D) = 1 - 0.84 = 0.16, and that P(A | not D) = 0.03 and P(A | D) = 0.97.
Using Bayes' theorem, we can compute:
P(not A | not D) = 1 - P(A | not D) = 1 - 0.03 = 0.97
P(not A | D) = 1 - P(A | D) = 1 - 0.97 = 0.03
Therefore,
P(A and not D) = P(not D) * P(A | not D) = 0.84 * 0.03 = 0.0252
P(not A and D) = P(D) * P(not A | D) = 0.16 * 0.03 = 0.0048
So the probability of an incorrect conclusion is:
P(A and not D) + P(not A and D) = 0.0252 + 0.0048 = 0.03
Therefore, the probability of an incorrect conclusion is 0.03, or 3% (rounded to the nearest tenth of a percent).
Why was this answer deleted prior?
What will be the result of substituting 2 for x in both expressions below?
Substituting for x in an expression means replacing the variable x with a specific value or expression. This is often done to evaluate the expression for that particular value or to simplify the expression.
What is the substituting for x in expressions?Substituting 2 for x in the first expression, we get:
[tex]1/2(2) + 4(2) + 6 - 1/2(2) - 2 = 1 + 8 + 6 - 1 - 2 = 12[/tex]
Substituting 2 for x in the second expression, we get:
[tex]2(2) + 2 - 1 = 5[/tex]
One expression equals 5 when substituting 2 for x, and the other equals 2 because the expressions are not equivalent.
Therefore, the first expression evaluated with x = 2 is 12, and the second expression evaluated with x = 2 is 5. Since they do not have the same value, the correct option is:
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The given question is incomplete. The complete question is given below:
What will be the result of substituting 2 for x in both expressions below? One-half x + 4 x + 6 minus one-half x minus 2 Both expressions equal 5 when substituting 2 for x because the expressions are equivalent. Both expressions equal 6 when substituting 2 for x because the expressions are equivalent. One expression equals 5 when substituting 2 for x, and the other equals 2 because the expressions are not equivalent. One expression equals 6 when substituting 2 for x, and the other equals 2 because the expressions are not equivalent.