An integer is a whole number that can be either positive, negative, or zero. In mathematics, a theorem is a statement that has been proven to be true using logic and reasoning. Theorem 1 states that the integer n = 33 is not divisible by 5.
To show that an integer n is divisible by 5, we need to find an integer k such that n = 5k. Let's apply this definition to each of the given integers.
(a) To show that n = 45 is divisible by 5, we need to find an integer k such that n = 5k. We can see that k = 9 satisfies this condition since 5k = 5(9) = 45. Therefore, 45 is divisible by 5.
(b) To show that n = -110 is divisible by 5, we need to find an integer k such that n = 5k. We can see that k = -22 satisfies this condition since 5k = 5(-22) = -110. Therefore, -110 is divisible by 5.
(c) To show that n = 0 is divisible by 5, we need to find an integer k such that n = 5k. We can see that k = 0 satisfies this condition since 5k = 5(0) = 0. Therefore, 0 is divisible by 5.
(d) To prove Theorem 1, we will use proof by contradiction. Let's assume that n = 33 is divisible by 5, which means there exists an integer k such that n = 5k. Then, we have 33 = 5k, which implies that k = 6.6. However, k must be an integer according to the definition of divisibility. Therefore, we have reached a contradiction, and our assumption that n = 33 is divisible by 5 must be false. Hence, Theorem 1 is proven.
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Seven playing cards are drawn from a deck without replacement. A success is recorded each time a card that shows a diamond is drawn. Check all that apply. 1. The outcome of each trial is independent of those of other trials. 2. There is a fixed number of n trials. 3. The probability of each possible outcome in any trial is the same from trial to trial. 4. Each trial has only two possible (mutually exclusive) outcomes. This example _________ a binomial experiment.
This example does not qualify as a binomial experiment because the conditions of a binomial experiment are not all met.
While there are only two possible outcomes (drawing a diamond or not), the other conditions are not satisfied. Specifically, the outcome of each trial is not independent of those of other trials because cards are drawn without replacement, and there is not a fixed number of n trials as the number of trials depends on how many cards are drawn until seven diamonds are obtained. Additionally, the probability of each possible outcome in any trial is not the same from trial to trial because the number of cards in the deck changes as cards are drawn.
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Geometry: find how much glass is needed to build.
The amount of glass needed = surface area of the triangular prism = 2,646 cm².
What is the Surface Area of a Triangular Prism?The glass has a triangular prism shape. Therefore, the amount of glass needed to build the showcase is calculated by finding the surface area of the image given.
Surface area = amount of glass needed = Perimeter of triangular face * length of prism + 2 * base area of triangular face
= (S1 + S2 + S3) * L + bh
We are given the variables as:
S1 = 15 cm
S2 = 15 cm
S3 = 24 cm
L = 45 cm
b = 24 cm
h = 9 cm
Plug in the values:
Surface area = (15 + 15 + 24) * 45 + 24 * 9 = 2,430 + 216
amount of glass needed = 2,646 cm²
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Triangle RST is similar to triangle RVW .
What is the value of d in millimeters?
The value of d in millimeters is 12 mm and this can be determined by using the similar triangle property.
How to calculate the valueTriangle RST is similar to triangle RVW.
The length of the segment RW = 10 mm
The length of the segment WT = 5 mm
The length of the segment TS = 18 mm
The following steps can be used in order to determine the value of d in millimeters:
The similar triangle property can be used in order to determine the value of d in millimeters.
The value will be:
= 10/15 × 18
= 12
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The polygon is composed of three rectangles. 4 ft 4 ft 2 ft 3 ft 4 ft 8 1 2 ft What is the area, in square feet, of the polygon?
For a polygon which is composed of three rectangles and dimensions are 4 ft× 2ft, 4 ft× 3 ft , 8 ft × 1.2ft. Area of polygon is 29.6 ft².
In geometry, a polygon is defined as the flat or plane surface, two-dimensional closed shape of boundaries. The sides of a polygon are also known as its edges. The points where two sides meet are called vertices (or corners) of a polygon. We have a polygon is composed of three rectangles.
The dimensions of rectangles are the following, 4 ft× 2ft, 4 ft× 3 ft , 8 ft × 1.2ft. We have to determine the area of polygon in square feet. The area of a polygon is defined as total space covered within the shape. The measurement is completed with square units. Rectangles are regular shape. The area of rectangle = length × width
Total area of polygon is equals to the sum of areas of three rectangles. So, area of polygon = 4 ft × 2 ft + 4 ft × 3 ft + 8 ft × 1.2 ft
= 8 ft² + 12 ft² + 9.6 ft²
= 29.6 ft²
Hence, required value is 29.6 ft².
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Find the probability that a randomly
selected point within the circle falls in the
red-shaded triangle.
24
13
10
13
P = [?]
Enter as a decimal rounded to the nearest hundredth.
Enter
The probability that the random;y selected point will be in the triangle = 0.33
How to solve for the probabilitysolve for area covered by the trangke
The area of the triangle is guven as 1/2 x b * h
b = base
h = height
The base = 10
The height = 24
The area = 1 / 2 x 10 x 24
= 240 / 2
= 120
Then we know that the complte angle of a cirle = 360 degrees
The probability that the random;y selected point will be in the triangle = 120 / 360
= 12 / 36
= 0.33
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Answer.
0.23
find area of triangle =120 then find area of circle= 530.66 then divide area of triangle by area of circle
the top of a silo is a hemisphere with a radius of 8 feet.the cylindrical body of the silo shares the same radius as the hemisphere and has a height of 40 feet.
A truck hauling grain To the silo has a rectangular container attached to the back that is 8' ft In length 5ft in Width and 4' ft height.
Determine the number of truck loads of grain required to fill an empty silo
help please
The number of truck loads of grain required to fill an empty silo is 51.97
How to solve for the truck loadsVolume of the hemisphere = 2/3)πr^3,
Volume of hemisphere would be
[tex]hemisphere = (2/3)\pi (8 ft)^3 = 268.08 ft^3[/tex]
Volume of cylinder = πr^2h
Then we will have
[tex]cylinder = \pi(8 ft)^2(40 ft) \\= 8046.72 ft^3[/tex]
Total volume
[tex]V_hemisphere + V_cylinder = 8314.80 ft^3[/tex]
[tex](8 ft)(5 ft)(4 ft) = 160 ft^3[/tex]
Number of truck loads
= 8314.80 ft^3 / 160 ft^3
= 51.97
Hence the number of truck loads of grain required to fill an empty silo is 51.97
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A playhouse is in the shape of a regular octagonal pyramid with a side length of 3 feet and a slant height of 12 feet. The wood used to build the walls of the playhouse costs $4 per square foot. What is the cost of the wood for the walls of the playhouse?
The cost of the wood for the walls of the playhouse is $1141.44.
To calculate the cost of the wood for the walls of the playhouse, we need to find the surface area of the walls and then multiply it by the cost per square foot.
The surface area of the walls of an octagonal pyramid can be calculated by finding the area of each trapezoidal face and adding them up. Since the side length of the pyramid is 3 feet and the slant height is 12 feet, we can use the Pythagorean theorem to find the height of each trapezoidal face:
h = √(12² - (3/2)²)
h = √(144 - 2.25)
h = √(141.75)
h ≈ 11.89 feet
The area of each trapezoidal face is:
A = 1/2 * (b1 + b2) * h
A = 1/2 * (3 + 3) * 11.89
A ≈ 35.67 square feet
There are 8 trapezoidal faces in the octagonal pyramid, so the total surface area of the walls is:
SA = 8 * A
SA ≈ 285.36 square feet
Finally, we can calculate the cost of the wood for the walls by multiplying the surface area by the cost per square foot:
Cost = SA * $4
Cost = 285.36 * $4
Cost = $1141.44
Therefore, the cost of the wood for the walls of the playhouse is $1141.44.
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Need help asap. Write a explicit formula for a^n, the n^th term of the sequence 33,30,27
The explicit formula of the sequence is -3n + 36.
How to find the explicit formula of a sequence?The sequence above is a arithmetic progression. Therefore, let's write the nth term of the sequence.
Hence,
33, 30, 27
a + (n - 1)d = nth term
where
a = first termn = number of termsd = common differenceTherefore,
a = 33
d = 30 - 33 = -3
n = number of term
Hence,
nth term = 33 + (n - 1)-3
nth term = 33 - 3n + 3
nth term = -3n + 36
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Bob had $110 in his bank account before writing a check to invest in his next big adventure. After writing the check, Bob found that he had a balance of -$24 in his account. How much money was on the check Bob wrote?
Show steps
$134
$110+$24=$134
if Bob had 110 in his account and found he had a -24 balance, you would need to add the two together to find the check amount written.
e. a 20 foot by 10 foot rectangular pool has been built. if 50 cubic feet of water is pumped into the pool per hour, write the water-level height (feet) as a function of time (hours).
To find the water-level height (feet) as a function of time (hours), we need to know the volume of the pool and how much water is being pumped in per hour.
The volume of the rectangular pool can be found by multiplying its length, width, and height:
Volume = Length x Width x Height
Since we know the dimensions of the pool are 20 feet by 10 feet, we can assume the height is 5 feet (half the length of the pool).
Volume = 20 ft x 10 ft x 5 ft = 1000 cubic feet
This means the pool can hold 1000 cubic feet of water.
If 50 cubic feet of water is pumped into the pool per hour, we can write the water-level height (h) as a function of time (t) as follows:
h(t) = (50t) / 1000
where t is the time in hours.
For example, after 1 hour, the water-level height would be:
h(1) = (50 x 1) / 1000 = 0.05 feet
After 2 hours, the water-level height would be:
h(2) = (50 x 2) / 1000 = 0.1 feet
And so on.
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Write the following absolute value function as a piecewise function.
please help
The absolute value function as a piecewise function is
f(x) = -(-x^2 + 9x - 18), x < 3 and x > 6f(x) = -x^2 + 9x - 18, 3 ≤ x ≤ 6Writing the absolute value function as a piecewise function.Given that
f(x) = |-x^2 + 9x - 18|
When the expression is factored, we have
f(x) = |-(x - 3)(x - 6)|
Set the expression in the absolute bracket to 0
This gives
-(x - 3)(x - 6) = 0
When the equation is solved for x, we have
x = 3 and x = 6
These values represent the boundaries of the piecewise function
So, we have
f(x) = -(-x^2 + 9x - 18), x < 3 and x > 6
f(x) = -x^2 + 9x - 18, 3 ≤ x ≤ 6
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PLEASE HELP THIS IS MY LAST QUESTION (07.05 MC)
The graph below represents the function f(x) = (x + 2)(x - 2)(x - A) which has a y-intercept of 12.
The missing value A is
-15
-10 -5
838
30
25
20
15
10
15
Step-by-step explanation:
The y-axis intercept occurs when x = 0
put in 0 for 'x' and compute the intercept as
(0+2)(0-2)(0-A) = 12
-4 ( -A) = 12
4A =12
A = 3
passengers need to validate their tickets on their own using a punching machine that creates holes on the ticket. transportation officials randomly travel around town and ask for the passengers' validated tickets. the tickets do not expire. in theory, the ticket needs to be inserted into the punching machine with the red arrow on top. in practice, this does not matter since the officials do not care about the direction. so, inserting the ticket with the red arrow on the bottom creates the same ticket. a fee evader wants to collect every possible validated ticket and use the appropriate one every time he/she travels. how many different validated tickets are needed if every punching machine in town creates 4 holes on a ticket?
There are 16 different validated tickets are needed if every punching machine in town creates 4 holes on a ticket
When a ticket is punched by a punching machine, it creates a hole in the ticket. In this case, each hole can either be punched or not punched, so there are 2 possibilities for each hole.
Since there are 4 holes on a ticket, the total number of possible combinations is calculated by multiplying the number of possibilities for each hole:
2 x 2 x 2 x 2 = 16
So, there are 16 possible combinations of holes on a ticket, which means that a fee evader would need 16 different validated tickets to cover all possible combinations. This assumes that each punching machine creates the same pattern of holes, which may not be the case in practice.
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2-1/3-2/+1 in its simplest fraction
Answer:
7/3
Step-by-step explanation:
2-1/3-2/+1 in its simplest fraction is equal to 7/3.
Evaluate the integral by interpreting it in terms of areas. 4/−3 (1 − x) dx
Answer:
[tex] \frac{2}{3} square \: units[/tex]
The flow rate y (m3/min) in a device used for air-quality measurement depends on the pressure drop x (in. of water) across the device’s filter. Suppose that for x values between 5 and 20, the two variables are related according to the simple linear regression model with true regression line y = –.12 + .095x.a. What is the expected change in flow rate associated with a 1-in. increase in pressure drop? Explain.b. What change in flow rate can be expected when pressure drop decreases by 5 in.?c. What is the expected flow rate for a pressure drop of 10 in.? A drop of 15 in.?d. Suppose σ = .025 and consider a pressure drop of 10 in. What is the probability that the observed value of flow rate will exceed .835? That observed flow rate will exceed .840?e. What is the probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in.?
The probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in. is .7602.
a. The expected change in flow rate associated with a 1-in. increase in pressure drop is the slope of the regression line, which is .095 m3/min per in. of water. This means that for each additional inch of pressure drop, we can expect the flow rate to increase by an average of .095 m3/min.
b. When pressure drop decreases by 5 in., we can expect the flow rate to decrease by an average of .095 * (-5) = -.475 m3/min.
c. For a pressure drop of 10 in., the expected flow rate can be calculated by plugging x = 10 into the regression line equation: y = -.12 + .095(10) = .838 m3/min.
d. To find the probabilities, we need to standardize the flow rate values using the formula z = (y - μ) / σ, where μ is the mean flow rate and σ is the standard deviation. For a pressure drop of 10 in., the expected flow rate is .838 m3/min, so
P(Y > .835) = P(Z > (.835 - .838) / .025) = P(Z > -.12) = .4522
P(Y > .840) = P(Z > (.840 - .838) / .025) = P(Z > .08) = .4681
where Z is a standard normal random variable.
e. We need to find the probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in. This can be done by subtracting the mean flow rate for each pressure drop from their respective observations, and then finding the probability that the difference is positive. Let Y_10 and Y_11 denote the flow rates for pressure drops of 10 in. and 11 in., respectively. Then the probability of interest is:
P(Y_10 - Y_11 > 0) = P((Y_10 - μ_10) - (Y_11 - μ_11) > -(μ_11 - μ_10))
where μ_10 and μ_11 are the mean flow rates for pressure drops of 10 in. and 11 in., respectively. Since the regression line is linear, we can find the mean flow rate for any given pressure drop x using the equation μ = -.12 + .095x. Therefore,
μ_10 = -.12 + .095(10) = .758 m3/min
μ_11 = -.12 + .095(11) = .853 m3/min
Substituting these values into the probability expression gives:
P(Y_10 - Y_11 > 0) = P((Y_10 - .758) - (Y_11 - .853) > -.095)
We know from part (a) that the standard deviation of the flow rate is σ = .095 m3/min per in. of water. Therefore, the standard deviation of the difference Y_10 - Y_11 is
σ_diff = sqrt(σ^2 + σ^2) = sqrt(2)*σ = .134 m3/min
Using the formula for a standardized normal variable, we have:
P((Y_10 - .758) - (Y_11 - .853) > -.095) = P(Z > (-.095 / .134)) = P(Z > -.71) = .7602
where Z is a standard normal random variable. Therefore, the probability that an observation on flow rate when pressure drop is 10 in. will exceed an observation on flow rate made when pressure drop is 11 in. is .7602.
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PLEASE HELP
Which of the following sentences is written in indicative mood?
If the coach would give a pep talk, then the team would play better.
The team plays much better after a pep talk from their coach.
Will the team play better after a pep talk from their coach?
If I were the coach, I'd give the team a pep talk.
The sentence written in the indicative mood is: "The team plays much better after a pep talk from their coach."
What is indicative mood ?The grammatical mood known as the indicative is employed to state or inquire about facts.
The verb forms in the indicative mood show that the action or state described is actually occurring or has already occurred. For instance, the statement "I am walking to the store" is suggestive since it is a factual statement describing an action that is now occurring.
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help me please please please
1) the mean, median, mode, and range of the set of data are given below.
What are the definition of the above terms?When considering a set of numbers, several measures can be used to describe the data. The mean, for example, is determined by adding all individual values together and dividing by the total number of elements in the set.
This value is representative of an average quantity among the group studied. On the other hand, if one were to arrange said values from smallest to largest, the median would represent the middle-most number in that list - or, if two middle numbers exist, their mean.
Range on the other hand is the variance between the largest and the smallest number in a data set.
Lastly but not least important is the mode, which indicates the most frequently appearing value within our dataset; or alternatively so noted as when there are multiple repetitions.
So here is the Mean, Median, Mode and Range for the given sets of data:
1)
Mean = (4.3 + 5.2 + 4.5 + 5.1 + 4.8 + 5.4 + 4.5 + 4.7 + 4.3 + 5.2 + 4.5 + 4.8 + 5.1) / 13
= 4.8
Mean ≈ 4.8
Median = when arranged in ascending order, the data se become:
4.3,4.3,4.5,4.5,4.5,4.7,4.8,4.8,5.1,5.1,5.2,5.2,5.4
Since there are 13 observation, 7th observation is the median.
4.3,4.3,4.5,4.5,4.5,4.7,| 4.8, | 4.8,5.1,5.1,5.2,5.2,5.4
hence median = 4.8
Note that where the number of data is even in number, the median become the average of the two middle numbers.
Mode - the number that occrs the highest is 4.5. It occurs thrice.
Range = Highest Data Value - Lowest Data Value
Range = 5.4 - 4.3
= 1.10
Using the above steps we derive the mean median, mode and range for the other data set:
2) 12.6, 12.8, 9.7, 10.4, 9.7, 10.8, 12.4, 12.8, 11.5, 10.4, 10.9, 12.8
Total of 12 number
Data in ascending order: 9.7,9.7,10.4,10.4,10.8,10.9,11.5,12.4,12.6,12.8,12.8,12.8
Mean = 11.4
Median = (10.9 +11.5)/2 = 11.2
Mode = 12.8
Range = 3.10
3)
-6, -13, -8, -3, -7, -10, 2, 0, -3, -5, 5, 7, -6, 2, 1, -6, -18
Data in ascending order; -12, -10, -8, -7, -4, -3, -2, -1, 0, 0, 0, 1, 2, 3, 4, 5, 7, 7
Mean = -1
Median = 0
Mode = 0
Range = 19
4) -6, -13, -8, -3, -7, -10, 2, o, -3, -5, 5, 7, -6, 2, 1, -6, -18
Data in ascending order: -18, -13, -10, -8, -7, -6, -6, -6, -5, -3, -3, 1, 2, 2, 5, 7
Mean = -4.25
Median = -5.5
Mode = -6
Range = 25
5) 0.24, 0.31, 0.43, 0.22, 0.34, 0.24, 0.35, 0.4, 0.18, 0.3, 0.29
Data in ascending order: 0.18, 0.22, 0.24, 0.24, 0.29, 0.3, 0.31, 0.34, 0.35, 0.4, 0.43
Mean = 0.3
Median = 0.3
Mode = 2.4
Range = 2.5
6) -0.6, 0.4, 0.2, -0.3, 0.1, -0.5, 0.2, 0.4, 1.1, -0.6, 0.7, o, 0.2, -1.3
Data in ascending order: -1.3, -0.6, -0.6, -0.5, -0.3, 0.1, 0.2, 0.2, 0.2, 0.4, 0.4, 0.7, 1.1
Mean = 0
Median = 0.2
Mode = 0.2
Range = 2.4
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Two widgets and five gadgets cost $57. One widget and three gadgets cost $32.70 How much does one gadget cost?
Answer:
$8.40
Step-by-step explanation:
2w + 5g = 57
w + 3g = 32.7
w = 32.7 - 3g
2(32.7 - 3g) + 5g = 57
65.4 - 6g + 5g = 57
8.4 = g
Answer: $8.40
HELP
The table represents a quadratic function.
x y
−6 23
−5 8
−4 −1
−3 −4
−2 −1
−1 8
0 23
What is the equation of the function?
y = (x + 3)2 − 4
y = (x − 3)2 + 4
y = 3(x + 3)2 − 4
y = 3(x − 3)2 + 4
Answer: y = (x + 3)2 − 4 is the equation of the function.
Step-by-step explanation:
4. If (a, b) = 1, prove that (a?, b2) = 1. = =
It has been proved that if (a, b) = 1, then (a², b²) = 1.
If I understand correctly, you want to prove that if (a, b) = 1, then (a², b²) = 1.
Co-prime numbers or relatively prime numbers are those numbers that have their HCF (Highest Common Factor) as 1. In other words, two numbers are co-prime if they have no common factor other than 1.
Since (a, b) = 1, it means that a and b are coprime, which means they have no common factors other than 1. Now, let's consider their squares, a², and b².
If a² and b² had a common factor other than 1, then this factor would also be a factor of a and b, which contradicts our initial assumption that (a, b) = 1.
Therefore, (a², b²) must also be equal to 1, proving that if (a, b) = 1, then (a², b²) = 1.
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Find an interval of t-values such that c(t) = (cos t, sin t) traces the upper half of the unit circle (in the counter-clockwise direction), interval = Note: Use lowercase "pi" for pi. Example answer: [0,1 ].
The interval of t-values such that c(t) = (cos t, sin t) traces the upper half of the unit circle (in the counter-clockwise direction) is [0, pi].
To see why this is the case, recall that the unit circle is given by the equation x^2 + y^2 = 1, where (x,y) are the coordinates of a point on the circle. The upper half of the unit circle corresponds to the set of points (x,y) where y is positive or zero. We want to find the values of t for which c(t) lies on the upper half of the unit circle.
Using the definition of c(t), we have c(t) = (cos t, sin t). The y-coordinate of c(t) is sin t, so we want, sin t to be positive or zero. Since sin t is positive in the first and second quadrants of the unit circle, and zero at t = 0 and t = pi, we have that c(t) traces the upper half of the unit circle when t is in the interval [0, pi].
To see that c(t) traces the upper half of the unit circle in the counter-clockwise direction, note that as t increases from 0 to pi, c(t) moves counterclockwise around the unit circle, starting at (1,0) and ending at (-1,0). Thus, the interval [0, pi] corresponds to one-half of a full counterclockwise rotation around the unit circle, which is exactly the upper half of the circle.
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exercise 1.3 introduces a study where researchers collected data to examine the relationship between air pollutants and preterm births in southern california. during the study air pollution levels were measured by air quality monitoring stations. length of gestation data were collected on 143,196 births between the years 1989 and 1993, and air pollution exposure during gestation was calculated for each birth. (a) identify the population of interest and the sample in this study. (b) comment on whether or not the results of the study can be generalized to the population, and if the findings of the study can be used to establish causal relationships.
The population of interest in this study is all births in southern California between the years 1989 and 1993. The sample in this study is 143,196 births for which length of gestation data and air pollution exposure during gestation were collected.
The results of this study cannot be generalized to the entire population of births in southern California beyond the years 1989 to 1993. However, the findings of the study can still provide valuable insights into the relationship between air pollutants and preterm births in this specific population and time period. It is also important to note that this study alone cannot establish causal relationships between air pollutants and preterm births, as other factors may contribute to preterm births that were not measured or accounted for in this study. Further research and analysis would be needed to establish causal relationships.
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WILL GIVE BRAINLIEST The following data shows the grades that a 7th grade mathematics class received on a recent exam. {98, 93, 91, 79, 89, 94, 91, 93, 90, 89, 78, 76, 66, 91, 89, 93, 91, 83, 65, 61, 77} Part A: Determine the best graphical representation to display the data. Explain why the type of graph you chose is an appropriate display for the data. (2 points) Part B: Explain, in words, how to create the graphical display you chose in Part A. Be sure to include a title, axis label(s), scale for axis if needed, and a clear process of how to graph the data. (2 points)
A) The best graphical representation for the given data is a histogram.
B) The histogram of the given data is illustrated below.
Part A:
A histogram is a type of bar graph that shows the frequency distribution of a set of continuous or discrete data. The given data is a set of discrete data, and a histogram is the most appropriate graph to display the distribution of these data.
Part B:
To create a histogram for the given data, we need to follow these steps:
In summary, to create a histogram for the given data, we need to provide a title, label the x and y-axes, choose an appropriate scale for the x-axis, plot the data, and add final touches to make the graph more informative and visually appealing.
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please help i need to get this work done
The solution to the polynomial division is:
3x³ + 7x² + 5x - 1 - 4/(2x - 3)
How to carry out polynomial long division?A long division polynomial is defined as an algorithm that is used in dividing polynomial by another polynomial of the same or a lower degree. The long division of polynomials is made up of the divisor, quotient, dividend, and the remainder as in the long division method of numbers.
We are given the polynomial functions as:
f(x) = 6x⁴ - 23x³ + 31x² - 17x - 1
g(x) = 2x - 3
Using polynomial long division we have:
3x³ + 7x² + 5x - 1
2x - 3|6x⁴ - 23x³ + 31x² - 17x - 1
- 6x⁴ - 9x³
-14x³ + 31x²
- -14x³ + 21x²
10x² - 17x
- 10x² - 15x
- 2x - 1
- -2x + 3
- 4
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Factor 12+54. Write your answer in the form a(b+c) where a is the GCF of 12 and 54
For the answer of factors of expression (12 + 54), in the form of a(b + c), where a is the GCF of 12 and 54 is equals to 6( 2 + 9).
In math, to factor a number means to express it as a product of (other) whole numbers, called its factors. For example, if 7x5 = 35, 7 and 5 are both factors. The divisors that give the remainder to be 0 are the factors of the number. We have an expression of numbers, 12 + 54. We have to write this expression in form of a( b + c), where a is GCF of 12 and 54. Now, we can write the factors of 12 and 54 are 12 = 2×2×3
54 = 2×3 ×3×3
The greatest common factor, GCF of 12 and 54 is 2×3 = 6. So, 12 + 54 = 6× 2 + 6×9
Taking out the common factor 6 from above expression, 6( 2 + 9) which is required form a( b + c). Hence, required expression is 6( 2 + 9).
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-k -> Find the maximum Likelihood Estimates of t When pre f(t) = (1-tjok ott For K=Ogl K
The maximum likelihood estimate of t is at the endpoint t = 0.
We have,
To find the maximum likelihood estimates of t, follow these steps:
1. Write down the likelihood function L(t) for the given pdf f(t).
The likelihood function is the same as the pdf, which is:
L(t) = (1 - t)^k
2. Take the natural logarithm of the likelihood function, ln(L(t)), to make it easier to work with:
ln(L(t)) = ln((1 - t)^k)
3. Use the properties of logarithms to simplify the expression:
ln(L(t)) = k x ln(1 - t)
4. Differentiate ln(L(t)) with respect to t to find the critical points that might correspond to the maximum likelihood estimate:
d(ln(L(t))) / dt = - k / (1 - t)
5. Set the derivative equal to zero and solve for t:
- k / (1 - t) = 0
Since k is nonzero, this equation implies that there is no solution for t in the interval [0, 1].
Thus, the maximum likelihood estimate of t does not occur at a critical point in the interval.
6. Since there are no critical points, we must check the endpoints of the interval, t = 0 and t = 1, to find the maximum likelihood estimate.
The likelihood function L(t) = (1 - t)^k has its maximum value at the endpoint where the derivative is positive.
In this case,
The derivative -k / (1-t) is positive when t = 0.
Thus, the maximum likelihood estimate of t is at the endpoint t = 0.
Thus,
The maximum likelihood estimate of t is at the endpoint t = 0.
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Q8 (6 points) Let x be a binomial random variable with n = 100 and p = 0.3. (a) Can we use the Poisson approximation to find P(30 < = x < 35)? Why? (b) Use the normal approximation to find P(30 < = x< 50) points) If x is a binomial random variable with n = 4 and P(0) = 0.0081, find P(3).
P(3) is approximately equal to 0.139.
(a) Yes, we can use the Poisson approximation to find P(30 < x < 35) because both np and n(1-p) are greater than or equal to 10, where n = 100 and p = 0.3. Therefore, the conditions for the Poisson approximation are satisfied.
Using Poisson approximation, we have:
λ = np = 100 x 0.3 = 30
P(30 < x < 35) ≈ P(X = 31) + P(X = 32) + P(X = 33) + P(X = 34)
= e^(-λ) * ([tex]λ^31[/tex] / 31!) + e^(-λ) * (λ^32 / 32!) + e^(-λ) * (λ^33 / 33!) + e^(-λ) * (λ^34 / 34!)
≈ 0.1885
(b) Using the normal approximation, we have:
µ = np = 100 x 0.3 = 30
σ = sqrt(np(1-p)) = sqrt(100 x 0.3 x 0.7) = 4.58
P(30 < x < 50) ≈ P((30 - µ)/σ < (x - µ)/σ < (50 - µ)/σ)
≈ P(-4.34 < Z < 4.34) [where Z is a standard normal random variable]
≈ 1
Therefore, P(30 < x < 50) is approximately equal to 1.
(c) Let x be a binomial random variable with n = 4 and P(0) = 0.0081.
We need to find P(3).
Let P(1) = q
Then, from the given information, we have:
P(0) = (1-q)^4 = 0.0081
Solving for q, we get:
q = 1 - (0.0081)^(1/4) ≈ 0.207
Now, using the binomial probability formula, we have:
P(3) = (4 choose 3) * q^3 * (1-q)^1
= 4 * 0.207^3 * 0.793
≈ 0.139
Therefore, P(3) is approximately equal to 0.139.
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Find volume of the solid
The volume of the cylinder is 803.9 ft².
Given is oblique cylinder, we need to find it volume,
Volume = π × radius² × height
The radius = 8 ft
The height = 4 ft
So,
The volume = 3.14 × 8² × 4
= 803.9 ft²
Hence, the volume of the cylinder is 803.9 ft².
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Please help, I need a fast answer.
Which shortcut can be used to prove . There may be more than one answer. Select all that apply.
The shortcut that can be used to prove ΔAET ≅ ΔFRP is ASA (option d).
The given information includes the measure of some angles and the fact that two sides of the triangles are congruent. To prove that two triangles are congruent, you need to show that all their corresponding sides and angles are congruent.
To determine which shortcut can be used to prove that ΔAET ≅ ΔFRP, we need to check which postulate applies to the given information.
We know that angle AET is congruent to angle FRP (given), AE is congruent to FR (given), and angle T and angle P are congruent (given).
Therefore, the shortcut that applies to this situation is the ASA postulate, which states that two angles and the included side of the triangles are congruent. Thus, we can conclude that ΔAET ≅ ΔFRP by the ASA postulate.
Hence the correct option is (d).
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