It is generally appropriate to calculate a probability about the difference in sample means when the sample sizes are large (typically n ≥ 30) and the populations are approximately normally distributed.
How did we arrive at this assertion?To determine if it is appropriate to calculate a probability about the difference in sample means, consider the assumptions and conditions for conducting a hypothesis test or constructing a confidence interval. The appropriateness of calculating a probability about the difference in sample means depends on the following factors:
1) Both population shapes are unknown. n1 = 50 and n2 = 100:
- It is generally appropriate to calculate a probability about the difference in sample means when the sample sizes are large (typically n ≥ 30) and the populations are approximately normally distributed. Since the population shapes are unknown in this case, it is difficult to assess this condition. However, the large sample sizes (n1 = 50 and n2 = 100) may suggest that it is reasonable to approximate the population distributions as normal. Therefore, calculating a probability about the difference in sample means could be considered.
2) Population 1 is skewed right and population 2 is approximately Normal. n1 = 50 and n2 = 10:
- In this case, the assumption of approximately normally distributed populations is violated for population 1, which is skewed right. When the population distributions are not approximately normal, it may not be appropriate to calculate a probability about the difference in sample means. The small sample size for population 2 (n2 = 10) may also limit the accuracy of any inference made based on this sample.
3) Both populations are skewed right. n1 = 5 and n2 = 10:
- Similar to the previous case, the assumption of approximately normally distributed populations is violated for both populations. Additionally, the small sample sizes (n1 = 5 and n2 = 10) may not provide sufficient information for reliable inferences. Therefore, it is generally not appropriate to calculate a probability about the difference in sample means in this situation.
4) Population 1 is skewed right and population 2 is approximately Normal. n1 = 10 and n2 = 50:
- Similar to case 2, the assumption of approximately normally distributed populations is violated for population 1, which is skewed right. In this case, the sample size for population 1 (n1 = 10) is also small, which may limit the accuracy of any inference made based on this sample. The larger sample size for population 2 (n2 = 50) might make it more reasonable to approximate the population distribution as normal. However, the violation of the assumption for population 1 suggests caution when interpreting the results. It is not generally appropriate to calculate a probability about the difference in sample means in this situation.
5) Both populations have unknown shapes. n1 = 50 and n2 = 100:
- Similar to case 1, the population shapes are unknown. However, the large sample sizes (n1 = 50 and n2 = 100) might suggest that it is reasonable to approximate the population distributions as normal. As mentioned before, calculating a probability about the difference in sample means could be considered in this case.
6) Both populations are skewed left. n1 = 5 and n2 = 40:
The assumption of approximately normally distributed populations is violated for both populations, as they are skewed left. Additionally, the small sample sizes (n1 = 5 and n2 = 40) may not provide sufficient information for reliable inferences. Therefore, it is generally not appropriate to calculate a probability about the difference in sample means in this situation.
Summarily, it is generally appropriate to calculate a probability about the difference in sample means when the sample sizes are large (typically n ≥ 30) and the populations are approximately normally distributed. However, when the population distributions are not approximately normal or when the sample sizes are small, it is generally not appropriate to calculate a probability about the difference in sample means.
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What are the vertex and range of y = |3x + 6| − 4?
A (−2, −4); −∞ < y < ∞
B (−2, −4); −4 ≤ y < ∞
C (0, −4); −∞ < y < ∞
D (0, −4); −4 ≤ y < ∞
For the given function:
Vertex is at (-2, -4) and range is −4 ≤ y < ∞
Hence, Option b is correct.
The given function is
y = |3x + 6| − 4
We can see that it is consist of absolute value function or mod function.
Since we know that,
An absolute value function is an algebraic function in which the variable is contained inside the absolute value bars.
The absolute value function is also known as the modulus function, and its most frequent form is f(x) = |x|,
where x is a real integer. In general, the absolute value function may be represented as f(x) = a |x - h| + k,
where a denotes how far the graph extends vertically, h represents the horizontal shift, and k represents the vertical displacement from the graph of f(x) = |x|.
If the value of 'a' is negative, the graph opens downwards; otherwise, it opens upwards.
The appropriate method of finding range and vertex both is to plot its graph:
Therefore after plotting graph we get,
Vertex is at (-2, -4)
And range is (-4 , ∞) ⇒ −4 ≤ y < ∞
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What are the approximate polar coordinates for the point with rectangular coordinates (–2, 4)? Give θ in degrees rounded to the nearest thousandth.
If I have $25. How many cheeseburgers can I get if they are 2.50 each?
Answer: 10 cheeseburgers
Step-by-step explanation: 25/2.5=10
Answer:
10
Step-by-step explanation:
To determine the number of cheeseburgers you can get with $25, you can divide the total amount of money by the cost of each cheeseburger.
$25 ÷ $2.50 = 10
Therefore, with $25, you can get 10 cheeseburgers if each cheeseburger costs $2.50.
What is the meaning of "R is a relation on X if [tex]R\subset X^{n}[/tex]"?
A relation R on a set X means , R is a subset of the Cartesian product Xⁿ, where n is the number of components of the relation.
Now, A relation R on a set X, where n is the arity or number of components of the relation, is a mathematical phrase that denotes that R is a subset of the Cartesian product Xⁿ
For example,
If X = 1, 2, 3, for instance, and R is a relation on X such that R = 1, 2, 3, then n=2.
We can see that R is a subset of X in this instance, which is composed of the elements (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), and (3,1), (3,2), (3,3).
Since each member of R is an ordered n-tuple consisting of n items that belong to X, it follows that if R is belong in Xⁿ, it is a relation on the set X.
Thus, A relation R on a set X means , R is a subset of the Cartesian product Xⁿ, where n is the number of components of the relation.
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its ixl please help
Use Juliana's text message data to answer the questions.
What was the mean number of text messages that Julianna sent per day
The mean number of text messages that Julianna sent per day is 7.
To calculate the mean number of text messages that Julianna sent per day, we need to sum up the number of text messages she sent each day and divide it by the total number of days.
Total number of text messages sent:
13 + 0 + 4 + 4 + 5 + 6 + 17 = 49
Total number of days: 7
Mean = Sum of all observations/number of observations
Mean =Total number of text messages sent/Total number of days
Mean number of text messages per day: 49 / 7
= 7
Therefore, 7 is the mean number of text messages that Julianna sent per day.
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Use julianna’s text message data to answer the questions.
Text messages sent:
Sun: 13 Mon: 0 tues: 4 wed: 4 thurs: 5 fri: 6 sat: 17
1. What was the mean number of text messages that julianna sent per day ?
A. 4
B.6
C.8
D.7
Determine the length of the missing side of the triangle. Round your answer to the
hundredths place.
Answer:
670
We can use the pythagorus theorum which says
The square of hypotenuse is equal to the square of perpendicular added to sq of base
Step-by-step explanation:
Use h2=p2+b2
H=6.708
Rounding to the hundredth place the answer is 670.8
Using Pythagoras Theorem (since the triangle is a right angled triangle)
[tex]( {hyp}^{2}) = ( {base}^{2} ) + ( {per}^{2} )[/tex]
[tex]( {hyp}^{2} ) = ( {3}^{2} ) + ( {6}^{2}) [/tex]
[tex]( {hyp}^{2} ) = 9 + 36[/tex]
[tex] {hyp}^{2} = 45[/tex]
Taking under root on both sides
[tex]hyp = 6.7[/tex]
Hence the missing side is 671
can you find the surface area of the prism of 3.5ft,4.5ft,2ft
Answer:
Surface Area = 63.5 ft².
Step-by-step explanation:
Length: 3.5 ft
Width: 4.5 ft
Height: 2 ft.
Surface Area = 2(Area of Base) + (Perimeter of Base) × Height
Area of The Base:Area of Base = Length × Width
Area of Base = 3.5 ft × 4.5 ft = 15.75 ft²
Perimeter of The Base:Perimeter of Base = 2(Length) + 2(Width)
Perimeter of Base = 2(3.5 ft) + 2(4.5 ft) = 7 ft + 9 ft = 16 ft
Substitute these values into the surface area formula:
Surface Area = 2(Area of Base) + (Perimeter of Base) × Height
Surface Area = 2(15.75 ft²) + (16 ft) × (2 ft)
Surface Area = 31.5 ft² + 32 ft²
Surface Area = 63.5 ft²
Therefore, the surface area of the given prism with dimensions 3.5 ft, 4.5 ft, and 2 ft is 63.5 ft².
Mean and Mean Absolute Deviation-Quiz-Level F
The dot plot shows the number of students who attended the first five meetings of a school's
Math Club.
Find the mean of the data,
Mean: 9 students
At the next meeting of the club, there are
15 students, How does including this value
with the data affect the mean?
The mean
increases
from 9 to ?
students.
5 6 7 8 9 10 11 12 13
Number of Students
The mean increases from 9 to 10 because the value of 15 is greater than the mean of 9. When a larger value is added to a set of data, the mean will increase.
How to calculate the meanThe mean of the data is 9 students because there are 45 students total and 5 meetings.
When 15 students attend the next meeting, the mean increases to 10.5 students because there are now 60 students total and 6 meetings.
Mean of the first 5 meetings:
There are 45 students total.
There are 5 meetings.
Therefore, the mean is:
= 45 / 5
= 9 students.
There are 60 students total.
There are 6 meetings.
Therefore, the mean is:
= 60/6
= 10 students.
The mean increases from 9 to 10 because the value of 15 is greater than the mean of 9. When a larger value is added to a set of data, the mean will increase.
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What the meaning of statement this?
Yes, the product of X and Y, that is X × Y is a set.
Given that, the product X×Y is a set because X×Y⊂PP(X∪Y).
A set is defined as a collection of elements or members, and X × Y meets this criteria as it is a collection of ordered pairs of elements from X and Y.
We can also prove this using set theory. The expression X × Y⊂PP(X∪Y) means that the set X × Y is a subset of the power set of the union of X and Y.
The power set of a set A is a set of all subsets of A, which means that PP(X∪Y) includes all of the possible combinations of the elements from X and Y.
So, the fact that X × Y⊂PP(X∪Y) proves that it is a subset of a set and thus is a set itself.
Yes, the product of X and Y, that is X × Y is a set.
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Question One:
If a raw score corresponds to a z-score of 1.75, what does that tell you about that score in relation to the mean of the distribution?
Question Two:
What if the raw score corresponds to a z-score of -0.85?
Question One:A positive z-score indicates that the raw score is above the mean, while a negative z-score indicates that the raw score is below the mean.
Question Two: , the raw score is relatively lower than the mean.
If a raw score corresponds to a z-score of 1.75, it tells us that the raw score is 1.75 standard deviations above the mean of the distribution. In other words, the raw score is relatively higher than the mean. The z-score provides a standardized measure of how many standard deviations a particular value is from the mean.
A positive z-score indicates that the raw score is above the mean, while a negative z-score indicates that the raw score is below the mean.
Question Two:
If a raw score corresponds to a z-score of -0.85, it tells us that the raw score is 0.85 standard deviations below the mean of the distribution. In other words, the raw score is relatively lower than the mean. The negative sign indicates that the raw score is below the mean.
To understand the meaning of a z-score, it is helpful to consider the concept of standard deviation. The standard deviation measures the average amount of variability or spread in a distribution. A z-score allows us to compare individual data points to the mean in terms of standard deviations.
In the case of a z-score of -0.85, we can conclude that the raw score is located below the mean and is relatively lower compared to the rest of the distribution. The negative z-score indicates that the raw score is below the mean and is within the lower portion of the distribution. This suggests that the raw score is relatively smaller or less than the average value in the distribution.
By using z-scores, we can standardize and compare values across different distributions, allowing us to understand the position of a raw score relative to the mean and the overall distribution.
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What is the mode of this data set?
{4, 15, 6, 11, 7, 4, 3, 14}
Answer:
4
Step-by-step explanation:
A mode is the number having the highest frequency, that is, the number which occurs the most times. (The number which occurs the most here is 4, there are two 4s. You only have one of the rest of the numbers.)
|x – 4| > –3 will have what type of solution set
Answer:
The inequality |x – 4| > –3 represents an absolute value inequality.
The absolute value of any real number is always non-negative, meaning it is greater than or equal to zero. Therefore, the left side of the inequality, |x – 4|, will always be greater than or equal to zero.
Since the right side of the inequality, -3, is also greater than or equal to zero, this means that the inequality |x – 4| > –3 holds true for all real numbers x. In other words, there are no restrictions on the value of x.
The solution set for this inequality is the set of all real numbers, often represented as (-∞, +∞).
Answer:
(−∞,∞)
Step-by-step explanation:
|x – 4| > – 3
Since |x – 4| is always positive and - 3 is negative, |x – 4| is always greater than - 3, so the inequality is always true for any value of x.
All real numbers
The result can be shown in multiple forms.
All real numbers
So, the answer is (−∞,∞)
From the observation deck of a skyscraper, Brandon measures a 45
angle of depression to a ship in the harbor below. If the observation deck is 1140 feet high, what is the horizontal distance from the base of the skyscraper out to the ship? Round your answer to the nearest tenth of a foot if necessary.
The horizontal distance from the base of the skyscraper out to the ship is 1140 m
What is the angle of depression?
Under the horizontal line, the angle of depression is measured, typically in degrees. It aids in figuring out how steep or incline the line of sight is in relation to the horizontal plane. The line of sight is steeply directed downward and increases with the angle of depression.
In many different disciplines, such as surveying, navigation, engineering, and physics, the angle of depression is frequently utilized.
We know that;
Tan 45 = x/1140
x = 1140 Tan 45
= 1140 m
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In a sample of 560 adults, 336 had children. Construct a 95% confidence interval for the true population proportion of adults with children.
Give your answers as decimals, to three places
< p
Answer:
[tex]\{0.559,0.641\}[/tex]
Step-by-step explanation:
[tex]\displaystyle CI_{95\%}=\frac{336}{560}\pm1.96\sqrt{\frac{\frac{336}{560}(1-\frac{336}{560})}{560}}\approx\{0.559,0.641\}[/tex]
(-0.68, 3.02) In y In x (1.07. -1.53) The variables x and y satisfy the equation y Ax-2, where A and p are constants. The graph of In y against In x is a straight line passing through the points (-0.68, 3.02) and (1.07,-1.53), as shown in the diagram. Find the values of A and p.
The calculated values of A and p that satisfy the equation y = Ax - 2p are A = -2.6 and p = -0.626
How to calculate the values of A and pFrom the question, we have the following parameters that can be used in our computation:
Points = (-0.68, 3.02) and (1.07,-1.53)
The equation is of the form
y = Ax - 2p
Using the given points, we have
3.02 = -0.68A - 2p
-1.53 = 1.07A - 2p
Subtract the eqations
-0.68A - 1.07A = 3.02 + 1.53
So, we have
-1.75A = 4.55
This gives
A = -2.6
Recall that
3.02 = -0.68A - 2p
So, we have
3.02 = 0.68 * 2.6 - 2p
Evaluate
2p = -1.252
Divide by 2
p = -0.626
Hence, the values of A and p are A = -2.6 and p = -0.626
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Question
The variables x and y satisfy the equation y = Ax - 2p, where A and p are constants. The graph of In y against In x is a straight line passing through the points (-0.68, 3.02) and (1.07,-1.53).
Find the values of A and p.
The number of hours that you play video games each day for five days are shown in the table. The mean amount of time that
you play video games each day of the week is 1.5 hours. How many total hours do you play video games on Friday and
Saturday
We play total of 4.5 hours of video games on Friday and Saturday.
How many hours is spent on video games on Friday and Saturday?An expression in math is a statement having minimum of two numbers or variables or both and an operator connecting them.
To get total hours played on Friday and Saturday, we will subtract the sum of hours played from the mean of 1.5 hours per day for the five days given.
The total hours played on Friday and Saturday will be:
= (Mean hours per day * 7) - (Sum of hours played on Sunday to Thursday)
= (1.5 * 7) - (1.75 + 1 + 0.5 + 1.5 + 1.25)
= 10.5 - 6
= 4.5 hours.
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Three boards are placed end to end to make a walkway. The first board is 3 feet 7 inches long, the second board is 5 feet 4 inches long, and the third board is 3
feet 10 inches long. How long is the walkway?
Write your answer in feet and inches. Use a number less than 12 for inches.
C
The walkway is 12 feet 9 inches long.
To find the total length of the walkway, we need to add the lengths of the three boards together.
The first board is 3 feet 7 inches long, which can be written as 3'7".
The second board is 5 feet 4 inches long, which can be written as 5'4".
The third board is 3 feet 10 inches long, which can be written as 3'10".
Now, let's add the lengths together:
3'7" + 5'4" + 3'10"
When adding feet and inches, we need to carry over any extra inches beyond 12 to the feet.
Adding the inches first:
7" + 4" + 10" = 21"
Now, let's add the feet:
3' + 5' + 3' = 11'
So, the total length of the walkway is 11 feet 21 inches.
We need to convert the inches to feet by dividing by 12:
11' + 21" ÷ 12 = 11' + 1'9" = 12'9"
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Quadrilateral A'B'C'D'is a translation of quadrilateral ABCD. What is the length
of B'C'?
A
60
A. 7 units
B. 6 units
OC. 4 units
D. 3 units
A'
D'
The length of B'C' from the given quadrilateral A'B'C'D' is 3 units. Therefore, option D is the correct answer.
A translation in math moves a shape left or right and/or up or down. The translated shapes look exactly the same size as the original shape, and hence the shapes are congruent to each other. They just have been shifted in one or more directions.
Here, AB = A'B' = 7 units
AD = A'D'= 6 units
DC = D'C'= 4 units
BC = B'C' = 3 units
Therefore, option D is the correct answer.
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Solve the following equation for B over the interval [0,2pi]
, giving exact answers in radian units. If an equation has no solution, enter DNE. Multiple solutions should be entered as a comma-separated list.
The solution of the equation for B over the interval [0,2π] is undefined
How to solve the equation for B over the interval [0,2π]From the question, we have the following parameters that can be used in our computation:
-√3tan(β) = tan(β)sin(β)
Divide both sides of the equation by tan(β)
so, we have the following representation
-√3 = sin(β)
Rewrite as
sin(β) = -√3
Take the arc sin of both sides
β = undefined
Hence, the solution of the equation for B over the interval [0,2π] is undefined
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Pure acid is to be added to a 10% acid solution to obtain 90L of 84% solution. How many liter of 10% solution should be in the mixture?
Answer:
16 liters of the 10% acid solution should be in the mixture.
Step-by-step explanation:
Let x represent the volume of the 10% acid solution to be added.
The volume of pure acid added would be (90 - x).
The equation to solve is: 0.1x + (90 - x) = 0.84(90)
Simplifying, we get: 0.1x + 90 - x = 75.6
Combining like terms, we have: -0.9x + 90 = 75.6
Subtracting 90 from both sides: -0.9x = -14.4
Dividing by -0.9: x = 16
Therefore, 16 liters of the 10% acid solution should be in the mixture.
A small liberal arts college in the Northeast has 200 freshmen. Sixty of the freshmen are education majors. Suppose thirty freshmen are randomly selected (without replacement).
Find the standard deviation of the number of education majors in the sample. Round your answer to two decimal places, if necessary.
The standard deviation of the number of education majors in the sample is 0.3.
What is the standard deviation of education majors?The standard deviation of the number of education majors in the sample is calculated as follows;
σ = √ [(N - n) x n(N - k) / ((N - 1) x N²)]
Where
N is the total population size = 200 freshmenn is the sample size = 30 freshmenk is the number of successes in the population = 60 education majorsThe standard deviation of the number of education majors in the sample is calculated as;
σ = √[(200 - 30) x 30(200 - 60) / ((200 - 1) x 200²)]
= √[(170 x 30 x 140 / (199 x 40000)]
= √(714000 / 7960000)
= 0.3
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I need to know the percentage of drivers who are at least 45. Using the table in the picture.
The percentage of drivers who are at least 45 is 62%
How to determine the percentage of drivers who are at least 45.From the question, we have the following parameters that can be used in our computation:
The table of values
From the table, we have
Age 45 = 62 percentile
When represented properly
So, we have
Age 45 = 62%
This means that the percentage of drivers who are at least 45 is 62%
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find the area of the triangle whoose side are 12,16 and 21 units
Answer:
[tex]A =[/tex] 95.45
Step-by-step explanation:
[tex]A=s(s﹣a)(s﹣b)(s﹣c)[/tex]
[tex]s=a+b+c[/tex]
Solving for A
A=1
4﹣[tex]a4+2(ab)2+2(ac)2﹣b4+2(bc)2﹣c4=1[/tex]
4-[tex]124+2·(12·16)2+2·(12·21)2﹣164+2·(16·21)2﹣214≈95.45123[/tex]
≈ 95.45123
PLEASE HELP QUICKLY 100 POINTS AND I WILL MARK BRAINLIEST!!
The length of the longest item that will fit in the shipping box is 26.3 inches. Now Use complete sentences to explain the process you would use to find the volume of the shipping box.
Answer:
The volume of the shipping box is 3434.6 in³ (to the nearest tenth).
The length of the longest item that will fit inside the shipping box is 26.8 inches (to the nearest tenth).
Step-by-step explanation:
The shipping box can be modelled as a cuboid.
A cuboid is a three-dimensional geometric shape with six rectangular faces and right angles between adjacent faces.
The volume of a cuboid can be calculated by multiplying its length (L), width (W), and height (H) together.
From the given diagram, the width of the cuboid is 16 inches and its height is 12 inches. Therefore, we need to find the measure of its length in order to calculate its volume.
As all sides of a cuboid have interior angles of 90°, and we have been given the face diagonal of the base (24 inches), we can use Pythagoras Theorem to calculate the length (L).
[tex]\begin{aligned}L^2+16^2&=24^2\\L^2+256&=576\\L^2&=320\\L&=\sqrt{320}\\L&=8\sqrt{5}\; \sf in\end{aligned}[/tex]
Substitute L = 8√5, W = 16 and H = 12 into the formula for the volume of a cuboid to calculate the volume of the shipping box:
[tex]\begin{aligned}\sf Volume&=\sf L \cdot W \cdot H\\&=8\sqrt{5} \cdot 16 \cdot 12\\&=128\sqrt{5} \cdot 12\\&=1526\sqrt{5}\\&=3434.60041...\\&=3434.6\; \sf in^3\end{aligned}[/tex]
Therefore, the volume of the shipping box is 3434.6 in³ to the nearest tenth.
[tex]\hrulefill[/tex]
In a cuboid, there are two types of diagonals: face diagonals and body diagonals.
Face Diagonals: These diagonals connect opposite corners of a face of the cuboid and lie entirely within that face.Body Diagonals: These diagonals connect opposite corners of the cuboid, passing through the interior of the cuboid and extending across multiple faces. Body diagonals are longer than face diagonals.The body diagonal of a cuboid is the longest line that can be drawn inside the cuboid. Therefore, to find the length of the longest item that will fit inside the shipping box, we need to calculate the body diagonal of the cuboid.
The formula for the body diagonal of a cuboid is:
[tex]\sf Body \;diagonal=\sqrt{L^2+W^2+H^2}[/tex]
Substitute L = 8√5, W = 16 and H = 12 into the formula to find the body diagonal of the cuboid (marked as a red dashed line on the given diagram):
[tex]\begin{aligned}\sf Body \;diagonal&=\sf \sqrt{L^2+W^2+H^2}\\&=\sqrt{(8\sqrt{5})^2+16^2+12^2\\&=\sqrt{320+256+144}\\&=\sqrt{720}\\&=26.8328157...\\&=26.8\; \sf in\;(nearest\;tenth)\end{aligned}[/tex]
Therefore, the length of the longest item that will fit inside the shipping box is 26.8 inches, to the nearest tenth.
Non Shaded Shaded
Area
Area
8
Find the radius
of the small circle
Answer:
The answer is 16pi or 50.3cm² to 1 d.p
Step-by-step explanation:
The non shaded=area of shaded
d=8
r=d/2=4
A=pir³
A=p1×4²
A=pi×16
A=16picm² or 50.3cm² to 1d.p
Answer:
3.45 cm (3 s.f.)
Step-by-step explanation:
We have been given a 5-sided regular polygon inside a circumcircle. A circumcircle is a circle that passes through all the vertices of a given polygon. Therefore, the radius of the circumcircle is also the radius of the polygon.
To find the radius of a regular polygon given its side length, we can use this formula:
[tex]\boxed{\begin{minipage}{6 cm}\underline{Radius of a regular polygon}\\\\$r=\dfrac{s}{2\sin\left(\dfrac{180^{\circ}}{n}\right)}$\\\\\\where:\\\phantom{ww}$\bullet$ $r$ is the radius.\\ \phantom{ww}$\bullet$ $s$ is the side length.\\\phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]
Substitute the given side length, s = 8 cm, and the number of sides of the polygon, n = 5, into the radius formula to find an expression for the radius of the polygon (and circumcircle):
[tex]\begin{aligned}\implies r&=\dfrac{8}{2\sin\left(\dfrac{180^{\circ}}{5}\right)}\\\\ &=\dfrac{4}{\sin\left(36^{\circ}\right)}\\\\ \end{aligned}[/tex]
The formulas for the area of a regular polygon and the area of a circle given their radii are:
[tex]\boxed{\begin{minipage}{6 cm}\underline{Area of a regular polygon}\\\\$A=\dfrac{nr^2\sin\left(\dfrac{360^{\circ}}{n}\right)}{2}$\\\\\\where:\\\phantom{ww}$\bullet$ $A$ is the area.\\\phantom{ww}$\bullet$ $r$ is the radius.\\ \phantom{ww}$\bullet$ $n$ is the number of sides.\\\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{6 cm}\underline{Area of a circle}\\\\$A=\pi r^2$\\\\where:\\\phantom{ww}$\bullet$ $A$ is the area.\\\phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Therefore, the area of the regular pentagon is:
[tex]\begin{aligned}\textsf{Area of polygon}&=\dfrac{5 \cdot \left(\dfrac{4}{\sin\left(36^{\circ}\right)}\right)^2\sin\left(\dfrac{360^{\circ}}{5}\right)}{2}\\\\&=\dfrac{5 \cdot \left(\dfrac{4}{\sin\left(36^{\circ}\right)}\right)^2\sin\left(72^{\circ}\right)}{2}\\\\&=\dfrac{\dfrac{80\sin\left(72^{\circ}\right)}{\sin^2\left(36^{\circ}\right)}}{2}\\\\&=\dfrac{40\sin\left(72^{\circ}\right)}{\sin^2\left(36^{\circ}\right)}\\\\&=110.110553...\; \sf cm^2\end{aligned}[/tex]
The area of the circumcircle is:
[tex]\begin{aligned}\textsf{Area of circumcircle}&=\pi \left(\dfrac{4}{\sin\left(36^{\circ}\right)}\right)^2\\\\&=\dfrac{16\pi}{\sin^2\left(36^{\circ}\right)}\\\\&=145.489779...\; \sf cm^2\end{aligned}[/tex]
The area of the shaded area is the area of the circumcircle less the area of the regular pentagon plus the area of the small central circle.
The area of the unshaded area is the area of the regular pentagon less the area of the small central circle.
Given the shaded area is equal to the unshaded area:
[tex]\begin{aligned}\textsf{Shaded area}&=\textsf{Unshaded area}\\\\\sf Area_{circumcircle}-Area_{polygon}+Area_{circle}&=\sf Area_{polygon}-Area_{circle}\\\\\sf 2\cdot Area_{circle}&=\sf 2\cdot Area_{polygon}-Area_{circumcircle}\\\\2\pi r^2&=2 \cdot \dfrac{40\sin\left(72^{\circ}\right)}{\sin^2\left(36^{\circ}\right)}-\dfrac{16\pi}{\sin^2\left(36^{\circ}\right)}\\\\2\pi r^2&=\dfrac{80\sin\left(72^{\circ}\right)}{\sin^2\left(36^{\circ}\right)}-\dfrac{16\pi}{\sin^2\left(36^{\circ}\right)}\\\\\end{aligned}[/tex]
[tex]\begin{aligned}2\pi r^2&=\dfrac{80\sin\left(72^{\circ}\right)-16\pi}{\sin^2\left(36^{\circ}\right)}\\\\r^2&=\dfrac{40\sin\left(72^{\circ}\right)-8\pi}{\pi \sin^2\left(36^{\circ}\right)}\\\\r&=\sqrt{\dfrac{40\sin\left(72^{\circ}\right)-8\pi}{\pi \sin^2\left(36^{\circ}\right)}}\\\\r&=3.44874763...\sf cm\end{aligned}[/tex]
Therefore, the radius of the small circle is 3.45 cm (3 s.f.).
HELP PLEASE 30 POINTS
Four transformations of the function f(x)=4x
are given below.
For each transformation, drag the expression that shows the result of that transformation into the box under it.
The expression that shows the result of that transformation are:
3f(x) = 3.4ˣ
f(3x)=4³ˣ
f(x+3)=4ˣ⁺³
f(x)+3=4ˣ+3.
The given function is f(x) =4ˣ.
We have to find the transformations applied to the function f(x).
Graph transformation involves modifying an existing graph or graphed equation to create a different version of the original graph.
f(x) =4ˣ
We have to find 3f(x), f(3x), f(x+3) and f(x)+3.
3f(x) = 3.4ˣ
f(3x)=4³ˣ
f(x+3)=4ˣ⁺³
f(x)+3=4ˣ+3.
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11. The height of a plant over 4 weeks is shown in the graph below.
Height of Plant (cm)
16
23
Week
What is the rate of growth of the plant, in centimeters per week?
A 2
B. 3
C. 8
D. 12
The rate of growth of the plant is 2 centimeters per week.
Calculate the change in height divided by the change in time (weeks) to find the plant's growth rate.
To calculate the rate of growth of the plant, we need to determine the change in height per week.
Given:
Initial height = 4 cm
Final height = 12 cm
Weeks = 4
To find the rate of growth, we can use the formula:
Rate of growth = (Final height - Initial height) / Weeks
Substituting the given values into the formula:
Rate of growth = (12 cm - 4 cm) / 4 weeks
Rate of growth = 8 cm / 4 weeks
Rate of growth = 2 cm/week
Therefore, the rate of growth of the plant is 2 centimeters per week.
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In one lottery, a player wins the jackpot by matching all five distinct numbers drawn in any order from the white balls (1
through 43) and matching the number on the gold ball (1 through 34). If one ticket is purchased, what is the probability
of winning the jackpot?
The probability of winning the jackpot with one ticket is P ( A ) = 1/34
Given data ,
To calculate the probability of winning the jackpot in the lottery, we need to determine the total number of possible outcomes (the sample space) and the number of favorable outcomes (winning outcomes).
Total number of possible outcomes:
For the white balls, there are 43 numbers to choose from, and we need to select 5 distinct numbers in any order. This can be calculated using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of options and r is the number of selections. In this case, we have 43 white balls and need to choose 5, so the number of possible outcomes for the white balls is:
C(43, 5) = 43! / (5! * (43 - 5)!) = 43! / (5! * 38!) = 43 * 42 * 41 * 40 * 39
For the gold ball, there are 34 numbers to choose from, and we need to select 1 number. So the number of possible outcomes for the gold ball is simply 34.
Therefore, the total number of possible outcomes is:
Total outcomes = (43 * 42 * 41 * 40 * 39) * 34
Number of favorable outcomes (winning outcomes):
To win the jackpot, we need to match all 5 distinct numbers from the white balls and the number on the gold ball. Since order doesn't matter for the white balls, we can use the combination formula again:
C(n, r) = n! / (r! * (n - r)!)
In this case, we have 43 white balls and need to choose 5, so the number of favorable outcomes for the white balls is:
C(43, 5) = 43! / (5! * (43 - 5)!) = 43 * 42 * 41 * 40 * 39
For the gold ball, there is only 1 winning number.
Therefore, the number of favorable outcomes is:
Favorable outcomes = (43 * 42 * 41 * 40 * 39) * 1
Probability of winning the jackpot:
The probability of winning the jackpot is the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes
Plugging in the values, we get:
Probability = [(43 * 42 * 41 * 40 * 39) * 1] / [(43 * 42 * 41 * 40 * 39) * 34]
Simplifying, we find:
Probability = 1 / 34
Hence , the probability of winning the jackpot with one ticket in this lottery is 1 in 34.
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The figure above shows a store's supply-demand graph for coffee makers. If the store sells $600 worth of coffee makers, which of the following is a valid possible price for them?
A. $15
B. $30
C. $40
D. $55
The possible price for the items if the store sells $600 is (c) $40
How to determine the possible price for the items?From the question, we have the following parameters that can be used in our computation:
The supply-demand graph
If the store sells $600, then there is a supply worth of $600
The equation of the supply line is calculated as
y = mx + c
Where
c = y = 0
i.e. c = 100
So, we have
y = mx + 100
Using another point on the graph, we have
30m + 10 = 400
So, we have
m = 13
This means that
y = 13x + 100
For a supply of 600, we have
13x + 100 = 600
So, we have
13x = 500
Divide by 13
x = 38.4
Approximate
x = 40
Hence, the possible price for the items is (c) $40
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