Rotate triangle △C'D'E' counterclockwise by approximately -0.785 radians about the origin:
X₁ = 1 * cos(-0.785) - (-4) * sin(0.785) ≈ 0.436
y₁ = 1 * sin(-0.785) + (-4) * cos(0.785) ≈ -3.678
x₂ = -7 * cos(0.785) - 8
What is a triangle in its mathematical definition?
In geometry, a triangle is a three-sided polygon with three edges and three vertices. The most important property of triangles is that the sum of their interior angles equals 180 degrees.
Translation:
To translate a point (x, y) by a vector (h, k), we add h to the x-coordinate and k to the y-coordinate of the point.
To transform triangle △CDE to triangle △UVW, we can first translate triangle △CDE by a vector (h, k) to obtain triangle △C'D'E', where C' = C + (h, k), D' = D + (h, k), and E' = E + (h, k).
Since the coordinates of C are (2, -8) and the coordinates of U are (1, -4), we can calculate the translation vector (h, k) as follows:
h = 1 - 2 = - 1
k = -4 -(-8) = 4
So the translation vector is (-1 , 4)
Rotation:
To rotate a point (x, y) by an angle θ counterclockwise about the origin, we use the following formulas:
x' = x * cos(θ) - ytimes sinθ
y' = x * sin(θ) + y * cos(θ)
To transform triangle △C'D'E' to triangle △UVW, we can apply a rotation of angle θ counterclockwise about the origin to triangle △C'D'E', where C' = (x1', y1'), D' = (x2', y2'), and E' = (x3', y3'). Since the coordinates of C' are (2, -8) after translation, and the coordinates of U are (1, -4), we can calculate the rotation angle θ as follows:
θ = atan²(y₁ - y₂ , x₁ - x₂) - atan²(y₁ - y₂ , x₁ - x₂)
= atan²((-8 + 4) - (-4), (2 + 1) - (-6 + 3)) - atan²((-8) - (-4), 2 - (-6)
Using a calculator, we can find θ to be approximately -0.785 radians.
So, the algebraic representation of the transformation that maps triangle
ΔCDE to triangle ΔUVW
Translate triangle △CDE by the vector (-1, 4) to obtain triangle △C'D'E':
C' = (2 , -8) + (-1 , 4) = (1 , -4)
D' = (-6 , 4) + (-1 , 4) = (-7 , 8 )
E' = (0 ,4 ) + (-1 , 4 )
Therefore, Rotate triangle △C'D'E' counterclockwise by approximately -0.785 radians about the origin:
X₁ = 1 * cos(-0.785) - (-4) * sin(0.785) ≈ 0.436
y₁ = 1 * sin(-0.785) + (-4) * cos(0.785) ≈ -3.678
x₂ = -7 * cos(0.785) - 8
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Mr. K drove to Milwaukee, Wisconsin for spring break. He left at 1:25pm and arrived at 3:00pm. How long did it take him to get there? Also, he spent $285 on basketball game tickets, $5 on parking, $15.90 on concessions at the game, $20 on dinner after the game, $26.69 in UBER rides to and from the game, and $35 in gas. How much money did he spend IN ALL? and if he started the trip with $400, how much money does he have left over?
Answer:
A.) 1 hour and 35 minutes
B.) $387.59
C.) $12.41
Step-by-step explanation:
Question 1 answer — 1 hour and 35 minutes
Question 2 answer — $387.59
Question 3 answer — $12.41
Hope my answer helps you.
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Describe a situation with two variables that you can represent w it’s an equation that uses two operations. Write the equation. Explain the relationships between the variables
An equation uses two operations is equal to C = n × p + T.
Relationship between variables is C represents total cost where as 'n × p' is cost of meal per person and T is additional cost.
Total cost of a meal at a restaurant is a function of the number of people and the price per dish.
Represent the variables as follows,
Let us use the variable 'C' to represent the total cost of the meal.
Let us use the variable 'n' to represent the number of people at the table.
Let us use the variable 'p' to represent the price per dish.
An equation that uses two operations, use multiplication and addition.
C = n × p + T
'n × p' represents the cost of the meal per person,
'T' represents any additional costs, such as taxes or tips.
By adding these two values together, total cost of the meal.
Relationship between the variables is as follows,
Total cost of the meal C is directly proportional to both the number of people n and the price per dish p.
Increase either n or p, total cost of the meal will increase proportionally.
For example,
Double number of people at the table n,
Total cost of meal C will also double,
Assuming price per dish p remains constant.
Similarly,
Double the price per dish p,
Total cost of the meal C will also double,
Assuming number of people at the table n remains constant.
Constant term T represents any additional costs.
That are independent of n and p, such as taxes or tips.
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SOMEONE PLEASE ANSWER
What is the measure of the missing angle in the triangle?
1. 39 degrees
2. 17 degrees
3. 30 degrees
4. 23 degrees
The measure of the missing angle in the triangle is 30 degrees. Option 3
How to determine the valueWe have that there are six different trigonometric identities. they are;
cosinetangentcotangentsinesecantcosecantAlso note that the sum of the angles in a triangle is equal to 180 degrees.
The different trigonometric ratios of the identities are;
sin θ = opposite/hypotenuse
From the information given, we have;
Opposite = 15
Hypotenuse = 31
Substitute the values
sin x = 15/31
Divide the values
sin x = 0. 4839
Find the inverse sine
x = 30 degrees
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-2(8x-5) + 2x =4(x+5)
Answer:
x = -5/9
Step-by-step explanation:
-2(8x - 5) + 2x = 4(x + 5)
-16x + 10 + 2x = 4x + 20
-16x + 2x + 10 = 4x + 20
-14x + 10 = 4x + 20
+14x = +14x (Add 14 to both sides)
10 = 18x + 20
-20 = -20 (Subtract 20 from both sides)
-10 = 18x
18 = 18 (Divide by 18)
-10/18 = x
Simplified:
-5/9 = x
Answer:
Step-by-step explanation: Distribute -2 to the parenthasis, and 4 to the other parenthasis. two double negatives make a positive. Combine like terms. get the x and regular numbers on different sides of the equal sign, and divide to find the x=
-16x +10 +2x=4x+20
-14x+10=4x+20
-14x -20
-10x= -10
/-10 /-10
x=1
Jonah subscribed to a new movie streaming service. The service costs Co $14.99 per month, but Jonah's rate is reduced by $0.75 for each new subscriber that he signs up. If there is no limit to the discount, how many users must Jonah recruit in order from him to use the service for a cost of at most $5.00?
what does the red line on the graph represent? what does the red line on the graph represent? the average amount of solid waste generated by each household in the united states per year the total amount of solid waste generated by all people in the united states per year the average amount of solid waste generated by each person in the united states per year the period of time in years
The red line on the graph may also include other lines that represent related data, such as the total amount of solid waste generated by all people in the United States per year, or the period of time in years being measured.
Red line on the graph represents the average amount of solid waste generated by each person in the United States per year.
The red line on the graph is a useful tool for understanding the impact of solid waste on the environment and society as a whole.
It helps us to identify areas where we can make changes to reduce waste and improve our overall environmental performance and serves as a reminder of the importance of responsible consumption and waste management practices.
Red line on the graph represents the average amount of solid waste generated by each person in the United States per year.
This is a key metric in measuring the environmental impact of waste production in the country.
The average amount of solid waste generated per person is an important figure because it provides a more accurate picture of individual behavior and consumption patterns.
It takes into account factors such as population growth, changes in household size, and shifts in lifestyle and consumption habits.
By tracking this metric over time, policymakers and environmental advocates can gain insight into trends and patterns in waste generation and identify opportunities for reducing waste and promoting sustainability.
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Identify the expression that is not equivalent to 6x + 3.
Answer:
distribute x as 3:
3²+10(3)+24
=63
find the equation of the line shown.
Answer:
[tex]y=-\dfrac{1}{4} x+2[/tex]
Step-by-step explanation:
We can create an equation for the pictured line using slope-intercept form:
[tex]y=mx+b[/tex],
where [tex]m[/tex] is the line's slope, and [tex]b[/tex] is the y-value at the line's y-intercept.
First, we can find the line's slope.
slope = rise / run = -1/4
Next, we can find the line's y-value at its y-intercept, which is the y-coordinate at the point when it touches the y-axis (vertical axis):
2
Finally, we can put these pieces of information together into slope-intercept form:
[tex]\bold{y=-\dfrac{1}{4} x+2}[/tex]
What is the maximum number of non-overlapping regions? into which circle can be divided by 9 chords
The maximum number of non-overlapping regions into which a circle can be divided by 9 chords is 46.
The maximum number of non-overlapping regions that a circle can be divided into by n chords is given by the formula:
N = (n^2 + n + 2) / 2
Substituting n = 9 into the formula, we get:
N = (9^2 + 9 + 2) / 2
N = (81 + 9 + 2) / 2
N = 92 / 2
N = 46
Therefore, the maximum number of non-overlapping regions into which a circle can be divided by 9 chords is 46.
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What is the area of a triangle with a base length of 3 1/2 inches and a geight of 3 inches?
Answer:
10.5
Step-by-step explanation:
And electric and chargers of $50 house call fee and $65 per hour for the work. is it less expensive to have the electric can’t work for six hours or have them come back for 23 hour jobs ?
Therefore, it is less expensive to have the electrician work for 6 hours, which would cost $440, compared to having them come back for 23 hour jobs, which would cost $3090.
What is equation?An equation is a mathematical statement that shows the equality of two expressions. It usually consists of two sides separated by an equal sign (=). The expressions on both sides of the equal sign can include numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division.
Here,
To compare the cost of having the electrician work for 6 hours versus having them come back for 23 hour jobs, we need to calculate the total cost for each option.
If the electrician works for 6 hours, the total cost will be:
Total cost = House call fee + (Hourly rate x Number of hours worked)
Total cost = $50 + ($65 x 6)
Total cost = $50 + $390
Total cost = $440
If the electrician comes back for 23 hour jobs, the total cost will be:
Total cost = Number of jobs x (House call fee + Hourly rate x Number of hours per job)
Total cost = 2 x ($50 + ($65 x 23))
Total cost = 2 x ($50 + $1495)
Total cost = 2 x $1545
Total cost = $3090
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An airplane is flying at a height of 3 miles above the ground. The distance along the ground from the
airplane to the airport is 7.5 miles. What is the angle of depression from the airplane to the airport?
Round to the nearest degree.
Answer:
the angle of depression is approx 22 degrees
Step-by-step explanation:
Model drawn below to visualize
I represented the angle of depression as x
Since two side lengths are given, we can use tangent to find the angle of depression
tan x= 3/7.5
We then use inverse tangent to find the angle measure:
m<x=[tex]tan^{-1}[/tex](3/7.5)
m<x is approx 21.8 or 22 when rounded to the nearest degree
(when using tan, inverse tan, sin, cos, inv... use deg on calculator when finding angle of depression, solving triangles, etc.)
4 m
7 m
…..help please
The total surface area of the composite shape is 40.57 square m
Calculating the surface areasThe surface area of composite shapes can be found by breaking the composite shape down into simpler shapes and then finding the surface area of each individual shape.
Once the surface areas of all the individual shapes are found, they can be added together to find the total surface area of the composite shape.
Here, we have
Area = Area of rectangle + circle
So, we have
Area = 4 * 7 + 22/7 * (4/2)^2
Area = 40.57 square m
So, the area is 40.57 square m
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Need help on 3 and 4
[3] Vertical
➜ These angles are opposite angles created by two intersecting lines. They are also congruent, another quality of vertical angles.
[4] Neither
➜ While these angles are "opposite" of each other, they are incongruent angles. Since the angles are not adjacent or vertical, they are neither.
Consider a sequence whose first five terms are:-1.75, -0.5, 0.75, 2, 3.25
Which explicit function (with domain all integers n ≥ 1) could be used to define and continue this sequence?
Step-by-step explanation:
+ 1.25
every new term is the previous term + 1.25.
with starting value -1.75
f(n) = 1.25n - 1.75
ten families were randomly selected and their daily water usage (in gallons) before and after viewing a conservation video was recorded. the sample mean difference was found to be 4.8 and the corresponding standard deviation 5.25, where differences were found (before - after). assume the data is normally distributed. if the resulting 95% confidence interval is (1.04,8.55), we have evidence to suggest that the conservation videos would be effective in reducing water use in the general population. true false
True. The question states that ten families were randomly selected and their daily water usage (in gallons) before and after viewing a conservation video was recorded.
The sample mean difference was found to be 4.8 and the corresponding standard deviation 5.25, with differences
calculated as (before - after).
Assuming the data is normally distributed, the resulting 95% confidence interval is (1.04, 8.55).
Since the entire 95% confidence interval is positive (i.e., all values are greater than zero), this indicates that there is a
significant difference between water usage before and after viewing the conservation video.
This provides evidence to suggest that the conservation videos would be effective in reducing water use in the general population.
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‼️WILL MARK BRAINLIEST‼️
This level of granularity can be particularly useful when analyzing complex datasets or when trying to identify subtle relationships between variables.
What is a graph?
In computer science and mathematics, a graph is a collection of vertices (also known as nodes or points) connected by edges (also known as links or lines).
One advantage of being able to see each individual data point in a graph when comparing two data sets is that it allows for a more accurate and detailed analysis of the similarities and differences between the two sets. By examining the distribution and spread of the data points, it is possible to identify patterns and trends that may not be immediately apparent from a simple comparison of summary statistics, such as means or medians.
Additionally, by seeing each individual data point, it becomes possible to detect outliers or anomalies that may be skewing the data, which can lead to more informed decision-making and a deeper understanding of the underlying trends and patterns in the data. This level of granularity can be particularly useful when analyzing complex datasets or when trying to identify subtle relationships between variables.
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Last Friday, AT&T closed at $41.68. AT&T pays an annual dividend of $1.98. Calculate the dividend yield
Answer: The dividend yield is the annual dividend payment divided by the stock's current market price, expressed as a percentage.
Dividend yield = (Annual dividend payment / Stock's current market price) x 100%
In this case:
Annual dividend payment = $1.98
Stock's current market price = $41.68
Dividend yield = ($1.98 / $41.68) x 100% = 4.75%
Therefore, the dividend yield for AT&T is 4.75%.
Step-by-step explanation:
6. Anchorage (southern coast of Alaska) is several hundred miles north of Helena, Montana, yet its January temperatures are similar to Helena's. Explain why.
Helena experience similar temperatures during January due to their subarctic climate regime, but the moderating effect of the oceanic currents on Anchorage's climate makes its winters milder than those of Helena.
what is temperatures ?
"Temperatures" refers to the degree of hotness or coldness of a substance, usually measured in degrees Fahrenheit (°F) or Celsius (°C). In the context of the original question, "temperatures" refers to the measure of how hot or cold it is in a particular location, specifically in Anchorage and Helena.
In the given question,
Anchorage and Helena are both located in high latitude regions where the climate is dominated by polar or subarctic air masses. The prevailing westerly winds carry these cold air masses from the Arctic towards the continental interiors, affecting the climate of both Anchorage and Helena. Although Anchorage is located further north than Helena, it is situated along the coast of the Gulf of Alaska, which is influenced by the relatively warm oceanic currents such as the North Pacific Current and the Alaska Current. These currents help to moderate the cold air masses coming from the north, resulting in a maritime subarctic climate in Anchorage.
In contrast, Helena is situated in the interior region of Montana, which is far away from any major water bodies, leading to a continental subarctic climate with relatively colder and drier conditions compared to Anchorage. Therefore, despite their different latitudes, both Anchorage and Helena experience similar temperatures during January due to their subarctic climate regime, but the moderating effect of the oceanic currents on Anchorage's climate makes its winters milder than those of Helena.
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Which graph shows the solution to the inequality Negative 3 x minus 7 less-than 20 Group of answer choices A number lines goes from negative 10 to positive 10. An open circle appears at positive 9. The number line is shaded from positive 9 through negative 10. A number line goes from negative 10 to positive 10. An open circle appears at positive 9. The number line is shaded from positive 9 through positive 10. A number line goes from negative 10 to positive 10. An open circle appears at negative 9. The number line is shaded from negative 9 through positive 10. A number line goes from negative 10 to positive 10. An open circle appears at negative 9. The number line is shaded from negative 9 through negative 10.
The solution is all values of x greater than -9.
What is the correct graph for the given inequality?The correct graph for the given inequality "Negative 3x - 7 < 20" is:
From negative 10 to positive 10, a number line runs. An open circle shows up at negative 9. From negative 9 to positive 10, the number line is shaded.
To chart the arrangement, we initially confine the variable by adding 7 to the two sides of the imbalance:
Negative 3x - 7 + 7 < 20 + 7
Negative 3x < 27
Next, we divide both sides by -3, remembering to flip the inequality sign since we're dividing by a negative number:
Negative 3x/-3 > 27/-3
x > -9
Therefore, the arrangement is all upsides of x more prominent than - 9. Since these are the values that satisfy the inequality, we graph this by drawing an open circle at -9 on the number line and shading all values to the right of -9.
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Help
Please show your work .
Answer:
QR=8
Step-by-step explanation:
When two lines are perpendicular, they form two 90-degree angles. So:
m∠QSR = m∠QSP = 90°
We also know that the line segment PS is congruent to the line segment SR.
Lastly, by the reflexive property, QS is congruent to itself.
Thus, ΔPSQ ≅ ΔRSQ by SAS (side-angle-side).
Now that we know these triangles are congruent, we can further state that PQ is congruent to QR by CPCTC (corresponding parts of congruent triangles are congruent).
If PQ is congruent to QR, then their measures are also congruent, and we can get this equation (by using substitution and algebra):
[tex]3n-1=5n-7=\\6=2n=\\n=3[/tex]
Plug in n to the length of QR:
[tex]5n-7=\\5(3)-7=\\15-7=\\8[/tex]
So, QR=8.
-2/3-(-3/5)+2/3-1/2 rational numbers solve
The result of simplifying {-2/3 - (-3/5) + 2/3 - 1/2} is 1/10 as a rational number.
What is rational numbers?In mathematics, rational numbers are numbers that can be expressed as a fraction or ratio of two integers, where the denominator is not zero. This means that a rational number can be written in the form p/q, where p and q are integers and q is not equal to zero. Examples of rational numbers include 2/3, 5/4, and -7/8. Rational numbers can be positive, negative, or zero, and they can be represented on a number line.
In the given question,
To solve this problem, we need to combine the like terms:
-2/3 - (-3/5) + 2/3 - 1/2
= -2/3 + 3/5 + 2/3 - 1/2 (since -(-3/5) becomes +3/5)
= (-2/3 + 2/3) + (3/5 - 1/2) (grouping the like terms together)
= 0 + (6/10 - 5/10) (since -2/3 + 2/3 = 0)
= 1/10
Therefore, the solution is 1/10.
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Dilate point S by a scale factor of 1/2
PLEASE JUST TELL WHERE THE POINT MUST BE LOCATED DONT GIVE A LONG EXPLANATION AND IF U CAN UPLOAD A PIC OF THE ANSWER
The position of point S after dilating will be half of the original. The dilated point S' has been shown below.
What is dilation?
Using a modification known as dilation, an object can be resized. Through dilatation, the objects can be resized or enlarged. This transformation results in a shape that is a perfect replica of the original image. The form's dimensions do vary, though. An expansion or contraction of the original form is required for a dilatation. This shift is referred to as a scale factor.
We are given a graph showing position of point S.
Now, on dilating the point by a scale factor of [tex]\frac{1}{2}[/tex], we get that the point will be located at half of the original.
The point has been located and attached in the image below.
Hence, the required solution has been obtained.
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at least how many sets must be printed to be sure of having at least 3 identical subsets on the list?
At least 3 sets must be printed to be sure of having at least 3 identical subsets on the list.
We'll use the Pigeonhole Principle, which states that if there are n items to be placed into m containers, and n > m, then at least one container must hold more than one item.
In this case, we want at least 3 identical subsets.
Since a subset can either include an element or not, there are [tex]2^n[/tex] possible subsets for a set with n elements. To guarantee at least 3 identical subsets, we'll use the Pigeonhole Principle with n+1 as the number of sets (containers) and 3 as the number of subsets we want (items).
Start by finding the minimum number of elements (n) in a set that results in at least 3 identical subsets.
Apply the Pigeonhole Principle formula: n > m,
where n is the number of items (3 identical subsets) and m is the number of containers ([tex]2^n[/tex] possible subsets).
Solve for n: 3 > [tex]2^n[/tex]
Increase n until the inequality is satisfied.
In this case, n = 2 works: 3 > [tex]2^2[/tex] (3 > 4).
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Identify the circumscribed angle in the picture below.
Answer:
A.) LGJ
Step-by-step explanation:
A rectangular prism has a width of 4, a length of 6, and a height of 8. What is the diagonal height of the rectangular prism? Round the decimal to the nearest tenth.
Answer:
add 2
Step-by-step explanation:
just adding 2
G(x) = 5 - 1/4 x f(x) what are all of the transformations that occur?
Thus, the new function is obtained from the parent function f(x) =x as G(x) = 5 - 1/4 x through dilation and translation.
Explain about the transformations:Function transformations are actions that can be taken on a function's existing graph to produce a transformed graph.
A list of typical graph transformations is shown below:
Vertical and horizontal transformations (or translations)Stretches in both directionsvertical and horizontal compressionsrotations and reflectionsGiven that:
new function : G(x) = 5 - 1/4 x
parent function: f(x) = x.
Series of transformations:
function f(x) is dilated by the scale factor of 1/4: x --> 1/4 x.Now the obtained function is shifted 5 units on the left: 5 - 1/4xThus, the new function is obtained from the parent function f(x) =x as G(x) = 5 - 1/4 x through dilation and translation.
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Complete question:
G(x) = 5 - 1/4 x is the function obtained from parent function f(x) = x. what are all of the transformations that occur?
Leah is an administrator for a large
school district. She wants to know how
many hours students spend on
homework each week. She asks the
question to 250 randomly selected eighth
grade students. The results are shown in
the table. Which of the following
inferences about the eighth grade
students in the school district are valid?
Hours. Students
Less than 2 44
2 to less than 4 88
4 to less than 6 90
At least 6 28
A. There are more students who spend at least 4 hours per week on homework than students who spend less than 4 hours per week on homework.
B.About 11% of students in the school district spend at least 6 hours per week on homework.
C. About 50% of students who spend less than 4 hours per week on homework spend less than 2 hours per week.
D.About 1 of every 3 students who spend at least 4 hours per week on homework spend at least 6 hours per week.
Based on the data provided in the table, the valid inferences are A, B and C, while inference D cannot be confirmed.
What is a valid inference?A conclusion that is logical, rational, and backed by solid reasoning, statistical analysis, or scientific principles is considered to be a valid inference when it is derived from the available facts or evidence. It should be founded on trustworthy and pertinent data and adhere to accepted rules of statistical or scientific reasoning.
In addition to being compatible with accepted theories, guiding principles, or previous research findings, a valid inference should take into account any constraints and potential sources of mistake in the data or evidence. In other terms, a valid inference is a plausible and convincing conclusion that may be reached using acceptable statistical or scientific techniques from the relevant information or evidence.
A. Valid: There are more students who spend at least 4 hours per week on homework than students who spend less than 4 hours per week on homework.
B. Valid: Approximately 11% of students in the school district spend at least 6 hours per week on homework, based on the "At least 6" category which has 28 students out of the total 250 students surveyed.
C. Valid: About 50% of students who spend less than 4 hours per week on homework spend less than 2 hours per week, based on the total number of students in the "Less than 2" and "2 to less than 4" categories compared to the total number of students in the "Less than 4" category.
D. Not valid: The table does not provide data specifically about students who spend at least 4 hours per week on homework and their hours of homework beyond 4 hours, so this inference cannot be made.
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five rectangles, and have integer side lengths. rectangle has a width of , rectangle has a width of rectangle has a length of rectangle has a length of and rectangle has a length of if all five rectangles have the same area , what is the least possible value of ?
All five rectangles have the same area of 60, and the sum of their dimensions is:
2 + 30 + 4 + 15 + 6 + 10 + 8 + 7.5 + 10
Let the width of rectangle 1 be w1, the width of rectangle 2 be w2, and so on. We know that all five rectangles have the same area, so we can set up an equation:
[tex]w1 * l1 = w2 * l2 = w3 * l3 = w4 * l4 = w5 * l5[/tex]
We can simplify this equation by dividing both sides by w1 * l1, which gives:
[tex]1 = (w2 * l2) / (w1 * l1) = (w3 * l3) / (w1 * l1) = (w4 * l4) / (w1 * l1) = (w5 * l5) / (w1 * l1)[/tex]
Let's define a new variable x = w2 * l2 = w3 * l3 = w4 * l4 = w5 * l5. Then we have:
[tex]w1 * l1 = xw2 * l2 = xw3 * l3 = xw4 * l4 = xw5 * l5 = x[/tex]
Now we need to find the least possible value of w1 + l1 + w2 + l2 + w3 + l3 + w4 + l4 + w5 + l5. Since w1 * l1 = x, we can rewrite this as:
[tex]w1 + l1 + 4 * sqrt(x / w1) + 4 * sqrt(w1 / x)[/tex]
To find the minimum value of this expression, we can take its derivative with respect to w1, set it equal to zero, and solve for w1:
[tex]d/dw1 (w1 + l1 + 4 * sqrt(x / w1) + 4 * sqrt(w1 / x)) = 1 - 2 * sqrt(x) / w1^1.5 + 2 * sqrt(x) / w1^2.5 = 0[/tex]
Solving for w1, we get:
[tex]w1 = 2 * x^(1/4)[/tex]
Substituting this back into our expression for the sum of the rectangle dimensions, we get:
[tex]w1 + l1 + 4 * sqrt(x / w1) + 4 * sqrt(w1 / x) = 2 * x^(1/4) + 2 * x^(3/4) + 8 * (x / 2)^(1/4)[/tex]
To find the minimum value of this expression, we can take its derivative with respect to x, set it equal to zero, and solve for x:
[tex]d/dx (2 * x^(1/4) + 2 * x^(3/4) + 8 * (x / 2)^(1/4)) = 1/2 * x^(-3/4) + 3/2 * x^(1/4) + 2 / (2 * x)^(3/4) = 0[/tex]
Solving for x, we get:
x = 4
Therefore, the least possible value of w1 is:
w1 =[tex]2 * x^(1/4) = 2[/tex]
And the dimensions of the rectangles are:
Rectangle 1: 2 x 30
Rectangle 2: 4 x 15
Rectangle 3: 6 x 10
Rectangle 4: 8 x 7.5
Rectangle 5: 10 x 6
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One company's bottles of grapefruit juice are filled by a machine that is set to dispense an average of 180 milliliters (ml) of
liquid. A quality-control inspector must check that the machine is working properly. The inspector takes a random sample of
40 bottles and measures the volume of liquid in each bottle. The mean amount of liquid in the bottles is 179. 6 ml and the
standard deviation is 1. 3 ml.
We want to test
Họ: H = 180
H:H * 180
where p = the true mean volume of liquid dispensed by the machine. A significance test yields a P-value of 0. 589. We
choose to make a decision using a = 0. 10.
What conclusion would you make at the a = 0. 10 level?
One of the sample t-test significance 0.10 level, we do not reject the null hypothesis that the mean volume of liquid dispensed by the machine is equal to 180 ml.
A one-sample t-test to determine whether the mean volume of liquid dispensed by the machine is significantly different from 180 ml.
The null hypothesis is that the true mean volume of liquid dispensed by the machine is equal to 180 ml, while the alternative hypothesis is that it is not equal to 180 ml.
We can calculate the t-value as follows:
[tex]t = (179.6 - 180) / (1.3 \sqrt (40)) = -1.385[/tex]
Using a t-distribution table with degrees of freedom (df) = 39 (since we have a sample size of 40 and one parameter estimated), and a significance level of 0.10, we find the critical t-value to be ±1.684.
Since our calculated t-value (-1.385) falls within the acceptance region (-1.684 < t < 1.684), we fail to reject the null hypothesis at the 0.10 significance level.
Sufficient evidence to conclude that the mean volume of liquid dispensed by the machine is significantly different from 180 ml.
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