The complement and the supplement of an angle of 68º are given as follows:
Complement: 22º.Supplement: 112º.How to obtain the complement of an angle?When two angles are complementary, the sum of the measures of the angles is of 90º, hence the complement of an angle is obtained subtracting 90º by the angle measure.When two angles are supplementary, the sum of the measures of the angles is of 180º, hence the supplement of an angle is obtained subtracting 180º by the angle measure.Hence the complement and the supplement of an angle of 68º are obtained as follows:
Complement: 90 - 68 = 22º.Supplement: 180 - 68 = 112º.More can be learned about complementary and supplementary angles at https://brainly.com/question/2046046
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geometry please help
A graph of triangle CAT is shown below.
A graph of triangle C'A'T, the image of ΔCAT after a reflection in the x-axis is shown below.
ΔC'A'T represents a rigid motion because the pre-image and the image are congruent.
What is a reflection over the x-axis?In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y). This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.
Next, we would apply a reflection over or across the x-axis;
(x, y) → (x, -y)
C (-1, 2) → (-1, -(2)) = C' (-1, -2)
A (-1, 4) → (-1, -(4)) = A' (-1, -4)
T (3, 2) → (3, -(2)) = T' (3, -2)
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If the rate of inflation is 2.6% per year, the future price p (1) (in dollars) of a certain item can be modeled by the following exponential function, where t is the
number of years from today.
p(t)=1200 (1.026)
Find the current price of the item and the price 8 years from today.
Round your answers to the nearest dollar as necessary.
Current price ?$
Price 8 years from today?$
a) Based on the exponential function, p(t)=1200 (1.026)^t, the item's current price with a rate of inflation of 2.6% per year is $1,200.
b) In 8 years, the future price of the item would be $1,473.53.
What is an exponential function?An exponential function is a mathematical equation depicting the relationship between one variable and another with a periodic constant rate of growth or decline.
Exponential functions are classified into two:
Exponential growth functionExponential decay function.Let the number of years from today = t
Annual inflation rate = 2.6%
Current price = $1,200
Time in years = 8 years
Exponential function:p(t)=1200 (1.026)^t
p(8) = 1,200(1.026)^8
= $1,473.53
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help
A grocery store clerk is placing cans of soup on an 8-ft long shelf. If the cans are lined up in a row, how many more of the smaller cans will fit on the shelf than the larger cans? Round to the nearest whole number.
There is 1 smaller can that will be put on the shelve.
Given that, a clerk is placing cans of soup on an 8-ft long shelf, we need to find how many more of the smaller cans will fit on the shelf than the larger cans,
Volume of small cans = 127.2 in³
Therefore,
127.2 = 3.14 × 4.5 × r²
r² = 127.2 / 14.13
r² = 9
r = 3
Similarly, the volume of the large can = 301.6 in³
301.6 = 3.14 × 6 × r²
r² = 16
r = 4
The number of small cans that can be out on the shelve = 8/3 ≈ 9
The number of larger cans that can be out on the shelve = 8/4 = 2
Hence, there is 1 smaller can that will be put on the shelve.
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johnathannes brahms destoryed or left many because of his work unplublished because
A. He didnt want to be sucessful
B. He thought they were too good to publish
C. He was a perfectionist and was never satisfied with them.
Answer asap for brainlist
Mary has the qualities of being a team leader.
Here, we have,
Team leaders upload any other stage of control. They’re employed to persuade and construct relationships, to make matters happen. Team chief’s to-do lists may be vast, however through categorizing them, it’ll come up with readability approximately the reason of your job.
A group chief has an outline of a set of humans, motivates, offers guidance and video display units performance. It is probably an professional identify alternate or a delegation exercising out of your management, however both way, being a group chief separates you out of your friends as a depended on individual to manipulate a assignment or institution of humans.
Team leaders and executives have exceptional responsibilities. Unlike managers, group leaders won’t have the authority to direct, alternate plans, put in force or construct their groups via hiring and firing. Their position is mostly a motivational and inspirational one inside an organization. They’re professional dating developers and mediators, liaising among the humans and management.
Thus this are the qualities of team leader.
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complete question;
Mary is a perfectionist . She loves participating on teams and often takes on too many tasks because she wants to ensure that they are done on time and on budget.
A. Roel organized his students' scores on the following line plot.
Students' Scores
00000
60
65
70
75
80
85
90
95
100
Based on this information, which of the following BEST describes the median student
score?
A 35
B. 80
C. 85
D. 90
Answer: - None of the above.
Step-by-step explanation:
The median is the middle value of the data set when arranged in order. In this case, we have an even number of values, so we need to take the average of the middle two values to find the median. The middle two values are 80 and 85. Therefore, the median student score is: (80 + 85) / 2 = 82.5
The closest answer choice to this value is C. 85, but it is not the correct answer. Therefore, the correct answer is none of the above.
Please help with this simple question (Large reward)
Answer:
H
Step-by-step explanation:
Firstly, G and J cannot be the answers because the beginning numbers are too big. If you look at the fractions, you can see 2 and 7/8 because 2 is the smallest of all of them. Now the answer is between F and H.
In F, the second number is 3 and 1/2.
In H, the second number is 3 and 1/4.
If you want to make it an improper fraction to visualize better, it would be 7/2 and 12/4=6/2. Or you can just compare to see which is smaller, 1/2 or 1/4. In this case, 3 and 1/4 is smaller. Therefore the answer is H.
Amelia gathered data about the heights of students at her school. Based on the displays, what inferences can you make about each sample?
Amelia can make inferences by using
The range of heightsThe shape/mean of the distributionThe presence of outliers, etc.What is the inferences about?Inferences from data on student heights include range determination based on minimum and maximum values, indicating greater variability with a larger range.
The average height of students can be determined using measures like mean, median, and mode. The distribution's shape can be determined by the histogram or box plot, indicating how the heights are distributed. inferences can be made by the use of height samples to gain insight into school population.
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Which equation represents a direct variation?
Oy - 4x = 8
Oy + 2 = 7x
Oy - 3x = 0
O y = 5x - 2
Answer:
The equation that represents a direct variation is y = 5x - 2.
In a direct variation, one variable is proportional to the other variable, which means that as one variable increases or decreases, the other variable also increases or decreases in direct proportion. The general form of an equation that represents direct variation is y = kx, where k is a constant of proportionality.
In this case, the equation y = 5x - 2 is in the form of y = kx, where k = 5. This means that as x increases or decreases, y will also increase or decrease proportionally, with a constant rate of change of 5. Therefore, this equation represents a direct variation.
Question 20 of 42
Which of the following is a type of data that is likely to be normally
distributed?
A. The weights of all the 5-year-olds in the United States
B. The outcomes of a fair die rolled 100 times
C. The number of students in a class who get a perfect score on a
test
D. The number of students in a class who get a B on a test
Answer: Option B, the outcomes of a fair die rolled 100 times, is likely to be normally distributed.
Step-by-step explanation:
Answer: Option B, The outcomes of a fair die rolled 100 times
Step-by-step explanation:
that would most likely be normally distributed.
hope it helped! Good luck!!! :)
Triangle ABC and triangle DEF are similar. What is the measure
of side DF?
B
E
9 cm
15 cm
D
с
? cm
25 cm
F
The measure of side DF is 15cm
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. For two traingles to be similar, the corresponding angles of the triangles must be equal.
And also, the ratio of corresponding sides of Similar triangles are equal
This means;
15 /25 = 9/DF
representing DF by x
15/25 = 9/x
15x = 25×9
15x = 225
divide both sides by 15
x = 225/15
x = 15cm
Therefore the measure of DF is 15cm
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(c) log12-(1/2log 9+1/3 log8) write a single form equation
The log expression in single equation is log 12 - log 3 + log 2.
What is the solution of the log equation?
The solution of the log equation is calculated by applying the following method.
log 12 - ¹/₂log 9 + ¹/₃log 8
we can simplify the equation as follows;
log 12 - log 3 ⁽² ˣ ¹/₂⁾ + log 2 ⁽³ ˣ ¹/₃⁾
= log 12 - log 3¹ + log 2¹
= log 12 - log 3 + log 2
Thus, the log equation or expression has been simplified in the form of single equation.
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what type of number is 15/10
Answer:
The number 15/10 is a fraction, specifically a rational number. It can be expressed as the quotient of two integers, where 15 is the numerator and 10 is the denominator. In this case, the fraction is equivalent to 1.5, which is a decimal representation of the rational number.
Step-by-step explanation:
Answer:
Fraction, 15/10 is written out as 15 Over 10
What types of polynomials do you think could factor using the relationship made
in the previous question?
If a is replaced with an x, and b is replaced with a 2, how would the equation
look, and what does it mean?
Create your own replacement of a and b and explain how your equations would
look and what would it mean?
The relationship in the previous question involves the difference of squares pattern, which applies to polynomials of the form:
a^2 - b^2
where a and b are any real numbers.
If we replace a with x and b with 2, the equation would look like:
x^2 - 2^2 = (x + 2)(x - 2)
This means that any polynomial of the form x^2 - 4 can be factored using the difference of squares pattern.
As for creating our own replacement of a and b, let's say we replace a with y and b with 3. Then the equation would look like:
y^2 - 3^2 = (y + 3)(y - 3)
This means that any polynomial of the form y^2 - 9 can be factored using the difference of squares pattern.
graphing transformations
Each of the graphs should be matched with the appropriate function as follows;
Graph 1: D. f(x) = (x - 2)² + 3
Graph 2: C. f(x) = -(x - 2)² - 3
Graph 3: B. f(x) = -(x - 3)² - 2
Graph 4: C. f(x) = (x - 3)² + 2
What is the general form of a quadratic function?In Mathematics and Geometry, the standard or general form of a quadratic function can be modeled and represented by using the following quadratic equation;
y = ax² + bx + c
Where:
a and b represents the coefficients of the first and second term in the quadratic function.c represents the constant term.Mathematically, the vertex form of a quadratic function is modeled by this equation:
f(x) = a(x - h)² + k
Where:
h and k represents vertex of the graph.a represents leading coefficient.In conclusion, we can logically deduce that each of the given quadratic function were either vertically translated or horizontally translated.
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Select all of the figures that show axis symmetry.
Answer: All of them
Step-by-step explanation:
All of them can be divided into two equal halves.
The figures that show axis symmetry are 1, 2, and 3.
What is the axis of symmetry?The axis of symmetry is a line, plane, or other axis that divides a geometric figure into two mirror-image halves that are congruent to each other.
A figure has axis symmetry if it looks the same after being rotated around an axis.
Out of the given figures, the ones that have axis symmetry are:
Sphere (has infinite axis symmetry)Cone (has rotational symmetry around its central axis)Cylinder (has rotational symmetry around its central axis)The figures that do not have axis symmetry are:
Cube (has only point symmetry)Rectangular cone (has no rotational symmetry)So, the figures that show axis symmetry are 1, 2, and 3.
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Each triangle in the quilting square pattern shows has a base and height of 2 inches how much fabric is needed to create one quilting square
Answer:
The amount of fabric needed to create one quilting square is 8 inches.
Answer:
The amount of fabric needed to create one quilting square is 8 inches.
Step-by-step explanation:
I need help with solving and graph each of the following inequalities
The solution and graph of each of the inequalities is shown below:
k ≥ 4v < 17x ≤ 6y > 8g > 3.9w > -7What is an inequality?In Mathematics and Geometry, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;
Greater than (>).Less than (<).Greater than or equal to (≥).Less than or equal to (≤).Next, we would determine the solution for each of the inequalities as follows;
k + 7 ≥ 11
k ≥ 11 - 7
k ≥ 4
5 > v - 12
5 + 12 > v
17 > v
v < 17 (flip).
-5x ≤ -30
x ≤ 6
y/4 > 2
y > 2(4)
y > 8
10g + 19 > 49
10g > 49 - 19
10g > 39
g > 3.9
-w - 7 > 0
-w > 7
w > -7
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5a-3b=7b find the ratio of a:b
(i need answers quick)
The ratio of a:b in 5a-3b=7b is 4:5.
We know that,
A ratio is an ordered pair of numbers a and b, written as a/b where b does not equal 0. A proportion is an equation in which two ratios are equal to each other. For example, if there are 2 boys and 6 girls, you can write the ratio as: 1 : 3 (for every boy there are 3 girls).
Ratios compare two numbers obtained by dividing. If one data point (A) is compared to another data point (B), it is expressed as A/B. It means that we are partitioning information A from information B. For example, if A is five and B is 10, your ratio would be 5/10, or 1/2.
For above information,
5a - 3b = 7b
5a = 7b- 3b
5a= 4b
So, a/b= 4/5
a:b is 4:5
Thus, the ratio of a:b in 5a-3b=7b is 4:5.
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URGENT! Can someone please help?
a. The missing values of the logarithm expression is log₃(40).
b. The missing values of the logarithm expression is log₅(8).
c. The missing values of the logarithm expression is log₂(1/25).
What is the missing of the logarithm expression?The missing values of the logarithm expression is calculated as follows;
(a). log₃5 + log₃8, the expression is simplified as follows;
log₃5 + log₃8 = log₃(5 x 8) = log₃(40)
(b). The log expression is simplified as;
log₅3 - log₅X = log₅3/8
log₅X = log₅8
X = 8
(c). The log expression is simplified as;
-2log₂5 = log₂Y
log₂5⁻² = log₂Y
5⁻² = Y
1/25 = Y
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∠1 and ∠2 are vertical angles. ∠2 has a measure of 31°.
What is the measure of ∠1?
Enter your answer in the box.
The measure of <1 is 31°.
What are vertical angles?Two angles are said to be vertical angles when they are formed by two straight lines intersecting at a point. This produces a series of angles of which some are vertical, thus they have equal measure.
In the information given in the question, it was given that <1 and <2 are vertical angles. This implies that they have equal measure, such that;
<2 = 31° (given)
So that;
<1 = <2 = 31°
Thus,
<1 = 31°
Therefore, it can be concluded that the measure of <1 is 31°.
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Chantel is deciding which of two similar wallets she will buy. Both are rectangular, but she wants the one that will best fit in her purse. The larger wallet is 18 centimeters long and the smaller one is 10 centimeters long. What is the area of the larger wallet if the area of the smaller one is 60 square centimeters?
The requried area of the larger wallet is 194.4 square centimeters.
Since the two wallets are similar, their corresponding sides are proportional. Let x be the width of the smaller wallet, then the width of the larger wallet is 18/10 times x, or 9/5 times x. Therefore, the dimensions of the larger wallet are 9/5 times the dimensions of the smaller wallet.
The area of the smaller wallet is x times 10 (length times width), or 10x square centimeters. We are given that this area is 60 square centimeters, so:
10x = 60
x = 6
Therefore, the width of the smaller wallet is 6 centimeters. The width of the larger wallet is 9/5 times 6, or 10.8 centimeters. The length of the larger wallet is 18 centimeters.
The area of the larger wallet is 10.8 times 18, or 194.4 square centimeters. Therefore, the area of the larger wallet is 194.4 square centimeters.
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What is the value of c in the equation?
15.3 = StartFraction c Over 2.1 EndFraction
13.2
17.4
32.13
321.3
Answer:
32.13
Step-by-step explanation:
The value of c in the equation 15.3 = c/2.1 is 32.13. To solve for c, you can cross-multiply and simplify the equation to c = 15.3 x 2.1 = 32.13.
The solution to the equation "the quotient of a number and 8 is 16" is 128. To solve for the number, you can multiply both sides by 8 to get "the number is 16 x 8 = 128."
To translate the sentence "the quotient of a number and 8 is 16" into an equation, you can use "x" to represent the number and write x/8 = 16.
Answer:
c or a
Step-by-step explanation:
i thing a
Claire got 17 out of 20 questions correct on her vocabulary quiz. What percent of the questions did she get wrong?
The percentage of the questions she got wrong is 15%
What is a Percentage?
Percentage is a number or ratio by dividing the value by the total value and then multiplying the result by 100.
How to determine this
Out of 20 questions, Claire got 17 questions
i.e the questions she got right = 17
The questions she got wrong = 20 - 17 = 3
To get the percentage she got wrong
= 3/20 *100%
= 300/20%
= 15%
Therefore, the percentage she got wrong is 15%
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Let p be the constant function p (z) = 1,000,000,000. Let q be the exponential function q (2) = 10
The product of the two functions is (pq) (z) = 10ª-2, where a is a constant. What is the value of a?
C2023 Illuminate Education, Inc.
P
C
[tex]\large \boxed{\mathrm{10}}[/tex]
The constant function p has a fixed output of one billion, while the exponential function q has a base of two and an output of 10. As the input increases by one, the output of q doubles. For example, q(3) = 20 and q(4) = 40. The rate of growth of q is much faster than p. In fact, q will eventually surpass p for large enough inputs. This demonstrates the power of exponential growth and highlights the importance of understanding the functions used in equations and models.
Answer:
10 or 1b
Step-by-step explanation:
Line ac and df are parallel they are cut by transversal Hj with your partner find the seven unknown angle measures in the diagram explain your reasoning. What do you notice about the angles with vertex B and the angles with vertex E
The measure of the unknown angles in the parallel lines AC and DF with transversal HJ are as follow,
m ∠DEH = 63° , m ∠BEF = 63° ,m ∠CBJ = 63°, m ∠CBE = 117° , m ∠FEH = 117° , m ∠BED = 117° , and m ∠ABJ = 117°.
Two lines AC and DF are parallel to each other in the attached figure
Transversal HJ cut lines AC and DF at point B and E respectively.
Measure of ∠ABE = 63°.
Using the result based on corresponding angles, alternate interior angles , and vertically opposite angles of a parallel lines we get,
Measure of ∠ABE ≅ Measure of ∠DEH ( corresponding angles)
⇒Measure of ∠DEH = 63°
Measure of ∠ABE ≅ Measure of ∠BEF ( alternate interior angles)
⇒Measure of ∠BEF = 63°
Measure of ∠ABE ≅ Measure of ∠CBJ ( vertically opposite angles)
⇒Measure of ∠CBJ = 63°
Using the result of linear pair angles,
Measure of ∠ABE + Measure of ∠CBE = 180°
⇒ Measure of ∠CBE = 180° -63°
⇒ Measure of ∠CBE = 117°
Measure of ∠CBE ≅ Measure of ∠FEH ( corresponding angles)
⇒Measure of ∠FEH = 117°
Measure of ∠CBE ≅ Measure of ∠BED ( alternate interior angles)
⇒Measure of ∠BED = 117°
Measure of ∠CBE ≅ Measure of ∠ABJ ( vertically opposite angles)
⇒Measure of ∠ABJ = 117°
Therefore, the measure of the seven unknown angles formed in the parallel line are given by m ∠DEH = 63° , m ∠BEF = 63° ,m ∠CBJ = 63°, m ∠CBE = 117° , m ∠FEH = 117° , m ∠BED = 117° , and m ∠ABJ = 117°.
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The above question is incomplete, the complete question is:
Lines AC and DF are parallel. They are cut by transversal HJ with your partner find the seven unknown angle measures in the diagram explain your reasoning. What do you notice about the angles with vertex B and the angles with vertex E.
In the attached figure.
A swimming pool is shaped like the figure below. Each end is a semicircle, and the length and width of the rectangle are 60 feet and 30 feet, respectively.
The area of the border is 105.53 feet².
We have,
The swimming pool is a combination of two shapes.
Rectangle:
Length = 60 feet
Width = 30 feet
Area = 60 x 30 = 1800 feet²
Semicircle:
Diameter = 30 feet
Radius = 15 feet
Area = π/2 r² = 1/2 x 3.14 x 15 x 15 = 353.25 feet².
Now,
The area of the swimming pool.
= 1800 + 353.25
= 2153.25 feet²
Now,
Without the border.
Area of the swimming pool.
= rectangle area + semicircle area
= 60 x (30 - 1) + 1/2 x π x (15 - 1)²
= 60 x 29 + 1/2 x 3.14 x 14 x 14
= 1740 + 307.72
= 2047.72 feet²
Now,
The area of the border.
= 2153.25 - 2047.72
= 105.53 feet²
Thus,
The area of the border is 105.53 feet².
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A man has 16 coins in his pocket, all of which are dimes and quarters. If the total value of his change is $2.50, how many dimes and how many quarters does he have?
Let d = number of dimes, q = number of quarters.
Step one:
A man has 16 coins in his pocket, all of which are dimes and quarters.
Set up the first equation for the number of dimes and quarters. Help PLS
Answer: 10 dimes and 6 quarters, see below for the equations.
Step-by-step explanation:
We know how much a dime is worth and how much a quarter is worth, and they give us the total number of coins and the total number of change.
This equation shows the total number of coins:
d + q = 16
This equation shows the worth of these coins:
0.10d + 0.25q = 2.5
Now, we can graph these equations. See attached. The point of intersection, where the lines cross, is our solution. In the graph, d = x and q = y.
This point is (10, 6) meaning he has 10 dimes and 6 quarters in his pocket.
Name 2 other categories of data you could use to better understand the physical fitness levels of these students.
Two other categories of data that could be used to better understand the physical fitness levels of students are:
Body composition data:
Cardiovascular fitness data:
We have,
Two other categories of data that could be used to better understand the physical fitness levels of students are:
Body composition data:
This data can include measurements such as body mass index (BMI), body fat percentage, and waist circumference.
By collecting body composition data for students, we can better understand their overall physical health and identify potential risk factors for chronic diseases related to physical fitness.
Cardiovascular fitness data:
This data can include measurements such as resting heart rate, maximal oxygen uptake (VO2 max), and cardiovascular endurance test results.
By collecting cardiovascular fitness data for students, we can better understand their cardiovascular health and fitness levels, as well as identify potential risk factors for heart disease and other related conditions.
Thus,
Two other categories of data that could be used to better understand the physical fitness levels of students are:
Body composition data:
Cardiovascular fitness data:
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HELP FASTER WILL GIVE BRAINLIEST!!!!!!!!! NOTICE ITS + AND THEN - IN EQUATION NOT TWO +, SO D ISNT CORRECT!!!!!!
Identify the correct graph of the system of equations.
3x + y = 12
x − 4y = 4
The graph shows a line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma 1. There is a second line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma 12.
The graph shows a line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma 1. There is a second line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma negative 12.
The graph shows a line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma negative 1. There is a second line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma 12.
The graph shows a line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma negative 1. There is a second line with an x-intercept at 4 comma 0 and a y-intercept at 0 comma negative 12.
The correct graph of the system of equations 3x + y = 12 and x - 4y = 4 is:
The graph shows a line with an x-intercept at (4, 0) and a y-intercept at
(0, -1). There is a second line with an x-intercept at (4, 0) and a y-intercept at (0, 12).
We have,
To graph the system of equations 3x + y = 12 and x − 4y = 4, we first need to rearrange them in slope-intercept form (y = mx + b) where m is the slope and b is the y-intercept:
3x + y = 12 => y = -3x + 12
x - 4y = 4 => y = (1/4)x - 1
Now we can see that the first equation has a slope of -3 and a y-intercept of 12, while the second equation has a slope of 1/4 and a y-intercept of -1.
Now,
The x-intercept is when y = 0.
So,
3x + y = 12
3x = 12
x = 4
And,
x - 4y = 4
x = 4
Thus,
The correct graph of the system of equations 3x + y = 12 and x - 4y = 4 is:
The graph shows a line with an x-intercept at (4, 0) and a y-intercept at
(0, -1). There is a second line with an x-intercept at (4, 0) and a y-intercept at (0, 12).
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The volume in a set of wine bottles is known to follow a normal distribution with mean μ ounces and standard deviation of σ = 0.5 ounces. You take a sample of the bottles and measure their volumes. How many bottles do you need to sample to construct a 90% confidence interval for μ with a margin of error at most 0.1 ounces?
We need to sample at least 18 wine bottles to construct a 90% confidence interval for the mean volume of the population, with a margin of error of at most 0.1 ounces.
To determine the sample size required for a given confidence interval and margin of error, we need to use the following formula:
[tex]n = (Z^2 * \sigma^2) / E^2[/tex]
Where:
n = sample size
Z = the z-score associated with the desired confidence interval (for 90% confidence, Z = 1.645)
σ = the standard deviation of the population (0.5 ounces in this case)
E = the desired margin of error (0.1 ounces in this case)
Plugging in the values, we get:
[tex]n = (1.645^2 * 0.5^2) / 0.1^2[/tex]
n = 17.1
Thus, we need to sample at least 18 wine bottles to create a 90% confidence interval for the population's mean volume, with a margin of error of no more than 0.1 ounces.
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We need to sample at least 69 bottles to construct a 90% confidence interval for the mean volume of the wine bottles with a margin of error at most 0.1 ounces.
We have,
To construct a confidence interval for the mean volume of the wine bottles, we need to use the formula:
Confidence interval = sample mean ± margin of error
Where,
Margin of error = z (σ/√(n))
Here, we want to construct a 90% confidence interval with a margin of error at most 0.1 ounces.
This means that the margin of error should be less than or equal to 0.1, and the confidence level is 90%, so the critical value of z is 1.645
(using a standard normal table or calculator).
We are given that the standard deviation is σ = 0.5 ounces.
We don't know the sample mean or the sample size (number of bottles), so we'll use the worst-case scenario to find the sample size that gives the largest margin of error.
The worst-case scenario is when the sample mean is equal to the population mean (μ) plus the margin of error (0.1 ounces), and the sample size is as small as possible (which gives the largest margin of error).
So, we have:
0.1 = 1.645 x (0.5/sqrt(n))
Solving for n, we get:
n = (1.645 x 0.5/0.1)²
n = 68.225
Rounding up to the nearest whole number, we get:
n = 69
Therefore,
We need to sample at least 69 bottles to construct a 90% confidence interval for the mean volume of the wine bottles with a margin of error at most 0.1 ounces.
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