Answers
When we evaluate the function f(x)=(1/4)^-x for x = -2, 1, and 2, we find that f(-2) = 1/16, f(1) = 4, and f(2) = 1/16.
To evaluate f(x)=(1/4)^-x for specific values of x, we can substitute the given values into the function expression and simplify the calculations.
Evaluating f(-2):
Substitute x = -2 into the function f(x):
f(-2) = (1/4)^-(-2) = (1/4)^2 = 1/16
Evaluating f(1):
Substitute x = 1 into the function f(x):
f(1) = (1/4)^-1 = 4^1 = 4
Evaluating f(2):
Substitute x = 2 into the function f(x):
f(2) = (1/4)^-2 = (1/4)^2 = 1/16
Therefore, the evaluations of f(-2), f(1), and f(2) are:
f(-2) = 1/16
f(1) = 4
f(2) = 1/16
In conclusion, when we evaluate the function f(x)=(1/4)^-x for x = -2, 1, and 2, we find that f(-2) = 1/16, f(1) = 4, and f(2) = 1/16. These values represent the corresponding outputs of the function for the given inputs.
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Related Questions
devide 255 in the ratio 3:7:7:10
Answers
Answer:5
Step-by-step explanation:
5098
1,300 aliens from Area 51 escaped at 2:00am on Wednesday, and the armed forces are able to capture half of them every four hours. How many aliens will remain free at 8:00pm on Wednesday?
Answers
Answer:
Let's start by figuring out how many aliens are captured every four hours. If half of the aliens are captured every four hours, that means the number of remaining aliens is also halved every four hours. So we can use exponential decay to model the number of aliens remaining after t four-hour periods:
N(t) = N₀ (1/2)^t
where N₀ is the initial number of aliens and t is the number of four-hour periods that have passed.
The initial number of aliens is 1,300, and we want to know how many remain after 14 four-hour periods (from 2:00am to 8:00pm, a total of 16 hours with 4-hour intervals):
N(14) = 1,300 (1/2)^14
N(14) ≈ 20.16
So there will be approximately 20 aliens remaining free at 8:00pm on Wednesday. Note that the actual number may be slightly different due to rounding, and also because we assumed that the capture rate remains constant throughout the 16-hour period.
Step-by-step explanation:
Let's start by figuring out how many aliens are captured every four hours. If half of the aliens are captured every four hours, that means the number of remaining aliens is also halved every four hours. So we can use exponential decay to model the number of aliens remaining after t four-hour periods:
N(t) = N₀ (1/2)^t
where N₀ is the initial number of aliens and t is the number of four-hour periods that have passed.
The initial number of aliens is 1,300, and we want to know how many remain after 14 four-hour periods (from 2:00am to 8:00pm, a total of 16 hours with 4-hour intervals):
N(14) = 1,300 (1/2)^14
N(14) ≈ 20.16
So there will be approximately 20 aliens remaining free at 8:00pm on Wednesday. Note that the actual number may be slightly different due to rounding, and also because we assumed that the capture rate remains constant throughout the 16-hour period. please give brainliest! :D
Help please and thank u
Answers
Answer:
A=6cm
D=11,66cm
Step-by-step explanation:
A=6cm
D=6²+10²=136
D=√136
D=11,66
Assume the probability of rain on any given independent day is 0.23. What is the average number of days it rains for the first time?
Answers
The probability that this song will be played on exactly 0 days out of 7 days, if The probability that a particular song is played is 0.23 is 0.008.
We have,
A combination is a choice of items from a group of different items, where the order of the choices is irrelevant.
Given:
The probability that a particular song is played is 0.23,
As can be seen that the problem is of combination, so we have to choose 5 out of 7,
Calculate the combination as shown below,
⁷C₅ = 7! / (5! × 2!) = 7 × 6 / 2
⁷C₅ = 7 × 6 / 2
⁷C₅ = 21
Calculate the probability as shown below,
P = ⁷C₅ × (0.23)⁵ × (0.77)²
P = 0.008
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complete question;
During a single day at radio station WMZH, the probability that a particular song is played is 0.23. What is the probability that this song will be played on exactly 0 days out of 7 days? Round your answer to the nearest thousandth.
QUESTIONS 1. Explain and illustrate the difference between and (4) 2. Explain and illustrate the difference between ratio and proportional reasoning. Do not copy definitions from any source but use your own understanding. (4) 3. Illustrate with a drawing or diagram what "fifth" means in each of the following instances: ● ordinal number ● unit fraction 4. Show with a diagram 3, using the following: 1.1 the area model 1.2 set mode 1.3 the length model (3) 2. A rectangular piece of paper has a perimeter of 54 cm. Its length is 1 of its width. Find its dimensions. (3) 3. If the given regular hexagon represents one whole: 3.1 Show three sixths. 3.2 Write the equivalence 6.1 above. 3 Represent+ in the figure given to prove that a set of two halves 6 make a whole. (4) 6.3.1 Draw a number line showing a whole made of six equal parts. Show the sum of two halves on the same number line. (1) (1) (2) (3)
Answers
The difference between 1/5 and 2/10 is that they are different representations of the same value.
The ratio is a comparison of two quantities while proportional reasoning involves finding an equivalent ratio.
"Fifth" can refer to the ordinal number of a position or the unit fraction 1/5.
The diagram can be used to illustrate fraction concepts.
How to solve1/5 is simple, but 2/10 can be reduced to 1/5 by dividing both numbers by 2. They are both 0.2 or 20%. 1/5 and 2/10 are different fractions with different purposes.
Use 1/5 for 5 parts and 1/5 (simplified from 2/10) for 10 parts. 1/5 and 2/10 = 0.2, ratio reasoning emphasizes ratios between quantities. A 2:3 boys-to-girls ratio compares the constant ratio between changing quantities.
Proportional reasoning maintains constant ratios between quantities, while ratio reasoning uses fractions or ratios to compare relationships. For instance, if distance doubles, so does speed.
Step 3/5: "Fifth" is the ordinal number for the object five places away in a sequence. See diagram. The fifth circle is 1/5 of a whole in unit fraction sequence. Imagine a rectangle divided into five equal parts, each shaded to show 1/5. Shaded part = 1/5 of whole.
To show 25/7 with area model, make 7x25 rectangle and split into 7 equal parts. 25/7 per part.
1.2 The set model:
To represent 25/7, divide 25 objects into 7 equal sets, resulting in 3 objects per set with 1 left over. This represents the fraction of 25/7.
Step 5/5
Let's assume the width of the rectangular piece of paper as x cm. Then the length of the paper would be:
Now we know that the perimeter of the rectangle is given by the formula:
So, for this rectangle:
54 = 2(5x + x)
Simplifying and solving for x, we get:
54 = 2(6x)
27 = 6x
x = 4.5 cm
Therefore, the width of the paper is 4.5 cm and the length is 5 times the width which is 22.5 cm
The difference between 1/5 and 2/10 is that they are different representations of the same value.
The ratio is a comparison of two quantities while proportional reasoning involves finding an equivalent ratio.
"Fifth" can refer to the ordinal number of a position or the unit fraction 1/5.
The diagram can be used to illustrate fraction concepts.
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62x2^5x=27 what is the answerr to this question that i cant figure out
Answers
Answer: hello am I solving for x?
Step-by-step explanation:question
Finding Probabilities
Every person has blood type O, A, B, or AB. A random
group of people are blood-typed, and the results are
shown in the table.
Blood Type
A
B
AB
Number of People
22
20
6
2
Use the table to determine the following probabilities.
The probability that a randomly chosen person from
this group has type B is [
The probability that a randomly chosen person from
this group has type AB is [
The probability that a randomly chosen person from
this group has type B or type AB blood is
Answers
The probability that a randomly chosen person from this group has type B is 0.4.
The probability of type AB is 0.12, and the probability of type B or type AB is 0.52.
We have,
The probability that a randomly chosen person from this group has type B is given by the number of people with blood type B divided by the total number of people in the group:
Probability of type B
= Number of people with type B / Total number of people
In this case, the number of people with blood type B is 20, and the total number of people is 50 (22 + 20 + 6 + 2).
Probability of type B = 20 / 50 = 0.4
Similarly, the probability that a randomly chosen person from this group has type AB is given by the number of people with blood type AB divided by the total number of people:
Probability of type AB = Number of people with type AB / Total number of people
In this case, the number of people with blood type AB is 6, and the total number of people is 50. Therefore,
Probability of type AB = 6 / 50 = 0.12
To find the probability that a randomly chosen person from this group has blood type B or type AB, we need to add the probabilities of both events:
Probability of type B or type AB = Probability of type B + Probability of type AB
Probability of type B or type AB = 0.4 + 0.12 = 0.52
Thus,
The probability that a randomly chosen person from this group has type B is 0.4, the probability of type AB is 0.12, and the probability of type B or type AB is 0.52.
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In 2004, a school population was 2122. By 2009 the population had grown to 2647.
1) How much did the population grow between the year 2004 and 2009?
Answers
The population grow by 24.7% between the years 2004 and 2009
Calculating how much the population grow between the yearsfrom the question, we have the following parameters that can be used in our computation:
Population in 2004 = 2122
Population in 2009 = 2647
The growth of the population between the years is calculated as
Percentage = Population in 2009/Population in 2004 - 1
Substitute the known values in the above equation, so, we have the following representation
Percentage = 2647/2122 - 1
Evaluate
Percentage = 24.7%
Hence, the population grow between the years by 24.7%
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As people are living longer and the world's population keeps increasing, there are more and more people ages 65 or older. Over the last 17 years, the number of people ages 65 or older has been growing exponentially, increasing about 2.7% every year. In 2017, there were approximately 656 million people ages 65 or older. If the trend continues, in what year will the population of people 65 or older surpass 1 billion?
Answers
Nt = N0 * (1 + r)^t
where Nt is the population at time t, N0 is the initial population, r is the annual growth rate as a decimal, and t is the number of years.
We know that N0 = 656 million, r = 0.027 (since the population is growing at a rate of 2.7% per year), and we want to find the year when Nt = 1 billion.
Substituting in these values, we get:
1 billion = 656 million * (1 + 0.027)^t
Simplifying:
1.524 = 1.027^t
Taking the natural log of both sides:
ln(1.524) = t * ln(1.027)
Dividing by ln(1.027):
t = ln(1.524) / ln(1.027)
Using a calculator:
t ≈ 23.1 years
Therefore, if the trend continues, the population of people ages 65 or older will surpass 1 billion in approximately 23.1 years from 2017.
Answer: 2040 (rounded to the nearest whole year).
A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 70 cm/s. Find the rate (in cm2/s) at which the area within the circle is increasing after each of the following.
(a) after 1 s
Answers
A = πr^2
where A is the area and r is the radius.
We are given that a circular ripple is created in a lake and it travels outward at a speed of 70 cm/s. This means that the radius of the circle is increasing at a rate of 70 cm/s.
(a) After 1 second, the radius of the circle is:
r = 70 cm/s × 1 s = 70 cm
Now we can use the formula for the area of a circle to find the rate at which the area within the circle is increasing:
A = πr^2
A = π(70 cm)^2
A ≈ 15,400 cm^2
To find the rate at which the area within the circle is increasing, we need to take the derivative of the area with respect to time:
dA/dt = 2πr(dr/dt)
where dA/dt is the rate at which the area is increasing, and dr/dt is the rate at which the radius is increasing (which we know is 70 cm/s).
Plugging in the values we have:
dA/dt = 2π(70 cm)(70 cm/s)
dA/dt ≈ 9,800 cm^2/s
So after 1 second, the area within the circle is increasing at a rate of approximately 9,800 cm^2/s.
Find x and the measurement of each angle.
Answers
Answer:
x = 8, m2 = 78, m4 = 102, m6 = 78, m7 = 78,
Step-by-step explanation:
The graph shows the cube root parent function.
-5
5
Which statement best describes the function?
A
Answers
The statement of the function is (b) when x > 0, the graph is positive
Describing the transformation of the function f(x)From the question, we have the following parameters that can be used in our computation:
The functions f(x) and g(x)
In the graph, we can see that
The graph of the function passes through (0, 0)
This means that
When x > 0, the graph is positive
When x < 0, the graph is negative
Hence, the statement of the function is (b)
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A rectangle floor is 9 yards long and 3 yards wide what is the area of the floor in square feet
Answers
Answer:
1. Find the area of the rectangle in square yards. In this case, the area is 9 yards * 3 yards = 27 square yards.
2. Convert square yards to square feet. There are 9 square feet in 1 square yard. So, the area of the rectangle in square feet is 27 square yards * 9 square feet/square yard = 243 square feet.
Therefore, the area of the rectangle floor in square feet is 243 square feet.
Answer:243
Step-by-step explanation:
(9x3) x (3x3)
27 x 9 = 243
Find the equation of the line.
Use exact numbers.
Answers
Answer:
y = - 3x + 7
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (0, 7) and (x₂, y₂ ) = (2, 1) ← 2 points on the line
m = [tex]\frac{1-7}{2-0}[/tex] = [tex]\frac{-6}{2}[/tex] = - 3
the line crosses the y- axis at (0, 7 ) ⇒ c = 7
y = - 3x + 7 ← equation of line
The equation of the line in fully simplified slope-intercept form is y = -2.8x + 7
Writing the equation of the line in slope-intercept form.The linear graph represents the given parameter
For the graph, we have the points
(0, 7) and (25, 0)
A linear equation is represented as
y = mx + c
Where
c = y when x = 0
So, we have
y = mx + 7
Using the point (2.5, 0) on y = mx + 7, we have
m(2.5) + 7 = 0
2.5m + 7 = 0
Evaluate
m = -2.8
So, we have
y = -2.8x + 7
Hence, the equation of the line in fully simplified slope-intercept form is y = -2.8x + 7
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sin( 3pi/4 ) =
O A. 1/2
OB. -√2/2
O C. √3/2
O D. √2/2
Answers
Answer:
sin( 3pi/4 ) = -√2/2
So, B.
Which of the following will have a smaller standard deviation, if nine observations are randomly drawn from a population?
The distribution of the individual observations, X
The distribution of the sample means,
x
¯
This cannot be determined from the information given.
Answers
The distribution of the sample means will have a smaller standard deviation
Identifying which will have a smaller standard deviationFrom the question, we have the following parameters that can be used in our computation:
Observations = 9
The population is not given
However by the theorem of the central limit; we understand that
The distribution of the sample means will always have a smaller standard deviation
This is because the sample mean selected from the population might not totally represent the whole population, but would have smaller standard deviation
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When you order a salad from Nelly's Deli, you can choose from 5 different base salads, 3 different proteins, and 6 different dressings. How many different combinations for a salad are possible?
Answers
Answer:
90
Step-by-step explanation:
This type of problem is an example of combinations. In this case, in one salad, it states that you can chose between 5 bases, 3 proteins, and 6 dressings.
In this problem, you would have to multiply all numbers together.
In this case, 5(bases)x3(protiens)x6(dressings)=90
I hope this helps you!
Corine and Shonda are reading the same book.
The rate at which Corine is reading the book can be represented by the equation y=32x,
where x
is the time in hours and y
is the number of chapters read.
Shonda's rate is represented by this graph.
A graph with the X-axis labeled Time in Hours and the Y-axis labeled Chapters Read. A line begins at the origin and passes through (1, 2), (2, 4), (3, 6), and continues on.
© 2017 StrongMind. Created using GeoGebra.
If the girls read the book at the same time, who will finish first? Why?
Select the option that correctly answers both questions.
Responses
Corine will finish first. Her ratio of chapters to hours is 32.
Shonda's rate of chapters to hours is 21,
and 32
is a greater slope than 21.
Corine will finish first. Her ratio of chapters to hours is Shonda's rate of chapters to hours is 2 over 1 textsf comma and 3 halves is a greater slope than
Shonda will finish first. Her ratio of chapters to hours is 21.
Corine's rate of chapters to hours is 32,
and 21
is a greater slope than 32.
Shonda will finish first. Her ratio of chapters to hours is Corine's rate of chapters to hours is 3 halves textsf comma and 2 over 1 is a greater slope than
Shonda will finish first. Her ratio of chapters to hours is 12.
Corine's rate of chapters to hours is 23,
and 12
is a greater slope than 23.
Shonda will finish first. Her ratio of chapters to hours is Corine's rate of chapters to hours is 2 thirds textsf comma and 1 half is a greater slope than
Corine will finish first. Her ratio of chapters to hours is 23.
Shonda's rate of chapters to hours is 12,
and 23
is a greater slope than 12.
Answers
If the girls read the book at the same time, who will finish first: A. Shonda will finish first. Her ratio of chapters to hours is 2/1. Corine's rate of chapters to hours is 3/2, and 2/1 is a greater slope than 3/2.
How to calculate or determine the slope of a line?In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = rise/run
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
By substituting the given data points into the formula for the slope of a line, we have the following;
Slope (m) = 2/1
Slope (m) = 2.
Based on the table, the slope is the change in y-axis with respect to the x-axis and for Shonda it is equal to 2/1 or 2. Therefore, Shonda would finish before Corine because 2/1 > 3/2.
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
Carson and his children went into a restaurant where they sell hamburgers for $6 each and tacos for $2.50 each. Carson has $65 to spend and must buy a minimum of 12 hamburgers and tacos altogether. If Carson decided to buy 8 tacos, determine the maximum number of hamburger that he could buy.
Answers
Which of the following would you see on a circle graph?
OA. Percentages
OB. Bars
C. Points
OD. A y-axis
Answers
A, percentages
You would see percentages on a circle graph, because it's how many parts of the whole that they take up.
Happy to help, have a great day! :)
when the football game started, there were twice as many eagle fans as spartan fans. At the end of the first quarter, 500 extra eagle fans came on busses. The total attendance at the game was 4790. How many people attended from each school?
Answers
Answer:
2860 Eagle fans and 1430 Spartan fans-------------------
Let E represent the number of Eagle fans and S represent the number of Spartan fans at the start of the game.
We can set up two equations using the given information:
1) E = 2S (twice as many Eagle fans as Spartan fans) 2) E + S + 500 = 4790 (total number, including the extra 500 fans)Now we can solve the equations.
From equation (1), we can replace E in equation (2), the solve for S:
(2S) + S + 500 = 4790 3S + 500 = 47903S = 4290 S = 1430Now find the number of Eagle fans using equation (1):
E = 2S E = 2(1430) E = 2860100 Points! Algebra question. Photo attached. Please show as much work as possible. Thank you!
Answers
A. The distributions of the periods is that using the five number summary is that
The 7th Period has higher minimum and lower maximumThe 3rd period is more spread (from the quartiles)The 7th Period has higher medianB. The box and whisker plot of the periods is attached
A. Comparing the distributions of the periodsFrom the question, we have the following parameters that can be used in our computation:
7th Period
66, 72, 77, 78, 78, 80, 82, 84, 84, 87, 88, 88, 89, 89, 90 92, 92, 93, 94, 94 96, 96
3rd Period
52, 60, 62, 64, 64, 65, 68, 71, 72, 74, 75, 76, 78, 79, 83, 84, 85, 88, 89, 93, 94, 97
The distributions of the periods will be compared using the five-number summaries
Using a graphing tool, the five-number summaries are:
7th Period
Minimum: 66Lower Quartile Q1: 79.5Median: 88Upper Quartile Q3: 92.25Maximum: 963rd Period
Minimum: 52Lower Quartile Q1: 64.75Median: 75.5Upper Quartile Q3: 85.75Maximum: 97B. Creating a box and whisker plot of the periodsThe box and whisker plot of the periods is added as an attachment
From the box and whisker plot attached, we can see the five-number summaries of the periods
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Urgent! A home décor store donated a percent of every sale to hospitals. The total sales were $8,400 so the store donated $672. What percent of $8,400 was donated to hospitals?
Answers
Answer:
here is the answer in the photo
Step-by-step explanation:
mark branliest
Of the $8,400 in sales, the store donated about 8% to hospitals.
To find the percentage of $8,400 that was donated to hospitals, we can set up a proportion using the given information.
Let's represent the percentage of the sales donated as x.
We can set up the proportion:
(x/100) = 672/8400
To solve for x, we can cross-multiply and then divide:
8400 * x = 672 * 100
8400x = 67200
x = 67200 / 8400
x = 8
Therefore, the store donated approximately 8% of $8,400 to hospitals.
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NO LINKS!!
will give brainliest!!!!!!!!!!!
Answers
The value of the side BC is equal to 10√21 in the simplest rationalized form using the trigonometric ratio of tangent.
What is trigonometric ratios?The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.
The basic trigonometric ratios includes;
sine, cosine and tangent.
Considering the tangent of angle 60°
tan 60° = BC/(10√7) {opposite/adjacent}
BC = 10√7 × tan 60° {cross multiplication}
BC = 10√7 × √3 {tan 60 = √3}
BC = 10√21
Therefore, the value of the side BC is equal to 10√21 in the simplest rationalized form using the trigonometric ratio of tangent.
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The question is in the first picture, the other two pictures should help you solve the question. Certain parts of it won't make sense in the pictures two and three since its a multiple part test. Please answer this as soon as possible if you can, thank you in advance
Answers
The logo that fits the requirement is the circle logo i.e. logo 1
Calculating the area and the width in meters?From the question, we have the following parameters that can be used in our computation:
The figures that represent the logos
For the circle logo, we have
Area = 3.14 * 1/2 * 1/2
Convert units to feet using the scale
Area = 22/7 * 1/2 * 1/2 * 7 * 7
Evaluate
Area = 38.5 sq feet
Also, we have
Width = 1/2 feet *7
Width = 3.5 feet
For the plus sign, we have
Area = 5 * 1/2 * 1/2
Convert units to feet using the scale
Area = 5 * 1/2 * 1/2 * 6 * 6
Evaluate
Area = 45 sq feet
Also, we have
Width = 3 * 1/2 * 6
Width = 9 feet
Hence, the logo that fits the requirement is the circle logo i.e. logo 1
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HELP DUE!! HELP ASAP!!
Answers
The time needed before the doctor administers the medication again is given as follows:
d) 22.2 hours.
How to model the situation?The exponential function giving the amount of the medicine in the patient's bloodstream after t hours is given as follows:
[tex]A(t) = 500(0.93)^t[/tex]
The medication will be applied when the amount is given as follows:
A(t) = 100.
Hence the time needed is obtained solving the exponential function as follows:
[tex]100 = 500(0.93)^t[/tex]
[tex](0.93)^t = \frac{100}{500}[/tex]
[tex](0.93)^t = 0.2[/tex]
The logarithm of base 10 is the inverse of the exponential, hence:
[tex]t = \frac{\log{0.2}}{\log{0.93}}[/tex]
t = 22.2 hours.
Which is the time needed before the doctor administers the medication again.
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100 Points! Algebra question. Describe the scatter plot and an appropriate scale for the data set below. Photo attached. Please show as much work as possible. Thank you!
Answers
Strength: TThe strength of the scatterplot refers to how closely the data points cluster around a line or curve. Direction: The direction of the scatterplot refers to the general trend or pattern exhibited by the data points.
Answers to the aforementioned questionsTo describe the scatterplot and determine an appropriate scale, we can analyze the given data set:
x: 0, 2, -5, 3, 1, -4, 8, -6, -1
y: -3, -1, -13, 3, -1, -11, 12, -15, -5
Strength: TThe strength of the scatterplot refers to how closely the data points cluster around a line or curve.
Direction: The direction of the scatterplot refers to the general trend or pattern exhibited by the data points.
Scale: The appropriate scale for the scatterplot depends on the range of values for the x and y variables. To determine the scale, we need to consider the minimum and maximum values for each variable.
For x: The minimum value is -6 and the maximum value is 8.
For y: The minimum value is -15 and the maximum value is 12.
Based on these ranges, an appropriate scale for both x and y could be chosen with intervals of 2 or 5 units, depending on the desired level of detail.
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50 Points! Multiple choice algebra question. Photo attached. Thank you!
Answers
Answer: Answer is seen below (sorry if its confusing, this is what I got though)
Step-by-step explanation:
Need help with this pls solve
Answers
Answer:
m<1 = 47 degrees
m<2 = 43 degrees
m<3 = 90 degrees
m<4 = 43 degrees
Step-by-step explanation:
Maximize z = 4x + 4y
Subject to
2x + 4y ≤ 28
5x+6y ≤ 50
x ≥ 0
y ≥ 0
Answers
The maximum value of z = 4x + 4y, subject to the given constraints, is 92/3, which occurs when x = 14/3 and y = 4.
To maximize the objective function z = 4x + 4y subject to the given constraints, we can solve this linear programming problem using the graphical method.
First, let's graph the feasible region determined by the constraints:
1. Graph the line 2x + 4y = 28:
Rewrite the equation in slope-intercept form: y = -0.5x + 7
Plot two points on the line: (0, 7) and (14, 0), and draw a line through them.
2. Graph the line 5x + 6y = 50:
Rewrite the equation in slope-intercept form: y = (-5/6)x + (25/3)
Plot two points on the line: (0, 25/3) and (10, 5/3), and draw a line through them.
3. Shade the region that satisfies the constraints:
Shade the region below the line 2x + 4y ≤ 28.
Shade the region below the line 5x + 6y ≤ 50.
The feasible region is the intersection of the shaded regions.
Next, we evaluate the objective function at the corner points of the feasible region to determine the maximum value.
The corner points of the feasible region are:
A: (0, 7)
B: (0, 25/6)
C: (14/3, 4)
D: (10, 5/3)
Evaluate the objective function at each corner point:
z(A) = 4(0) + 4(7) = 28
z(B) = 4(0) + 4(25/6) = 50/3
z(C) = 4(14/3) + 4(4) = 92/3
z(D) = 4(10) + 4(5/3) = 140/3
The maximum value of z is obtained at z(C) = 92/3.
Therefore, the maximum value of z = 4x + 4y, subject to the given constraints, is 92/3, which occurs when x = 14/3 and y = 4.
For more details regarding constraints, visit:
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