a) f(x) is a PDF. b) the marginal of X and Y is (y/2 + 1) / 2 c) the covariance of X and Y is: -1/18
What is meant by PDF?
In probability theory, a probability density function (PDF) is a function that describes the relative likelihood for a continuous random variable to take on a given value.
What is covariance?
Covariance is a statistical measure that quantifies the degree to which two random variables are linearly associated.
According to given information:(a) To show that f(x) is a probability density function (PDF), we need to show that it satisfies the following two conditions:
Non-negativity: f(x) is non-negative for all x in its domain.
Normalization: The integral of f(x) over its domain is equal to 1.
The domain of f(x) is given by the inequality 0 ≤ x + 2y ≤ 2. To find the integral of f(x) over its domain, we need to integrate it with respect to y from (0-x/2) to (2-x/2), and then integrate the result with respect to x from 0 to 2:
∫(0 to 2) ∫(0-x/2 to 2-x/2) (x+y) dy dx
Solving the inner integral with respect to y, we get:
∫(0 to 2) [xy + [tex]y^2[/tex]/2] |_0-x/[tex]2^{(2-x/2)[/tex] dx
= ∫(0 to 2) ([tex]x^2[/tex]/4 - [tex]x^3[/tex]/12 + 1) dx
= [[tex]x^3[/tex]/12 - [tex]x^4[/tex]/48 + x] |_[tex]0^2[/tex]
= 2 - 2/3 + 2 = 8/3
Since the integral is finite and positive, the first condition of non-negativity is satisfied. To satisfy the normalization condition, we divide the function by the integral:
f(x) = (x+y) / (8/3)
Therefore, f(x) is a PDF.
(b) To find the marginal of X, we integrate f(x,y) over the range of y:
f(x) = ∫(0-x/2 to 2-x/2) (x+y) / (8/3) dy
= (x/2 + 1) / 2
Similarly, to find the marginal of Y, we integrate f(x,y) over the range of x:
f(y) = ∫(0 to 2) (x+y) / (8/3) dx
= (y/2 + 1) / 2
(c) To find the covariance of X and Y, we use the formula:
Cov(X, Y) = E[XY] - E[X]E[Y]
To find E[XY], we integrate xy*f(x,y) over the range of x and y:
E[XY] = ∫(0 to 2) ∫(0-x/2 to 2-x/2) xy*(x+y)/(8/3) dy dx
= ∫(0 to 2) [[tex]x^3[/tex]/6 - [tex]x^4[/tex]/24 + [tex]x^2[/tex]/4] dx
= 2/3
To find E[X] and E[Y], we integrate xf(x) and yf(y) over their respective ranges:
E[X] = ∫(0 to 2) x*(x/2+1)/2 dx
= 7/3
E[Y] = ∫(0 to 2) y*(y/2+1)/2 dy
= 7/6
Therefore, the covariance of X and Y is:
Cov(X, Y) = E[XY] - E[X]E[Y] = 2/3 - (7/3)*(7/6) = -1/18
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A farmer wants to estimate the number of moles in a local area she catches and tags 30 moles, then releases them back into the local area a few days later the farmer catches another sample of the miles and finds that half of them have a tag estimate the total number of moles in the local area
The estimated total number of moles in the local area is 4.
The farmer needs to use the capture-recapture method to estimate the total number of moles in the local area. This involves tagging a population sample, releasing them back into the population, and then capturing another sample to determine the proportion of tagged individuals in the second sample. Using this proportion, the total population can be estimated using the following formula:
total population = (number in first sample * number in the second sample) / number of tagged individuals in the second sample
In this case, the farmer tagged 30 moles in the first sample and found that half of the second sample had tags. We can use these values to estimate the total number of moles:
total population = (30 * 2) / 15
total population = 4
Note that the actual population size may be larger or smaller than this estimate due to factors such as migration, birth, and death rates.
To correctly estimate the number of moles in a substance, the farmer needs to use the grams to moles formula, which is:
n = m / M
where n is the number of moles, m is the mass of the substance in grams, and M is the molar mass in grams per mole . To calculate the molar mass of a substance, the farmer can sum the molar masses of its component atoms
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The line plot displays the cost of used books in dollars.
A horizontal line starting at 1 with tick marks every one unit up to 9. The line is labeled Cost in Dollars, and the graph is titled Cost of Used Books. There is one dot above 2, 4, 8, and 9.There are two dots above 6 and 7. There are three dots above 3.
Which measure of center is most appropriate to represent the data in the graph, and why?
The mean is the best measure of center because there are no outliers present.
The mean is the best measure of center because there are outliers present.
The median is the best measure of center because there are no outliers present.
The median is the best measure of center because there are outliers present.
Using the median, which is not affected by outliers, would provide a better representation of the typical or central value of the data.
What is median?
Median is a measure of central tendency that represents the middle value in a dataset when the values are arranged in order of magnitude. To find the median, you need to arrange the values in order from smallest to largest and then find the middle value.
Based on the given information, the median is the best measure of center because there are outliers present. The reason for this is because the presence of outliers can greatly affect the mean and make it a less representative measure of central tendency.
In this case, the dots above 6, 7, and 3 may be considered as outliers since they are far from the other data points.
Therefore, using the median, which is not affected by outliers, would provide a better representation of the typical or central value of the data.
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Leonardo da Vinci wrote, "When each of two squares touch the same circle at four points, one is double the other." Explain why the statement is true.
In the right triangle, find the length of the side not given. Give an exact answer and an approximation to three decimal
K
places
a=15, b=15
the exact value of C is
the approximate value of c is
PART A: A can of cat food measures 1" tall and a diameter of 3.5". What is the volume of cat food in the can? To solve Give your answer in cubic inches. Round to the nearest hundredth.
PART B: Cat food is sold by ounces (weight).
If the can holds 5.8 ounces, write a ratio to show cubic inches (your answer from slide 3) to ounces.
The volume of cat food in the can is approximately 9.63 cubic inches.
What is the volume of a cylinder?
The volume of a cylinder can be found using the formula:
V = πr²h
where V is the volume, r is the radius of the circular base, h is the height of the cylinder, and π is the constant pi (approximately equal to 3.14).
To find the volume of the can of cat food, we can use the formula for the volume of a cylinder, which is V = πr²h, where r is the radius (half the diameter) and h is the height.
The radius of the can is 3.5/2 = 1.75 inches, and the height is 1 inch. So, the volume of the can is:
V = π(1.75)²(1)
= 9.62 cubic inches
≈ 9.63 cubic inches (rounded to the nearest hundredth)
Therefore, the volume of cat food in the can is approximately 9.63 cubic inches.
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can someone please help thank you
The probability of choosing a password with no Y's and O's is 20/30 = 2/3, or approximately 0.667. The two probabilities are the same, so neither is greater than the other.
Describe Probability?Probability is a measure of the likelihood or chance that an event will occur. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
In mathematical terms, probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes. For example, if you flip a fair coin, the probability of getting heads is 0.5, because there is one favorable outcome (heads) out of two possible outcomes (heads or tails).
To calculate the probability of choosing a password with no Y's or no O's, we need to first count the number of valid passwords without Y's or O's and then divide it by the total number of possible passwords.
Number of valid passwords without Y's:
There are 5 choices for the first character (B, R, O, G, P), since Y is not allowed.
There are 4 choices for the second character (any of the remaining 5 letters except Y), since we cannot repeat the first letter.
So the total number of valid passwords without Y's is 5 x 4 = 20.
Number of valid passwords without O's:
There are 5 choices for the first character (Y, B, R, G, P), since O is not allowed.
There are 4 choices for the second character (any of the remaining 5 letters except O), since we cannot repeat the first letter.
So the total number of valid passwords without O's is 5 x 4 = 20.
The total number of possible passwords is the number of ways to choose 2 characters out of 6, without repetition. This is given by the formula:
6P2 = 6! / (6 - 2)! = 6 x 5 = 30
Therefore, the probability of choosing a password with no Y's is 20/30 = 2/3, or approximately 0.667.
Similarly, the probability of choosing a password with no O's is also 20/30 = 2/3, or approximately 0.667.
The two probabilities are the same, so neither is greater than the other.
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3. What is the solution to 4(y – 3) + 19 = 8(2y + 3) + 7?
A.
B.
를
-
C. -2
D. 2
Answer:
The answer to your problem is, C. -2
Step-by-step explanation:
To solve : 4(y - 3) +19=8(2y + 3) + 7?
Open the bracket:
4y - 12 + 19 = 16y + 24 + 7
Collect like terms
4y - 16y = 24 + 7 + 12 - 19
-12y = 24
Divide both sides by - 12 to isolate y
-12y / - 12 = 24 / - 12
y = -2
Thus the answer to your problem is, C. -2
For first step you do not really need to “ open the bracket “ just solve it.
Without making any calculations, which distribution of data has the largest standard deviation?
O 1, 1, 1, 1, 4, 4, 7,7,7,7
O 1, 1, 1, 4, 4, 4, 4,7,7,7
O 1, 1, 4, 4, 4, 4, 4, 4, 7, 7
O 1, 4, 4, 4, 4, 4, 4, 4, 4, 7
There are two boxes containing only white and green pens. Box A has 3 green pens and 12 white pens. Box B has 6 green pens and 2 white pens. A pen is randomly chosen from each box. List these events from least likely to most likely. Event 1: choosing a green or white pen from Box B. Event 2: choosing a white pen from Box B. Event 3: choosing a blue pen from Box A. Event 4: choosing a green pen from Box A. Least likely Event, Event, Event, Most likely Event
The events ranked from least likely to most likely are:
Event 3: Choosing a blue pen from Box AEvent 2: Choosing a white pen from Box BEvent 4: Choosing a green pen from Box AEvent 1: Choosing a green or white pen from Box BWhat is probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
To rank the events from least likely to most likely, we need to consider the probability of each event occurring.
Event 3: Choosing a blue pen from Box A
This event is the least likely because there are no blue pens in Box A. The probability of choosing a blue pen is 0.
Event 2: Choosing a white pen from Box B
This event is more likely than Event 3 because there are white pens in Box B. The probability of choosing a white pen from Box B is 2/8 or 1/4.
Event 4: Choosing a green pen from Box A
This event is more likely than Event 2 because there are green pens in Box A. The probability of choosing a green pen from Box A is 3/15 or 1/5.
Event 1: Choosing a green or white pen from Box B
This event is the most likely because it includes both a green and a white pen in Box B. The probability of choosing a green or white pen from Box B is 6/8 or 3/4.
Therefore, the events ranked from least likely to most likely are:
Event 3: Choosing a blue pen from Box A
Event 2: Choosing a white pen from Box B
Event 4: Choosing a green pen from Box A
Event 1: Choosing a green or white pen from Box B
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The ratio of campers to counselors at a summer camp is 8:1,which means…
Answer: There are 8 campers for every 1 counselor.
Step-by-step explanation:
A ratio is relation in number between two things. 8:1 means that there are 8 ___s for every 1 ____.
What is the surface area of a right circular cone with a diameter of 8 m, height of 6.9 m, and a slant height of 18.5 m? Round your answer to the nearest whole.
The surface area of a right circular cone can be found using the formula:
A = πr²+ πrl
where r is the radius of the base, l is the slant height, and π is approximately 3.14159.
In this case, the diameter of the base is 8 m, so the radius is 4 m. The slant height is 18.5 m. The height is not needed for this calculation.
Substituting the given values, we get:
A = π(4 m)² + π(4 m)(18.5 m)
A = π(16 m²) + π(74 m²)
A = π(90 m²)
Using the approximation π ≈ 3.14159, we get:
A ≈ 282.74 m²
Therefore, the surface area of the right circular cone is approximately 283 m² rounded to the nearest whole.
give branliest
you have to split the equation to get the answer for absolute value.
|x-5|=2
Step-by-step explanation:
When x -5 is a positive
x -5 = 2
x = 7
when x -5 is a negative
x-5 = -2
x = 3
I need help with this question it’s algebra 2 .
The true statements are:
C) -5 is a solution to f(x).
E) f(x) = 2x^2 + 9x - 5
F) 0 is a solution to f(x).
I) 5 is a solution to f(x).
K) -3 is a solution to f(x).
L) (-5,0) is an x-intercept of f(x).
How do we calculate?From the given factors of f(x), we know that:
f(x) = 3 * (2x-1) * (x+5)
Using the information, we can calculate the statements are true:
A) 1 is a solution to f(x).
Not necessarily true. We need to substitute x=1 into the expression for f(x) to check if it equals zero.
B) f(x) = 6x^2 - 33x - 15
Not true. The correct expression for f(x) is given above.
C) -5 is a solution to f(x).
True. Substituting x=-5 into f(x) gives us: f(-5) = 3*(-11)*0 = 0.
D) 3 is a solution to f(x).
Not necessarily true. We need to substitute x=3 into the expression for f(x) to check if it equals zero.
E) f(x) = 2x^2 + 9x - 5
True. This is the expanded form of f(x) using the given factors.
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Can someone help me ASAP it’s due tomorrow. I will give brainliest if it’s all done correctly.
Answer all parts!!
Part A: Sample space of randomly selecting 2 lollipops with replacement:
{GG, GC, GL, CG, CC, CL, LG, LC, LL}
Part B: Sample space of randomly selecting 2 lollipops without replacement:
{GC, GL, CG, CL, LG, LC}
What is the experiment about!The experiment of randomly selecting 2 lollipops without replacement shows dependent events because the probability of drawing the second lollipop depends on what the first lollipop was.
For example, if the first lollipop drawn is grape, then the probability of drawing another grape lollipop is decreased because there is only one left in the bag.
The experiment of randomly selecting 2 lollipops with replacement shows independent events because each lollipop can be chosen without affecting the probability of choosing the other lollipops.
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5.66. what is the probability that an irs auditor will catch only 2 income tax returns with illegitimate deduc-tions if she randomly selects 5 returns from among 15 returns, of which 9 contain illegitimate deductions?
The probability that an irs auditor will catch only 2 income tax returns with illegitimate deduc-tions = 0.24
Let us assume that X be the number of returns containing illegitimate deductions in the sample.
Here, X has a hypergeometric distribution.
We need to find the probability that an IRS auditor will catch only 2 income tax returns with illegitimate deduc-tions.
Here, N = 15, r = 9, n = 5
So, P (X = 2) = (⁹C₂ × ⁶C₃) / (¹⁵C₅)
We know that the combination formula:
⁹C₂ = 9! / (2! × 7!)
= 36
⁶C₃ = 6! / (3! × 3!)
= 20
¹⁵C₅ = 15! / (5! 10!)
= 3003
P (X = 2) = (⁹C₂ × ⁶C₃) / (¹⁵C₅)
= (36 × 20) / (3003)
= 0.24
Therefore, the required probability is: 0.24
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Las calificaciones de Emma al finalizar el semestre son las siguientes: 9, 10, 10, 6, 7, 8, 9, 6, ¿cuál es su promedio?
In ΔJKL, the measure of ∠L=90°, KJ = 41, JL = 40, and LK = 9. What ratio represents the cosine of ∠J?
Given:
[tex]\angle\text{L}=90^\circ[/tex]
[tex]\text{KJ}=41[/tex]
[tex]\text{JL}=40[/tex]
[tex]\text{LK}=9[/tex]
To find the cosine of angle J
By using cosine ratio,
[tex]\text{cos J}=\dfrac{\text{adjacent side}}{\text{hypotenuse}}[/tex]
[tex]\text{cos J}=\dfrac{\text{JL}}{\text{JK}}[/tex]
[tex]\text{cos J}=\dfrac{\text{40}}{\text{41}}[/tex]
The ratio of cosine of angle J is 40/41.
HELPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP PLAASE Kiran stakes his kite into the ground. The kite is on a string that is 18 ft long and makes a 30 degree angle with the ground. How high is the kite? Explain or show your thinking by filling in the spots below.
Answer:
blank1: 9
blank2: 30°
blank3: 60°
blank4: 90°
blank5: half
Step-by-step explanation:
The 30°-60°-90° triangle has some great shortcuts associated with it (there are math reasons, trig reasons for these shortcuts)
One of these great shortcuts is that the short leg is half the hypotenuse (or the hypotenuse is double the short leg)
For your question, the kite string, 18ft, is the hypotenuse. The kite height is half that--9ft.
see image.
Factor 25z^2 - 81
ANSWER FAST WILL GIVE BRAINLIEST
Answer: B= (5z+9) (5z-9)
Step-by-step explanation:
Answer:
Step-by-step explanation:
Step 1: Use the sum-product pattern
25z² - 81 =
25z² + 45z - 45z - 81 =
Step 2: Common factor from the two pairs
(25z² + 45z) + (−45z − 81) =
Step 3: Rewrite in Factored Form
5z (5z + 9) − 9 (5z + 9) =
(5 - 9) (5 + 9)
Solution:
B: (5 + 9) (5 - 9)
I really need help with this
The dilated triangle has vertices
P' (-3, 2), Q' (1, 4), and R' (1, -2).How to find the coordinates after dilation'The coordinates of the vertex of the triangle before dillation is given as
P (-6, 4 )
Q (2, 8)
R (2, -4)
To dilate a figure by a scale factor of 1/2, we need to multiply the coordinates of each point by 1/2.
The coordinates of P are (-6, 4). Multiplying by 1/2, we get:
(-6 * 1/2, 4 * 1/2) = (-3, 2)
The coordinates of Q are (2, 8). Multiplying by 1/2, we get:
(2 * 1/2, 8 * 1/2) = (1, 4)
The coordinates of R are (2, -4). Multiplying by 1/2, we get:
(2 * 1/2, -4 * 1/2) = (1, -2)
Therefore, the dilated triangle has vertices P'(-3, 2), Q'(1, 4), and R'(1, -2).
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The Area of a Rectangle is 40. The width is 2 less than 3
times the length. Find the width.
O 10
O 12
O 15
O 14
Based on the above, the width of the rectangle is 10.
What is the width?Let's use "w" to represent the width and "l" to represent the length of the rectangle.
From the problem, we know that the area of the rectangle is 40, so we can write:
Area = length × width
40 = l × w
We also know that the width is 2 less than 3 times the length, so we can write:
width = 3l - 2
Now we can substitute this expression for "w" in the equation for the area: 40 = l × (3l - 2)
Expanding the right side gives: 40 = 3l² - 2l
Rearranging and dividing by 2 gives:
3l²- 2l - 40 = 0
(3l + 10)(l - 4) = 0
Therefore, l = 4 (because the length cannot be negative), and we can find the width using the expression we found earlier:
width = 3l - 2
width = 3(4) - 2
width = 10
So the width of the rectangle is 10. Therefore, the answer is 10.
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6 less than a number raised to the fifth power
Answer:
let n be the number
6 less than the number
= (n-6)^5 = 32
(n-6)^5 = 2^5
since powers are equal we equate the bases
n-6 =2
n=8
What is the probability that Karen's second card will be a black card?
The probability that Karen's second card will be a black card is 3/650.
How to determine probability?After Sandra deals Karen a red seven, there are 51 cards remaining in the deck, of which 26 are black cards. So the probability of Karen drawing a black card on her second draw depends on whether or not the first card she drew was black or red:
If the first card Karen drew was black, then there are 25 black cards left in the deck out of 50 cards, since one card has already been drawn. So the probability of Karen drawing a black card on her second draw is 25/50 = 1/2.
If the first card Karen drew was red, then there are still 26 black cards left in the deck out of 50 cards, since one card has already been drawn. So the probability of Karen drawing a black card on her second draw is 26/50 = 13/25.
Since there are 26 red cards and 26 black cards in a standard deck of playing cards, the probability of Karen being dealt a red seven by Sandra is 2/52 = 1/26.
To find the overall probability of Karen being dealt a red seven and then drawing a black card on her second draw, multiply the probability of these two events occurring:
P(red seven and black card) = P(red seven) × P(black card on second draw | red seven)
P(red seven) = 1/26
P(black card on second draw | red seven) = 13/25
P(red seven and black card) = (1/26) × (13/25) = 13/650
Therefore, the probability of Karen's second card being a black card, given that Sandra dealt her a red seven, is 13/650.
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The complete question:
Sandra and Karen are playing a game of cards with a standard deck of playing cards. Sandra deals Karen a red seven. What is the probability that Karen’s second card will be a black card?
a machine that is programmed to package 1.60 pounds of cereal is being tested for its accuracy in a sample of 40 cereal boxes, the sample mean filling weight is calculated as 1.62 pounds. the population standard deviation is known to be 0.06 pounds. find the 95% confidence interval for the mean.
The 95% confidence interval for the mean is (1.6048, 1.6352).Hence, option (d) is the correct answer.
As given, a machine that is programmed to package 1.60 pounds of cereal is being tested for its accuracy in a sample of 40 cereal boxes, the sample mean filling weight is calculated as 1.62 pounds. The population standard deviation is known to be 0.06 pounds. We are required to find the 95% confidence interval for the mean. Here are the steps to solve this problem:
The formula to find the confidence interval is as follows;
Lower limit = x - zα/2 (σ/√n)
Upper limit = x + zα/2 (σ/√n)
Where,
x= sample mean
zα/2 = z-value of the level of significance
σ = population standard deviation
n = sample size
We are given;
x = 1.62 pounds
σ = 0.06 pounds
n = 40
We need to find the z-value of the level of significance, which can be found using the z-table or by using the calculator.Using the z-table, we get the z-value at 95% confidence interval as zα/2 = 1.96
Substituting the values, we get
Lower limit = 1.62 - 1.96(0.06/√40)
Upper limit = 1.62 + 1.96(0.06/√40)
Lower limit = 1.6048, Upper limit = 1.6352
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Math for School: Practice & Problem Solving (LMS graded)
Challenge Abag contains pennies, nickels, dimes, and quarters. There are 50 coins in all of the coins, 10%
han pennies. There are 2 more nickels than pennies How much money does the bag contain?
The bag contains $4.60 in coins.
What are coins?
Let's use P, N, D, and Q to represent the number of pennies, nickels, dimes, and quarters, respectively, in the bag.
We know that the total number of coins is 50, so:
P + N + D + Q = 50
We also know that 10% of the coins are pennies, so:
P = 0.1(50)
P = 5
There are 2 more nickels than pennies, so:
N = P + 2
N = 5 + 2
N = 7
Now we can use this information to find the number of dimes and quarters. Since there are 50 coins in total, we can substitute the values we have found for P and N into the first equation:
P + N + D + Q = 50
5 + 7 + D + Q = 50
12 + D + Q = 50
D + Q = 38
We also know the values of P and N, so we can find the total value of all the coins in the bag:
Total value = (value of pennies) + (value of nickels) + (value of dimes) + (value of quarters)
Total value = (5 cents x 5) + (7 cents x 5) + (10 cents x D) + (25 cents x Q)
Total value = 25 + 35 + 10D + 25Q
We can substitute D + Q = 38 into this equation to get:
Total value = 25 + 35 + 10(D + Q)
Total value = 60 + 10(38)
Total value = 460 cents, or $4.60
Therefore, the bag contains $4.60 in coins.
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Every year on her birthday, Addie measured her height. On her 5th birthday, she was 44 ⅘ inches tall. Each year, Addie grew ⅖ inch. How tall was Addie on her 12th birthday?
Answer:
Step-by-step explanation:
a₁ = 20, r=-3, n = 6
Oa. 4860
Ob. -4860
Oc. -300
Od. -360
Find An for each geometric sequence
Answer:
-4860
Step-by-step explanation:
The formula for finding the nth term (An) of a geometric sequence is;
An = a₁ * r^(n-1)
where a₁ is the first term, r is the common ratio, and n is the number of terms.
Plugging in the given values, we have;
An = 20 * (-3)^(6-1)
An = 20 * (-3)^5
An = 20 * (-243)
An = -4860
Therefore, the answer is (Ob) -4860
For each of the figures write an absolute value equation that has the following solution set.
To write an absolute value Equation for a given solution set, we need to use the appropriate formula depending on the nature of the solution set.
To write an absolute value equation for a given solution set, we need to first understand what absolute value is. Absolute value is a mathematical function that returns the positive value of a given number, irrespective of its sign. For example, the absolute value of -5 is 5, and the absolute value of 5 is also 5. Now, let's consider the given figures and their respective solution sets:
Figure 1: The solution set for this figure is {2, -2}. To write an absolute value equation for this solution set, we can use the formula |x| = a, where a is the value of the solution set. Therefore, the absolute value equation for this figure is |x| = 2.
Figure 2: The solution set for this figure is {5}. To write an absolute value equation for this solution set, we can use the formula |x - b| = a, where b is the midpoint of the solution set and a is the distance from the midpoint to one of the solutions. Therefore, the absolute value equation for this figure is |x - 5| = 0.
Figure 3: The solution set for this figure is {-3, 3}. To write an absolute value equation for this solution set, we can again use the formula |x| = a, where a is the value of the solution set. Therefore, the absolute value equation for this figure is |x| = 3.
In conclusion, to write an absolute value equation for a given solution set, we need to use the appropriate formula depending on the nature of the solution set.
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the ruler shows a 7 inch segment divided into 2 equal parts.What is the length of one of those part
Answer:
the ruler showed 7 inch segment
Step-by-step explanation:
have an amazing weekend
Answer:
7÷2= 3.5
Step-by-step explanation:
The length of each part is 3.5
At the Golf Classic Open, the professional golfer. Gunter Money hit a chip shot from the rough that just skimmed the top of a 90 foot pine tree and went right into the hole, 220 feft away, on the green for an eagle on the 12th fairway.
Use regression to find an equation for the path of the golf ball?
Use matrices to find the equation for the golf balls path?
The assumptions include that the path of the ball can be modeled by a parabolic function, that data on the trajectory of the ball is needed, and that matrices can be used to solve for the constants a, b, and c. Finally, the inverse of the matrix can be computed using matrix algebra.
What is a matrix?A matrix is a rectangular array of letters, numbers, or expressions in mathematics that are organized in rows and columns. In addition to representing and transforming data in statistics, physics, and computer science, matrices are also used to represent and convert linear equations and transformations in linear algebra.
We need to make some assumptions and approximations in order to develop an equation for the flight of the golf ball because the actual path of the ball is influenced by a number of variables, including wind, temperature, humidity, and spin. The idea that a parabolic function may simulate the ball's trajectory is one that is frequently held.
We would need information on the ball's trajectory, such as height and distance at various points along the trip, in order to perform regression. Regression analysis cannot be used to derive an equation for the trajectory of the golf ball in the absence of this data.
To use matrices, we can set up a system of equations based on the parabolic function:
[tex]y = ax^2 + bx + c[/tex]
where x is the distance traveled by the ball, y is the height of the ball, and a, b, and c are constants to be determined.
We can set up three equations based on the following assumptions:
At the starting point (x=0), the height of the ball is zero.
At the point where the ball hit the tree (x = d), the height of the ball is the height of the tree (90 feet).
At the landing point on the green (x = D), the height of the ball is zero.
Using these assumptions, we can set up the following system of equations:
c = 0 (from assumption 1)
[tex]ad^2 + bd + c[/tex] = 90 (from assumption 2)
[tex]aD^2 + bD + c[/tex] = 0 (from assumption 3)
To solve for the constants a, b, and c, we can use matrix algebra to write the system in matrix form:
[tex][0 0 1] [c] [0][/tex]
c[tex][d^2 d 1] * [b][/tex] = [90]
[tex][D^2 D 1][/tex] [a] [0]
Multiplying the matrices, we get:
[0 0 1] [c] [0]
[tex][d^2 d 1][/tex] * [b] = [90]
[tex][D^2 D 1][/tex] [a] [0]
Simplifying, we get:
[c] [0]
[b] = [90] * [tex][d^2 d 1]^-1 * [d^2 d 1][/tex]
[a] [0]
where the inverse of the matrix [tex][d^2 d 1][/tex] can be computed using matrix algebra. Once we have the values of a, b, and c, we can plug them into the equation [tex]y = ax^2 + bx + c[/tex] to get the equation for the path of the golf ball.
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