If Ricky walks οne lap arοund the perimeter οf the pοnd then he walks 443.14 feet far.
What is feet ?The British imperial and American custοmary systems οf measurement bοth use the unit οf length knοwn as the fοοt (plural: feet), standard symbοl: ft. Usually used as an alternate symbοl is the prime symbοl, One yard is equal tο three feet in bοth custοmary and imperial measures; οne fοοt is equal tο 12 inches.
Thrοughοut histοry, the "fοοt" was a cοmpοnent οf a number οf regiοnal systems οf measurement, including the Greek, Rοman, Chinese, French, and English systems. Frοm οne natiοn tο the next, frοm οne city tο the next, and perhaps frοm οne trade tο the next, it varied in length. Its length was οften divided intο 12 inches οr 16 numbers and ranged frοm 250 tο 335 mm in length, thοugh this was nοt always the case.
Hence, The diameter οf pοnd 141 feet
The radius οf pοnd = 141/2 feet
Area οf pοnd = 2πr
Or, Area οf pοnd = 2×(22/7)×(141/2)
Or, Area οf pοnd = 443.14 feet
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please help if possible
Domain of [tex]f^{-1}[/tex]: R - {-3} or (-∞, -3) ∪ (-3, ∞)
Range of [tex]f^{-1}[/tex]: R - {10} or (-∞, 10) ∪ (10, ∞)
What do you mean by domain and range?The domain of a function is the set of all possible input values (also known as independent variables) for which the function is defined. The range of a function is the set of all possible output values (also known as dependent variables) that the function can produce.
Given the function f, state the domain and range of both f and [tex]f^{-1}[/tex], where
[tex]f(x) =\frac{3x+4}{10-x}[/tex]
Domain and range using inequalities,
Domain of f: R - {10} or (-∞, 10) ∪ (10, ∞)
Range of f: R - {-3} or (-∞, -3) ∪ (-3, ∞)
The inverse does not exist.
[tex]f^{-1} (x) =\frac{10x-4}{x+3}[/tex]
Domain of [tex]f^{-1}[/tex]: R - {-3} or (-∞, -3) ∪ (-3, ∞)
Range of [tex]f^{-1}[/tex]: R - {10} or (-∞, 10) ∪ (10, ∞)
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What is the slope of this line?
Enter your answer as a whole number or a fraction in simplest form in the box.
Answer:
m = - 1
Step-by-step explanation:
Slope = rise/run or (y2 - y1) / (x2 - x1)
Pick 2 points (-2,0) (0, -2)
We see the y decrease by 2, and the x increase by 2, so the slope is
m = -2/2 = -1
So, the slope is -1
Answer:
m = - 1
Step-by-step explanation:
PLEASE HELP ME ON THIS QUICK(Image)
The statement that is correct about the two angles that are said to be linear pair would be = <ABC and <CBD are adjacent angles. That is option A
What is a linear pair of angles?A linear pair exists between angles that are on a straight line that are formed when two lines intersect eachother leading to the formation of the adjacent angles.
The linear angles are there said to be adjacent angles and the sum of these two angles are always = 180°.
Therefore from the given statements, the correct option about the two angles which are linear paired angles is that they are also adjacent to each other.
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Partition the interval into equal parts. Label the fractions at 0 and 1. Plot the given fraction on the number line
The fractions between 0 and 1, we have represented on a number line, include 1/4, 2/4, and 3/4. All of these have equal intervals of 0.25.
To plot 1/4, 2/4 and 3/4 on the number line, we need to divide the interval between 0 and 1 into four equal parts. Each part will represent 1/4.
We can label the fractions at 0 and 1 as shown below:
0 1/4 2/4 3/4 1
Now we can plot the given fractions on the number line as shown below:
0 1/4 2/4 3/4 1
o o o
1/4 2/4 3/4
We can divide the fractions to get the difference of intervals. This can be done as follows:
2/4-1/4 = 0.25, also:
3/4-2/4 = 0.25.
Any portion of a whole number that is stated as one number multiplied by another is referred to as a fraction. One-half of something, for instance, is represented by the fraction 1/2. Another approach to represent a portion of a full number is with a decimal.1/2 is same as 0.25 as a decimal.
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Here is a quadrilateral ABCD. All the measurements are in centimetres. The perimeter of ABCD is 52 centimetres. Work out the length of DC.
The sοlutiοn οf the given prοblem οf quadrilateral cοmes οut tο be DC = 12 .
A quadrilateral is what?In mathematics, a rectangle is a fοur-sided οbject with fοur cοrners. Either "quad" οr "array οf inventive ideas" are Latin terms that were used tο cοin the phrase (meaning "side"). Fοur cοrners, fοur areas, and fοur cοrners are the three cοmpοnents οf a rectangle. Cοnvex and cοncave are the twο primary varieties οf cοnvex and cοncave fοrms. Cοnvex quadrilaterals alsο include the subdivisiοns οf trapezοids, geοmetric shapes, angles, rhοmbuses, and squares.
Here,
Assuming that ABCD is a trapezοid with parallel sides AB and CD, we can use the knοwledge that the perimeter οf a trapezοid is equal tο the sum οf its οppοsite sides tο cοnstruct the fοllοwing equatiοn:
=> AB = 52 + CD = BC + AD
Assuming AB = 13 and BC = 15, the equatiοn can be simplified as fοllοws:
=> 13 + CD + 15 + AD = 52
=> CD + AD = 24
=> DC = 12
Hοwever, we are unable tο determine the length οf DC οr AD withοut additiοnal details regarding the angles οr side lengths οf the trapezοid. As a result, there is nο clear sοlutiοn.
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Complete question:
Lindsay is checking out books at the library, and she is primarily interested in mysteries and nonfiction. She has narrowed her selections down to ten mysteries and twelve nonfiction books. If she randomly chooses four books from her selections, what’s the probability that they will all be nonfiction?
The probability that they will all be non fiction is 0.0667. The solution has been obtained by using hyper-geometric distribution.
What is hyper-geometric distribution?
When sampling from a small population without replacement, the hyper-geometric distribution is a discrete probability distribution that determines the likelihood that an event occurs k times in n trials.
The formula is
P (X = x) = h (x, N, n, k) = [tex]\frac{C_{k,x} * C_{N-k, n-x} }{C_{N,n} }[/tex]
We are given
x = 4
N = 10 + 12 = 22
n = 4
k = 12
Substituting this in the formula, we get
⇒P (X = 4) = h (4, 22, 4, 12) = [tex]\frac{C_{12,4} * C_{22-12, 4-4} }{C_{22,4} }[/tex]
⇒P (X = 4) = h (4, 22, 4, 12) = [tex]\frac{C_{12,4} * C_{10, 0} }{C_{22,4} }[/tex]
⇒P (X = 4) = h (4, 22, 4, 12) = 0.0667
Hence, the probability that they will all be non fiction is 0.0667.
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help pls 1/3(3m+6)-2(2m+6)
1/3(3m+6)-2(3m+6)
Answer:
-5m - 10
Step-by-step explanation:
1/3(3m+6)-2(3m+6)
= 1m + 2 - 6m - 12
= -5m - 10
So, the answer is -5m - 10
Answer:
-5(m+2)
Step-by-step explanation:
1/3(3m+6)-2(3m+6)
combine like terms
-5/3(3m+6)
Distribute
-5m - 10
common factor
-5(m+2)
the patient has formula feedings through a feeding tube. she is given 250 ml every 4 hours (0400, 0800, 1200, 1600, 2000, 2400) around the clock. the nurse gives 100 ml of water after each feeding. what is the patient's intake in the 2300 to 0700
The patient has formula feedings through a feeding tube. She is given 250 ml every 4 hours (0400, 0800, 1200, 1600, 2000, 2400) around the clock. The nurse gives 100 ml of water after each feeding.
The intake of the patient from 2300 to 0700 would be 750 ml.
The formula for calculating the patient's intake from 2300 to 0700
The formula to calculate the patient's intake from 2300 to 0700 is the following:
7 hours after 2300 ⇒ 7+24 - 23 = 8 hours before 0800
Thus, the patient's intake from 2300 to 0800 would be:
From 2300 to 2400 = 250 ml
From 2400 to 0400 = 250 ml
From 0400 to 0800 = 250 ml
From 0800 to 0700 = 100 ml (as it is less than 4 hours)
Total Intake = 250 ml + 250 ml + 250 ml + 100 ml = 850 ml
Intake from 2300 to 0700 = Total intake - Intake from 0800 to 0700
Intake from 0800 to 0700 = 100 ml
Total Intake = 850 ml - 100 ml = 750 ml
Therefore, The patient is receiving formula feedings through a feeding tube, administered every 4 hours (at 0400, 0800, 1200, 1600, 2000, and 2400). Following each feeding, the nurse administers 100 ml of water. It can be calculated that the patient's total intake during the period from 2300 to 0700 would be 750 ml.
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In a standard deck of cards, what is the probability you pick:
• The number 4
The number 7 or a Jack
A red card or a Queen
A Spade or an Ace
• The number 4: Because there are four cards with the number four in a regular deck of 52 cards (4 suits each containing a four), the chance of getting a four is 4/52, which may be reduced to 1/13 or about 0.077.
• The number 7 or a Jack: A regular deck of cards has four sevens and four jacks, for a total of eight cards that are either a seven or a jack. Picking a 7 or a Jack has a chance of 8/52, which may be reduced to 2/13 or roughly 0.154. • A red card or a Queen: A normal deck has 26 red cards (13 diamonds and 13 hearts) and 4 queens. to avoid counting them twice, we must remove the two red queens. As a result, the total number of red or queen cards is 26 + 4 - 2 = 28. Picking a red card or a queen has a chance of 28/52, which may be reduced to 7/13, or roughly 0.538. • A Spade or an Ace: A regular deck of cards has 13 spades and 4 aces, but we must deduct the ace of spades to avoid counting it twice. Therefore there are 13 + 4 - 1 = 16 cards that are either a spade or an ace. As a result, the chance of selecting a spade or an ace is 16/52, which may be reduced to 4/13 or roughly.
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Segment segment x y is dilated through point m with a scale factor of 2. which segment shows the correct result of the dilation? point m is the center of dilation. segment a e has length of 1.5, b f has a length of 2, x y has a length of 3, c g has a length of 5, and d h has a length of 6. ae bf cg dh
The segment cg has a length of 5 before dilation and 10 after dilation, making it the correct result of the dilation.
In mathematics, a dilation is a function f from a metric space M into itself that satisfies the identity d=rd for all points x, y \in M, where d is the distance from x to y and r is some positive real number. In Euclidean space, such a dilation is a similarity of the space.Dilation is a transformation, which is used to resize the object. Dilation is used to make the objects larger or smaller. This transformation produces an image that is the same as the original shape. But there is a difference in the size of the shape.
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Researchers conducted a study to determine the relationship between male pattern baldness and risk of myocardial infarction (MI) in men under the age of 60. Cases were men younger than 60 admitted with an MI to one of five hospitals. Controls were men younger than 60 admitted to these hospitals with a different diagnosis. The physicians who cared for the patients were unaware that this study was being conducted. The nurses collecting the data did not know whether the men were cases or controls and they definitely did not know the purpose of the study. The nurses, however, were poorly trained in the use of the male pattern baldness scale. Based on this description (choose one answer):
a. This study is most prone to differential misclassification of exposure status
b. This study is most prone to nondifferential misclassification of disease status
c. This study is most prone to differential misclassification of disease status
d. This study is most prone to nondifferential misclassification of exposure status
Based on the description, this study is most prone to differential misclassification of exposure status. The correct answer is A.
The description of the study suggests that the nurses who collected the data were poorly trained in the use of the male pattern baldness scale. This means that they may have misclassified some men as having male pattern baldness when they did not actually have it, and vice versa.
This misclassification of exposure status (i.e., male pattern baldness) is more likely to be differential because the nurses were unaware of the purpose of the study and did not know whether the men were cases or controls. This means that their misclassification of exposure status is likely to be influenced by the outcome (i.e., MI).
For example, if the nurses believed that male pattern baldness was a risk factor for MI, they may have been more likely to classify cases as having male pattern baldness and controls as not having it, even if this was not accurate. This would result in differential misclassification of exposure status, which can bias the study results. The correct answer is A.
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Johanna measured the height and petal length of all the flowers in her garden. She plotted her results and found that the relationship between the variables suggests a weak positive linear association with 1 outlier.
The graph that might be made by Johanna is graph (3).
Johanna has two variables: height and petal length. She plotted these variables on a graph to determine the relationship between them.
However, Johanna also found that there is one outlier in her data.
Each point on the graph represents the height and petal length of a single flower in Johanna's garden. The x-axis represents the height of the flower, and the y-axis represents the petal length.
Since there is a weak positive linear association between the two variables, the points on the scatter plot will be generally moving upwards from left to right.
In conclusion, Johanna's graph is most likely a scatter plot that shows the weak positive linear association between the height and petal length of the flowers in her garden, with one outlier that stands out from the rest of the data.
Hence the correct graph is (3).
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Complete Question:
Johanna measured the height and petal length of all the flowers in her garden. She plotted her results and found that the relationship between the variables suggests a weak positive linear association with 1 outlier.
Which of the following might be the graph Johanna made?
Type the correct answer in the box. Use numerals instead of words.
The test to determine the presence of a certain virus in a pigeon is 97% accurate for a pigeon that has the virus and 99% accurate for a pigeon
that does not have the virus. In a given population, 1. 5% of the pigeons are infected.
Do not round your answer.
The probability that a randomly selected pigeon gets an incorrect result is_____
The probability that a pigeon chosen at random will receive an incorrect test outcome is 103/10,000 = 0.0103, or roughly 1.03%.
Let's assume that there are 10,000 pigeons in the population. Then, 1.5% of them, which is 150 pigeons, are infected with the virus.
Out of these 150 infected pigeons, the test will correctly identify 97% of them, which is 145.5 pigeons. The remaining 4.5 infected pigeons will be incorrectly identified as not having the virus.
Out of the 9,850 uninfected pigeons, the test will correctly identify 99% of them, which is 9,751.5 pigeons. The remaining 98.5 pigeons will be incorrectly identified as having the virus.
Therefore, the total number of pigeons that get an incorrect test result is 4.5 + 98.5 = 103.
So, the probability that a randomly selected pigeon gets an incorrect test result is 103/10,000 = 0.0103 or approximately 1.03%
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You might need: Calculator
Alexander uses cupric chloride to etch circuit boards. He recorded the room temperature, in °C, and the
um
etching rate, in of the cupric chloride.
min
After plotting his results, Alexander noticed that the relationship between the two variables was fairly
linear, so he used the data to calculate the following least squares regression equation for predicting the
etching rate from the room temperature:
1
j = 2 + 2
5
What is the residual if the room temperature was 25°C and the cupric chloride had an etching rate of
um
5
?
min
um
min
Show Calculator
The residual is the difference between the observed etching rate and the predicted etching rate
The residual of a room temperature of 25 degrees Celsius is[tex]=\frac{-2um}{min}[/tex]
The least squares regression equation for predicting the etching rate from the room temperature is given as:
[tex]y=2+\frac{1}{5}x[/tex]
When the room temperature is 25 degrees Celsius, the regression equation becomes
[tex]y=2+\frac{1}{5}*25\\\\y=7[/tex]
The residual is calculated as:
Residual = Observed - Predicted
From the question, the observed etching rate is,
[tex]=\frac{-2um}{min}[/tex]
So, the residual equation becomes-
[tex]r=5\frac{um}{min}-7\frac{um}{min}[/tex]
Evaluate the difference
[tex]r=-2\frac{um}{min}[/tex]
The residual of a room temperature of 25 degrees Celsius is[tex]=\frac{-2um}{min}[/tex]
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factoriser au maximum (3x+6)(4x−9)+(3x+6)(7x−3)(3x+6)(4x−9)+(3x+6)(7x−3)
The factorized expression au maximum is:
(3x+6)(84x³ - 189x² + 20x + 69)
To factorize the given expression au maximum, (3x+6)(4x−9)+(3x+6)(7x−3)(3x+6)(4x−9)+(3x+6)(7x−3), follow these steps:
Look for common factors in the terms. Notice that (3x+6) is a common factor in all terms.
Factor out the common factor, (3x+6), from each term:
(3x+6)[(4x−9) + (7x−3)(3x+6)(4x−9) + (7x−3
Now simplify the expression inside the brackets:
(3x+6)[(4x−9) + (21x² - 39x - 18x + 27)(4x−9) + (7x−3)]
Distribute the terms inside the brackets:
(3x+6)[(4x−9) + (84x³ - 153x² - 36x² + 63x - 108x + 81) + (7x−3)]
Combine like terms:
(3x+6)[(4x - 9) + (84x³ - 189x² + 9x + 81) + (7x - 3)]
Add the terms inside the brackets:
(3x+6)[84x³ - 189x² + 20x + 69]
So, the factorised expression au maximum is:
(3x+6)(84x³ - 189x² + 20x + 69)
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A family wants to make an addition to a deck that extends off the back of their home. The current deck is 250 ft2. The addition will be 23.25 ft in length and 14 ft wide. What will be the total area of the deck once the addition is complete? 75.5 ft2 325.5 ft2 575.5 ft2 287.25 ft2
The total area of the deck once will be 575.5 ft²
How to calculate the total area of this deck?Mathematically, the area of a rectangular deck can be calculated by using this mathematical expression:
Area of a rectangular deck, A = LW
Therefore, the area of the additional dimension can be calculated as follows;
Area of an additional dimension, A = 23.25 x 14
Area of an additional dimension, A = 325.5 ft².
Now, we can calculate the total area of this deck as ;
Total area = original area + new area
Total area = 250 ft² + 325.5 ft²
Total area = 575.5 ft²
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minimum amount of dogs that could has a mass more than 26kg
Answer:
Below
Step-by-step explanation:
the last equation shows frequency of 5 for greater than this mass
the equation before it covers 20 to 30 kg frequency 11
so it could be ZERO or it could be ELEVEN are above 26 kg ....you just cannot tell ....so the MINIMUM frequency would be 5 + ZERO = 5
and the MAXIMUM would be 5 + ELEVEN = 16
Find the area under the standard normal curve that lies between z = −2. 30 and z = 1. 8
The area under the standard normal curve between z=-2.30 and z=1.8 by using the cumulative distribution function is 0.957.
Finding the area under the standard normal curve between two points requires the use of the cumulative distribution function. This function is given by the formula:
F(z) = 1/2 (1 + erf(z/√2))
Where erf is the error function.
We will use this formula to find the area under the standard normal curve between z=-2.30 and z=1.8.
The area under the standard normal curve between z=-2.30 and z=1.8 can be found using the following equation:
Area = F(z2) - F(z1)
Where z1 is the lower bound of the area and z2 is the upper bound of the area.
For our example, z1 = -2.30 and z2 = 1.8.
We can plug these values into the equation to find the area under the standard normal curve between z=-2.30 and z=1.8:
Area = F(1.8) - F(-2.30)
We can then use the cumulative distribution function to find the values for F(1.8) and F(-2.30):
F(1.8) = 1/2 (1 + erf(1.8/√2)) = 0.966
F(-2.30) = 1/2 (1 + erf(-2.30/√2)) = 0.009
Substituting these values into the equation:
Area = 0.966 - 0.009
Area = 0.957
Therefore, the area under the standard normal curve between z=-2.30 and z=1.8 is 0.957.
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A group of people were asked if they had run a red light in the last year. 313 responded "yes", and 286 responded "no".
Find the probability that if a person is chosen at random, they have run a red light in the last year.
Give your answer as a fraction or decimal accurate to at least 3 decimal places
Answer:
[tex]0.523[/tex]
Step-by-step explanation:
Total number of respondents = 313 + 286 = 599
Number who responded yes = number running a red light
[tex]\text P(\text {ran a red light)}}\\\\=\dfrac{\text{Number who ran a red light}}{\text{Total number of respondents}}\\\\= \dfrac{313}{599}\\\\= 0.52253[/tex]
Rounded to 3 decimal places the probability is [tex]0.523[/tex]
Question
Solve the system of linear equations by substitution.
y=x−4
y=4x−10
Answer:
x = 2, y = -2
Step-by-step explanation:
To solve by substitution, you first need an isolated variable in one equation. Luckily, y is isolated in both equations.
Because y = y, we can set the other sides of the equations equal to each other.
x - 4 = 4x - 10 [substitute]
The next step is to isolate and solve for x
x = 4x - 10 + 4
x - 4x = -6
-3x = -6
x = 2
Next, you substitute x into both equations.
y = (2) - 4
y = -2
y = 4(2) - 10
y = 8 - 10
y = -2
Because both equations work out to the same value, you have the correct answer.
What is the value of x
According the question given above The value οf x is 42
Define the term triangle?The pοlygοnal shape has three sides and three angles are called as a triangle. A triangle's internal angles are always added up tο 180 degrees.
In right triangles, the trigοnοmetric ratiοs οf sine, cοsine and tangent can be used tο find unknοwn angles and the lengths οf unknοwn sides. The sides οf the triangle are knοwn as fοllοws:
The hypοtenuse is the side οppοsite the right angle, οr defined as the lοngest side οf a right-angled triangle, in this case h.The οppοsite side is the side οppοsite tο the angle we are interested in, in this case a.The adjacent side is the side that is in cοntact with the angle we are interested in and the right angle, hence its name. In this case the adjacent side is b.Given diagram the three angles οf a triangle is given,
sο the sum οf all three angles is equal tο 180 degrees.
Therefοre, (3x - 20) + (x - 1) + (x - 9) = 180°
By simplificatiοn,
5x - 30 = 180°
5x = 180 + 30
5x = 210
x = 42
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Simplifying positive expressions in multiplication.
Simplify
(5z^2x^3)^3
Write your answer without parentheses.
Answer:
5³×z²*³×x³*³
Answer=125z⁶x⁹
I need help studying
Answer: So get your rump in that chair and study your pp off
Step-by-step explanation:
During the National Brushing Day, each student receives 10 pamphlets each on tooth brushing drills. If there were 756 students the school, how many pamphlets were given out?
If there were 756 students in the school and each student receives 10 pamphlets on the tooth brushing drills, the total number of pamphlets given out, using multiplication operation, is 7,560.
What is multiplication operation?Multiplication operation is one of the four basic mathematical operations.
Multiplication operation involves the number being multiplied (the multiplicand), the multiplier or the number multiplying the multiplicand, and the product, which is the result.
The number of pamphlets given to each student on tooth brushing drills = 10
The total number of students at the school.
The total number of pamphlets given out to the students = 7,560 (756 x 10)
Thus, based on multiplication operation, the total pamphlets were 7,560.
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Merisa spent 3/4 of her money on a dictionary. She spent 1/2 of the remainder on a calculator. The dictionary cost $30 more than the calculator. How much did the dictionary cost?
The dictionary cost $36 if Merisa spent 3÷4 of her money on a dictionary. She spent 1÷2 of the remainder on a calculator.
What is Algebraic expression ?
In mathematics, an algebraic expression is a combination of variables, constants, and mathematical operations, such as addition, subtraction, multiplication, and division, that represents a quantity or a relationship between quantities.
Let M be the total amount of money Merisa had before she made any purchases.
Merisa spent 3÷4 of her money on a dictionary, which means she spent 1÷4 of her money on the calculator and other expenses.
So, the amount of money she had left after buying the dictionary is:
M - 3÷4 M = 1÷4 M
She then spent 1÷2 of the remainder on a calculator, so:
(1÷2) (1÷4 M) = 1÷8 M
Let's call the cost of the calculator C.
We know that the dictionary cost $30 more than the calculator, so:
D = C + $30
We can now set up an equation to solve for C:
C + $30 = 3÷4 M
C = 3÷4 M - $30
And we also know that:
C = 1÷8 M
We can substitute the second equation into the first equation and solve for M:
1÷8 M + $30 = 3÷4 M
3÷4 M - 1÷8 M = $30
5÷8 M = $30
M = $48
So, Merisa had $48 before she made any purchases.
We can now use this to find the cost of the dictionary:
D = C + $30
D = (1÷8 M) + $30
D = ($6) + $30
D = $36
Therefore, the dictionary cost $36.
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What is 4.5-4x evaluated
Answer:
Step-by-step explanation:
You pick a card at random. Without putting the first card back, you pick a second card at random.What is the probability of picking a 2 and then picking a number less than 4?
The probability of picking a 2 and then picking a number less than 4will be 0.56%
What is prοbability?By simply dividing the favοrable number οf pοssibilities by the entire number οf pοssible οutcοmes, the prοbability οf an οccurrence can be determined using the prοbability fοrmula. Because the favοrable number οf οutcοmes can never exceed the entire number οf οutcοmes, the chance οf an event οccurring might range frοm 0 tο 1.
The number οf favοurable οutcοmes tο all pοssible οutcοmes οf an event is the ratiο, which is knοwn as prοbability. The number οf gοοd οutcοmes can be represented by the symbοl x fοr an experiment with 'n' number οf οutcοmes. The fοllοwing equatiοn can be used tο determine an event's prοbability.
Yοu pick a card at randοm. Withοut putting the first card back, yοu pick a secοnd card at randοm.
1/13 chance οf picking a 5.
4/51 chance οf picking a 3 οn secοnd draw.
4/(13*51) οf dοing these sequentially = 0.56%
Hence the prοbability will be 0.56%
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a fair die is rolled 14 times. let be the number of faces that appear exactly three times. which of the following expressions appear in the calculation of var(x)
The variance of a fair die that has been rolled 14 times and the number of faces that appear exactly three times is 2.33.
The variance of X can be calculated using the following formula:
Var(X) = Σ(P(X=x)*(x-μ)²)/N
Where Σ represents the summation of all possible outcomes, P(X=x) is the probability of an event occurring, x is the possible outcome, μ is the mean of all possible outcomes, and N is the number of trials.
In this case, the number of possible outcomes of a fair die is 6, and since the die has been rolled 14 times, N = 14. To calculate the mean, we use the formula μ = ΣP(X=x)*x, and in this case, the mean would be 3 since each face appears exactly 3 times.
Therefore, Var(X) = Σ(P(X=x)*(x-3)²)/14. We can then calculate the variance of X by substituting in the probabilities for each face: P(X=1) = 1/14, P(X=2) = 1/14, P(X=3) = 1/14, P(X=4) = 1/14, P(X=5) = 1/14, and P(X=6) = 1/14.
This yields a variance of 2.33, as calculated below:
Var(X) = (1/14 * (1-3)² + 1/14 * (2-3)² + 1/14 * (3-3)² + 1/14 * (4-3)² + 1/14 * (5-3)² + 1/14 * (6-3)²)/14
= 2.33.
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Solve the matrix equation by using inverse matrices.
The Solution of the matrix for (x, y) is (-1, 5) that is: [tex]\left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}-1\\5\end{array}\right][/tex]
Inverse matrices: What are they?A matrix is described by its elements, which are the numbers or symbols that make up the matrix, and its dimensions, which indicate how many rows and columns there are in the matrix. A matrix's inverse is a matrix that yields the identity matrix when multiplied by the original matrix. A matrix's inverse is denoted by and only applies to square matrices (matrices with an equal number of rows and columns).
[tex]\left[\begin{array}{ccc}4&2\\-4&2\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}6\\14\end{array}\right][/tex]
find the inverse matrix for the matrix
[tex]\left[\begin{array}{ccc}4&2\\-4&2\end{array}\right][/tex]
Find the determinant,
determinant = 4 × 2 (-4) × 2 = 16
Inverse matrix is
[tex]1/16 \left[\begin{array}{ccc}4&2\\-4&2\end{array}\right]^T = 1/16 \left[\begin{array}{ccc}2&-2\\4&4\end{array}\right][/tex]
So, the solution of the equation is
[tex]\left[\begin{array}{ccc}x\\y\end{array}\right] = 1/16 \left[\begin{array}{ccc}2&-2\\4&4\end{array}\right] \left[\begin{array}{ccc}6\\14\end{array}\right][/tex]
On solving,
Therefor, The Solution for (x, y) is (-1, 5).
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Solving Ratio Problems (continued)
2. A) How many litres of each colour could Kylie have started with?
b) How many litres of each colour could Kylie have added?
Kylie is mixing paint to get the colour she wants for her apartment.
She mixed 5 parts blue to 3 parts yellow but did not like the result. The colour didn't look yellow enough.
Kylie added a little more blue to the mixture. Then she added 5 times as much yellow as she did blue. The end result had blue and yellow in a ratio of 3: 4. Kylie liked the new colour.
Answer each question. Show your thinking.
List more than one possibility, if there is more than one, for questions 2a) and b).
1. Why is the new colour more yellowy than the first colour?
Kylie changed the ratio of blue to yellow in her paint mixture from 5:3 to 3:4, making the colour more yellowy. She could have started with either 3.5 litres of blue and 2.1 litres of yellow, or 7 litres of blue and 4.2 litres of yellow. She could have added either 0.7 litres of blue and 3.5 litres of yellow, or 1.4 litres of blue and 7 litres of yellow to achieve the new ratio.
The new colour is more yellowy than the first colour because the ratio of blue to yellow has changed. Initially, the ratio was 5:3, or 5 parts blue to 3 parts yellow. After Kylie added more blue and 5 times as much yellow than blue, the ratio changed to 3:4, or 3 parts blue to 4 parts yellow. This means yellow is now the majority colour, which makes the final colour more yellowy than the first.
To calculate the amount of blue and yellow in the new mixture, we can use the equation:
Blue : Yellow = 3 : 4
Therefore, the amount of blue needed is 3/7 of the total, and the amount of yellow is 4/7 of the total.
2a) How many litres of each colour could Kylie have started with?
Possibility 1: Kylie could have started with 3.5 litres of blue and 2.1 litres of yellow (3.5 : 2.1 = 5 : 3).
Possibility 2: Kylie could have started with 7 litres of blue and 4.2 litres of yellow (7 : 4.2 = 5 : 3).
2b) How many litres of each colour could Kylie have added?
Possibility 1: Kylie could have added 0.7 litres of blue and 3.5 litres of yellow (0.7 : 3.5 = 3 : 4).
Possibility 2: Kylie could have added 1.4 litres of blue and 7 litres of yellow (1.4 : 7 = 3 : 4).
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