a) a = 2.72 makes the statement P(x ≤ a) = 0.24 true.
b)a = 3.77 makes the statement P(x > a) = 0.41 true.
c) a = 3.37 makes the statement P(2.29 ≤ x ≤ a) = 0.36 true.
Part A:
For a uniform probability distribution with range [2,5], the probability density function is:
f(x) = 1/(5-2) = 1/3 for 2 <= x <= 5
a) To find the value of a such that P(x ≤ a) = 0.24, we need to solve the following equation:
∫[2,a] f(x) dx = 0.24
Simplifying the equation, we get:
(a-2)/(5-2) = 0.24
a-2 = 0.72
a = 2.72
Therefore, a = 2.72 makes the statement P(x ≤ a) = 0.24 true.
b) To find the value of a such that P(x > a) = 0.41, we need to solve the following equation:
∫[a,5] f(x) dx = 0.41
Simplifying the equation, we get:
(5-a)/(5-2) = 0.41
5-a = 1.23
a = 3.77
Therefore, a = 3.77 makes the statement P(x > a) = 0.41 true.
c) To find the value of a such that P(2.29 ≤ x ≤ a) = 0.36, we need to solve the following equation:
∫[2.29,a] f(x) dx = 0.36
Simplifying the equation, we get:
(a-2.29)/(5-2) = 0.36
a-2.29 = 1.08
a = 3.37
Therefore, a = 3.37 makes the statement P(2.29 ≤ x ≤ a) = 0.36 true.
Part B:
Let D be the event that a family owns a dog, and let C be the event that a family owns a cat.
Given information:
P(D) = 0.35
P(C | D) = 0.20
P(C) = 0.32
We need to find:
P(D)
P(D | C)
Using Bayes' theorem, we can write:
P(D | C) = P(C | D) * P(D) / P(C)
Substituting the given values, we get:
P(D | C) = (0.20 * 0.35) / 0.32 = 0.21875
Therefore, the conditional probability that a randomly selected family owns a dog given that it owns a cat is 0.21875.
To find P(D), we can use the following formula:
P(D) = P(D | C) * P(C) + P(D | C') * P(C')
where C' represents the complement of C, i.e., the event that a family does not own a cat.
Since the probability of owning a cat or not owning a cat covers all possibilities, we have:
P(D) = P(D | C) * P(C) + P(D | C') * (1 - P(C))
Substituting the given values, we get:
P(D) = (0.20 * 0.35) + P(D | C') * (1 - 0.32)
We do not have enough information to directly calculate P(D | C'), so we need to use the fact that the probabilities must add up to 1. Thus:
P(D | C') = 1 - P(C' | D)
Using Bayes' theorem, we can write:
P(C' | D) = P(D | C') * P(C') / P
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What is the awnseress
Answer:
<
Step-by-step explanation:
because 42.18 is smaller than 42.7
Plot Points & Graph Exponential Function
Whats the y=2^x+1 -8?
The graph of y = 2ˣ +1 -8 is the curve of the exponential function y = 2ˣ shifted downward by 8 units and shifted to the left by 1 unit along the x-axis.
The exponential function y = 2ˣ represents a curve that grows rapidly as x increases.
The graph of y = 2ˣ +1 -8 is the same curve shifted downward by 8 units and shifted to the left by 1 unit along the x-axis.
To plot some points for the graph, choose a few x-values and then calculate the corresponding y-values using the function:
When x = -2, y = 2⁻¹ - 8 = -8.5
When x = -1, y = 2⁰ - 8 = -7
When x = 0, y = 2¹ - 8 = -6
When x = 1, y = 2² - 8 = -4
When x = 2, y = 2³ - 8 = 0
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What is the value of x?
The volume of the right cone below is 5677 units³. Find the value of x.
Answer:
21 units
Step-by-step explanation:
You want the height (x) of a cone with base radius 9 units and volume 567π cubic units.
VolumeThe volume of a cone is given by the formula ...
V = 1/3πr²h
Solving for the height, and using x for the height, we find ...
x = 3V/(πr²) = 3(567π units³)/(π(9 units)²) = 21 units
The height of the cone is 21 units.
An amount of 22,000 is borrowed for 5 years at 3.5% interest, compounded annually. If the loan is paid in full at the end of that period, how much must be paid back? Use the calculator provided and round your answer to the nearest dollar.
The requried, if the loan is paid in full at the end of the 5-year period, the borrower would have to pay back $26,130 (rounded to the nearest dollar).
The formula for the future value of an investment with compound interest is:
[tex]FV = PV * (1 + r)^n[/tex]
Where PV is the present value (the amount borrowed), r is the annual interest rate (as a decimal), and n is the number of compounding periods. In this case, since the interest is compounded annually for 5 years, n = 5.
Substitute in the values given, we get:
[tex]FV = 22,000 * (1 + 0.035)^5[/tex]
FV = 22,000 * 1.19416
FV = 26,129.09 = 26,130
Therefore, if the loan is paid in full at the end of the 5-year period, the borrower would have to pay back $26,130 (rounded to the nearest dollar).
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The volume of a cylinder is 225T cm³ and its height is 9 cm.
What is the length of the cylinder's radius?
o 7 cm
o 10 cm
o 5 cm
9 cm
Answer:
The answer for radius is 5cm
Step-by-step explanation:
Volume of Cylinder =pir²h
225pi=pir²×9
225=9r²
divide both sides by 9
9r²/9=225/9
r²=25
√r²=√25
r=5cm
Will mark as brainliest!!! Due is an hour
Using the graph provided, the value f (-1) is -3; option B.
Using the graph provided, the value of f (-4) is -4; option C.
Using the graph provided, the value of x is when f(x)=3 is 2; option C.
What are functions?A function from one set X to another allocates exactly one element of the other set Y to each element of X.
The set X is known as the function's domain, while the set Y is known as the function's codomain.
The 4 categories of functions are:
Constant Function: a polynomial function of degree zeroLinear function - a polynomial function of degree oneQuadratic function - a polynomial function of degree twoCubic function - a polynomial function of degree threeLearn more about functions at: https://brainly.com/question/10439235
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You are on a Ferris wheel that has a radius of 80 feet and the bottom of the wheel is 3 feet above the ground. The Ferris wheel starts when you get on at the bottom and rotates counter- clockwise and has a period of 2 minutes. Create a parametric function to model your location on the Ferris wheel at a given time.
The parametric function for your location on the Ferris wheel at any given time:
x(t) = 80 * sin(2π * (t/2))
y(t) = 80 * cos(2π * (t/2)) + 3
To create a parametric function that models your location on the Ferris wheel at a given time, we need to come up with equations that describe your horizontal and vertical positions as functions of time.
Let's start with the horizontal position. Since the Ferris wheel rotates counter-clockwise, we know that your position on the wheel will increase as time goes on. We can express this as:
x(t) = 80cos(2πt/120)
Here, t represents time in seconds, and the factor of 2π/120 ensures that the function completes one full cycle (i.e. one trip around the wheel) in 120 seconds, or 2 minutes. The cosine function gives us a smooth, periodic curve that oscillates between -80 and 80, corresponding to your position on either side of the wheel's center.
Next, let's consider your vertical position. We know that you start at a height of 3 feet above the ground, and as the wheel rotates, your height will vary sinusoidally over time. We can express this as:
y(t) = 80sin(2πt/120) + 3
Here, the sine function gives us a smooth, periodic curve that oscillates between 77 and 83 feet (i.e. the radius of the wheel plus or minus 3 feet).
So, putting it all together, our parametric function for your location on the Ferris wheel at a given time t is:
(r(t), θ(t)) = (80cos(2πt/120), 80sin(2πt/120) + 3)
Here, r(t) and θ(t) represent your radial distance from the center of the wheel and the angle you've rotated from the starting position, respectively. This parametric function describes a smooth, periodic curve that traces out your path on the Ferris wheel as it rotates counter-clockwise.
Given the information, we know the Ferris wheel has a radius of 80 feet, the bottom is 3 feet above the ground, it rotates counter-clockwise, and has a period of 2 minutes.
To create a parametric function, we need two equations, one for the x-coordinate (horizontal) and one for the y-coordinate (vertical). Let's denote the time variable as t, measured in minutes.
1. X-coordinate (horizontal position):
Since the Ferris wheel rotates counter-clockwise, we can use the following equation for the x-coordinate:
x(t) = 80 * sin(2π * (t/2))
2. Y-coordinate (vertical position):
To account for the bottom of the Ferris wheel being 3 feet above the ground, we need to add 3 to the vertical equation:
y(t) = 80 * cos(2π * (t/2)) + 3
Now, we have the parametric function for your location on the Ferris wheel at any given time:
x(t) = 80 * sin(2π * (t/2))
y(t) = 80 * cos(2π * (t/2)) + 3
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(Chapter 13) If |r(t)| = 1 for all t, then r'(t) is orthogonal to r(t) for all t.
The statement is true. This means that r'(t) is orthogonal (perpendicular) to r(t) for all t.
If |r(t)| = 1 for all t, then r(t) is a unit vector for all t. Differentiating both sides of this equation with respect to t, we get:
|r(t)|' = 0
Using the chain rule and the fact that the magnitude of a vector is the square root of the dot product of the vector with itself, we have:
|r(t)|' = (r(t) · √r(t))
= (2r(t) · r'(t)) / (2|r(t)|)
= r(t) · r'(t) / |r(t)|
= r(t) · r'(t)
Since |r(t)|' = 0, we have:
r(t) · r'(t) = 0
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Find the standard deviation, o, of the data. 3, 4, 5, 7, 10, 12, 15 x = 8 Variance (2) = 17.1 o = [?] Round to the nearest tenth. Standard Deviation Enter
The standard deviation of the data-set is given as follows:
s = 4.14.
How to calculate the mean and the standard deviation of the data-set?The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.
The data-set in this problem is given as follows:
3, 4, 5, 7, 10, 12, 15.
Hence the mean is:
Mean = (3 + 4 + 5 + 7 + 10 + 12 + 15)/7
Mean = 8.
The sum of the differences squared for the data-set is given as follows:
SS = (3 - 8)² + (4 - 8)² + (5 - 8)² + (7 - 8)² + (10 - 8)² + (12 - 8)² + (15 - 8)²
SS = 120.
The standard deviation is given by the square root of the sum of the differences squared divided by the cardinality, hence:
s = sqrt(120/7)
s = 4.14.
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a fertilizer mixture contains 2 parts nitrogen,3 parts potash and 2 parts phosphate by mass. How many kilogrammes of potash are in a bag of fertilizer that weighs 49 kilogrammes
There are 21 kilograms of potash in a bag of fertilizer that weighs 49 kilograms.
What is the mass of the potash?The total number of parts of the fertilizer is calculated as;
total parts = is 2+3+2
total parts = 7
The mass of potash in a bag of fertilizer is calculated by using the fraction of the mixture that is potash, and multiply it by the total weight of the bag.
The fraction of the mixture that is potash =3/7.
The mass is calculated as follows;
mass = (3/7) x 49 kg
mass = 21 kg
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Please help! 40 points! Convert the following function from vertex form to standard form. Show your work.
f(x) = 3(x — 8)^2 — 160
The required standard form of the function is f(x) = 3x² - 48x + 32.
The vertex form of a quadratic function is:
f(x) = a(x - h)² + k
Where (h, k) is the vertex of the parabola and a is a coefficient that determines the shape of the parabola.
In the given function, we can see that the vertex is at (8, -160) and a is 3. So we have:
f(x) = 3(x - 8)² - 160 (vertex form)
Expanding the square, we get:
f(x) = 3(x - 8)(x - 8) - 160
f(x) = 3(x² - 16x + 64) - 160
Distributing the 3, we get:
f(x) = 3x² - 48x + 192 - 160
Simplifying, we get:
f(x) = 3x² - 48x + 32
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TRUE OR FALSE the critical value of a hypothesis test is based on the researcher's selected level of significance.
Answer:
the answer is true. may I get brainliest
TRUE. The critical value of a hypothesis test is based on the researcher's selected level of significance.
The critical value of a hypothesis test is based on the researcher's selected level of significance. The level of significance represents the probability of rejecting a true null hypothesis, and it determines the critical value, which is the point beyond which we reject the null hypothesis. Therefore, the higher the level of significance, the lower the critical value, and the easier it is to reject the null hypothesis.
True, the critical value of a hypothesis test is based on the researcher's selected level of significance. The level of significance determines the threshold for rejecting or failing to reject the null hypothesis, and the critical value is a point on the test statistic's distribution that corresponds to this threshold.
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m(nx^2-y)/z=5n m=6 x=-2 y=-3 z=-5 find n
0.3673 is the value of the variable n.
The given expression is [tex]\frac{m(nx^2-y)}{z}=5n[/tex]
First, let's plug the given values into the equation:
[tex]\frac{6(n2^2-(-3))}{-5}=5n[/tex]
Simplifying:
[tex]\frac{6(n4+3)}{-5}=5n\\6(4n+3)=-25n\\24n+25n=18\\n=18/49[/tex]
n = 0.3673
Therefore, n is approximately equal to 0.3673.
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The table below gives the Uk population in
percentage according to a census.
England-83%
Wales-5%
Scotland-9%
N Ireland-3%
(a) Represent this information by a pie
chart.
b) Find the population of England if the
total population of the UK is 60 m.
The information as shown in a pie chart is found in the attachment.
The population of England is 49.8 million.
What is the population of England?The population of England is calculated as a percentage of the total population of the United Kingdom.
The data table given shows that England has 83% of the UK population and the total population of the UK is 60 million.
Hence, the population of England is calculated as follows:
The population of England = 83% of 60 million
The population of England = (83/100) x 60,000,000
The population of England = 49,800,000
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-1/6 divided by -7/4 times 0.5
Answer:
[tex]\frac{1}{21}[/tex]
Step-by-step explanation:
To start, let's divide [tex]-\frac{1}{6}[/tex] by [tex]-\frac{7}{4}[/tex]. To do this, we're going to have to flip around the second number and multiply it by the first one (this is how to divide fractions).
[tex]-\frac{1}{6}[/tex] × [tex]-\frac{4}{7} = \frac{4}{42} =\frac{2}{21}[/tex]
Now, it's time for us to multiply that result by 0.5. As you may already know, 0.5 is equivalent to [tex]\frac{1}{2}[/tex], so let's use that value instead.
[tex]\frac{2}{21}[/tex] × [tex]\frac{1}{2} = \frac{2}{42} = \frac{1}{21}[/tex]
So, our final answer is [tex]\frac{1}{21}[/tex]. Let me know if you need more help!
Answer:0.00298
Step-by-step explan
Solve the equation 4x² + 4x - 9 = 0 and round
the roots to the nearest hundredth.
Answer:
x = 1.08 and x = -2.08
Step-by-step explanation:
We can solve this equation using the quadratic formula.
In order to understand the quadratic formula, we must first realize that the equation is currently in standard form and the general formula for the standard form of a quadratic equation is[tex]ax^2+bx+c=0[/tex]
We must also remember that the quadratic formula can have two solutions since you can have a positive square (e.g., 4 * 4 = 16) and a negative square (e.g., -4 * -4 = 16)Thus, in the equation given, 4 is our a value, 4 is (also) our b value and -9 is our c value.
The formula for the positive solution of the quadratic formula is
[tex]x=\frac{-b+\sqrt{b^2-4ac} }{2a}[/tex]
The formula for the negative solution of the quadratic formula is
[tex]x=\frac{-b-\sqrt{b^2-4ac} }{2a}[/tex]
Positive solution:
[tex]x=\frac{-4+\sqrt{4^2-4(4)(-9)} }{2(4)}\\ \\x=\frac{-4+\sqrt{160} }{8}\\ \\x=1.08113883\\\\x=1.08[/tex]
Negative solution:
[tex]x=\frac{-4-\sqrt{4^2-4(4)(-9)} }{2(4)}\\ \\x=\frac{-4-\sqrt{160} }{8}\\ \\x=-2.08113883\\\\x=-2.08[/tex]
If A=QR, where Q has orthonormal columns, what is the relationship between R and QT?
The upper triangular matrix R is invariant under multiplication by the transpose of Q. This relationship is sometimes referred to as the "QR factorization identity".
If A=QR, where Q is an n×n matrix with orthonormal columns and R is an n×n upper triangular matrix, then we can express A as:
A = QR = Q(QT)R
Since Q has orthonormal columns, its transpose QT is its inverse. Therefore:
Q(QT)R = I_n R = R
where I_n is the n×n identity matrix. So we can see that R is equal to Q(QT)R, which is the product of Q and the transpose of Q. This product is equal to the identity matrix times R, so we can say that:
R = QT R
In other words, the upper triangular matrix R is invariant under multiplication by the transpose of Q. This relationship is sometimes referred to as the "QR factorization identity".
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Surface area and volume please I need help
The Volume of the solid = 601.83 m³
The Surface area of the solid = 266.9 m²
How to Find the Surface Area and Volume?The solid is composed of a cylinder and a cone, therefore, we would need the following formulas:
Volume of cone = 1/3 * πr²h
Volume of Cylinder = πr²h
Surface area of cone = πrl
Surface area of cylinder = 2πr(h + r)
Volume of the solid = 1/3 * πr²h + πr²h
radius (r) = 5 m
height of cone (h) = 5 m
Height of cylinder (h) = 6 m
π = 3.14
Plug in the values:
Volume of the solid = 1/3 * 3.14 * 5² * 5 + 3.14 * 5² * 6
Volume of the solid = 130.83 + 471 = 601.83 m³
Surface area of the solid = πrl + 2πr(h + r) - 2(πr²)
r = 5 m
l = 5m
π = 3.14
h = 6 m
Plug in the values:
Surface area of the solid = 3.14 * 5 * 5 + 2 * 3.14 * 5(6 + 5) - 2(3.14 * 5²)
= 78.5 + 345.4 - 157
= 266.9 m²
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I need this quick i only have 2 minutes
Answer:
[tex]2 \frac{1}{3} + 1 \frac{3}{4} [/tex]
[tex]2 \frac{4}{12} + 1 \frac{9}{12} = 3\frac{13}{12} = 4 \frac{1}{12} [/tex]
The correct answer is A.
An oatmeal bar in the shape of a rectangular prism has a base area of 4 square inches and a height of 4 inch.
What is the volume of the oatmeal bar?
OA. 9 3/4 cubic inches
OB. 9 3/16 cubic inches
OC. 6 12/16 cubic inches
OD. 6 15/16 cubic inches
The oatmeal bar has a volume of 16 cubic inches.
What is the volume of the oatmeal bar?A rectangular prism is simply a three-dimensional solid shape which has six faces that are rectangles.
The volume of a rectangular prism is expressed as;
V = w × h × l
V = base area × height
Where w is the width, h is height and l is length
To find the volume of the rectangular prism oatmeal bar, we need to multiply the base area by the height.
So, the volume of the oatmeal bar can be calculated as:
Volume = Base Area × Height
Volume = 4 in² × 4 in
Volume = 16 in³
Therefore, the volume of 16 cubic inches.
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Which description best fits the distribution of the data shown in the histogram?
A. uniform
B. skewed right
C. approximately bell-shaped
D. skewed left
Answer:
We know that, if in a distribution one tail is longer than the other, the distribution is skewed.
Here in the given histogram, it has a long right tail which is in the positive direction on the x axis. So it is right skewed distribution. That is also known as positive skewed distribution.
Step-by-step explanation:
So here we have got the required answer. Option C is correct.
What is the most specific classification of 9/3
The most specific classification of the fraction 9/3 is: Rational Numbers
How to identify rational numbers?A rational number is defined as a number that is expressed as the ratio of two integers, where the denominator should not be equal to zero,
The fraction we are given as: 9/3
Now, we know that 9 and 3 are classified as integers and as such we can easily classify them as rational numbers.
Thus, the best specific classification of 9/3 is really rational numbers.
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the standard deviation of the sampling distribution is called the standard error of the median. true false
The standard deviation of the sampling distribution is called the standard error of the mean, not the median. The standard error of the median is a different measure that is used to estimate the variability of the median of a sample.
When we take a sample from a population, the values of the sample statistics, such as the mean, median, standard deviation, etc., will vary from sample to sample. The distribution of the sample statistics is called the sampling distribution.
The standard deviation of the sampling distribution of the mean is called the standard error of the mean. It is a measure of the variability of the sample means from sample to sample.
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your favorite cereal has a little prize in each box. there are 5 such prizes. each box is equally likely to contain any one of the prizes. so far, you have been able to collect 2 of the prizes. what is the probability that you will get the third different prize in the next box you buy?
The probability of getting a different prize in the next box you buy is: P(A) = 3 / 5 = 0.6 or 60%.
The probability of getting a different prize in the next box you buy depends on the number of prizes left and the total number of prizes.
Since there are 5 prizes in total and you have already collected 2, there are 3 remaining prizes. The probability of getting a different prize in the next box you buy is 3 out of 5, or 60%. This is because each box is equally likely to contain any one of the prizes, and you have already collected 2 out of the 5 prizes.
To calculate the probability, you can use the formula: P(A) = number of favorable outcomes / total number of possible outcomes In this case, the number of favorable outcomes is 3 (the number of remaining prizes) and the total number of possible outcomes is 5 (the total number of prizes). Therefore, the probability of getting a different prize in the next box you buy is: P(A) = 3 / 5 = 0.6 or 60%
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e ohio lottery has a game called pick 4 where a player pays $1 and picks a four-digit number. if the four numbers come up in the order you picked, then you win $3900. a) write the probability distribution for a player's winnings. fill in the table below. for the computer to grade this one correctly make sure that your x values are from smallest to largest.
The probability of winning $3,899 is 0.0001, which is a very small probability, but still possible.
To write the probability distribution for a player's winnings in the Pick 4 game, we need to consider all the possible outcomes and their probabilities.
There are a total of 10,000 possible four-digit numbers that can be drawn in the game. Since the player has to match the numbers in the exact order, there is only one winning combination for each four-digit number. Therefore, the probability of winning is 1/10,000.
To calculate the player's winnings, we need to subtract the $1 cost of playing from the $3,900 prize. Thus, the player's net winnings can be calculated as follows:
Net Winnings = $3,900 - $1 = $3,899
The probability distribution for the player's winnings can be summarized in the following table:
| Winnings (x) | Probability (P) |
|--------------|-----------------|
| $0 | 0.9999 |
| $3,899 | 0.0001 |
Note that the table shows the possible winnings (x) in ascending order, as requested. The probability of winning $0 is 0.9999, which means that the player is most likely to lose their $1 bet.
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show that the following number is rational by writing it as a ratio of two integers. 52.470817081708
Answer: To show that the number 52.470817081708 is rational, we need to express it as a ratio of two integers.
Let x = 52.470817081708. We can write x as a sum of its integer part and fractional part:
x = 52 + 0.470817081708
To convert the fractional part to a fraction, we can multiply both numerator and denominator by a power of 10 that will eliminate the decimal point. In this case, we can multiply by 10^12:
0.470817081708 = 470817081708 / 10^12
Therefore, we can write:
x = 52 + 470817081708 / 10^12
We can simplify the fraction by dividing both numerator and denominator by their greatest common divisor:
x = 52 + 163717 / 3125000
So we have expressed x as a ratio of two integers, 163717 and 3125000. Therefore, 52.470817081708 is a rational number.
The number 52.470817081708 is rational because it can be expressed as a ratio of two integers: 52470817081708
To show that 52.470817081708 is rational, we need to write it as a ratio of two integers.
First, we can see that the number has a repeating pattern of digits after the decimal point, which tells us that it can be expressed as a fraction. To do this, we'll count the number of decimal places in the repeating pattern, which is 12 in this case.
Next, we'll use the following formula to convert the repeating decimal to a fraction:
x = a + b/(10^n - 1)
Where:
- x is the repeating decimal
- a is the non-repeating part of the decimal (in this case, it's just the whole number 52)
- b is the repeating part of the decimal (in this case, it's the digits 470817081708)
- n is the number of digits in the repeating pattern (in this case, it's 12)
So plugging in our values, we get:
52.470817081708 = 52 + 470817081708/(10^12 - 1)
Simplifying the denominator, we get:
52.470817081708 = 52 + 470817081708/999999999999
To write this as a ratio of two integers, we'll simplify the fraction:
52.470817081708 = 52 + 235408540854/499999999999
So the number 52.470817081708 can be written as the fraction 52 + 235408540854/499999999999, which is a ratio of two integers. Therefore, we have shown that 52.470817081708 is rational.
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Using set notation, write out the sample space for each of the following random experiments.(a) A coin is tossed three times in a row. The observation is how the coin lands ( or ) on each toss.(b) A basketball player shoots three consecutive free throws. The observation is the result of each free throw for success, for failure).(c) A coin is tossed three times in a row. The observation is the number of times the coin comes up tails.(d) A basketball player shoots three consecutive free throws. The observation is the number of successes.
A coin is tossed three times in a row. The observation is how the coin lands (H for heads or T for tails) on each toss. The sample space, using set notation, is:
S = {(H, H, H), (H, H, T), (H, T, H), (H, T, T), (T, H, H), (T, H, T), (T, T, H), (T, T, T)}
A basketball player shoots three consecutive free throws. The observation is the result of each free throw (S for success, F for failure). The sample space, using set notation, is:
S = {(S, S, S), (S, S, F), (S, F, S), (S, F, F), (F, S, S), (F, S, F), (F, F, S), (F, F, F)}
A coin is tossed three times in a row. The observation is the number of times the coin comes up tails. The sample space, using set notation, is:
S = {0, 1, 2, 3}
A basketball player shoots three consecutive free throws. The observation is the number of successes. The sample space, using set notation, is:
S = {0, 1, 2, 3}
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T/F : The cofactor expansion of det A along the first row of A is equal to the cofactor expansion of det A along any other row
True. The cofactor expansion of the determinant of a matrix A along any row or column will yield the same result.
The cofactor expansion of the determinant of a matrix A along a row or a column is given by the formula:
```
det(A) = a1j * C1j + a2j * C2j + ... + anj * Cnj
```
where `aij` is the element in the ith row and jth column of A, and `Cij` is the (i,j)-cofactor of A.
The (i,j)-cofactor of A is defined as `(-1)^(i+j) * Mij`, where `Mij` is the determinant of the (n-1) by (n-1) matrix obtained by deleting the ith row and jth column of A.
To see why the cofactor expansion is independent of the row or column chosen, consider the formula for the determinant of a matrix obtained by transposing A:
```
det(A^T) = det([a11, a21, ..., an1],
[a12, a22, ..., an2],
...,
[a1n, a2n, ..., ann])
```
By the cofactor expansion along the first row of A^T, we have:
```
det(A^T) = a11 * C11' + a12 * C12' + ... + a1n * C1n'
```
where `Cij'` is the (i,j)-cofactor of A^T.
Now note that `Cij' = (-1)^(i+j) * Mji`, where `Mji` is the determinant of the (n-1) by (n-1) matrix obtained by deleting the jth row and ith column of A. But this is precisely the (j,i)-cofactor of A. Therefore, we have:
```
det(A^T) = a11 * C11 + a21 * C21 + ... + an1 * Cn1
```
which is the cofactor expansion of det A along the first column of A. Since the transpose of a matrix has the same determinant as the original matrix, we conclude that the cofactor expansion of det A along any row is equal to the cofactor expansion along any other row.
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in right triangle ABC, m
Answer:
In right triangle ABC, right angled at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM.