The formula for the standard matrix for the linear transformation is as follows: beginbmatrix cosfracpi2 & -sinfracpi2 & 0 sinfracpi2 & & cosfracpi2 & 0 & & -1 endbmatrix
The following linear transformation rotates the points by[tex]$fracpi$[/tex]radians in the direction of rotation before reflecting the points through the line. We begin by examining the rotational component of the transformation to determine the standard matrix. A matrix beginbmatrix cosfracpi2 & -sinfracpi2 & 0 sinfracpi2 & cosfracpi2 & 0 0 & 0 & 1 endbmatrixan be used to describe the rotation by radians. We can use the matrix beginbmatrix 1 & 0 & 0 0 & -1 & 0 0 & 0 & 1 endbmatrixfor the reflection component. We obtain the final standard matrix for the linear transformation by multiplying these two matrices. $$beginbmatrix cosfracpi2 & -sinfracpi2 & 0 sinfracpi2 & cosfracpi2 & 0 0.
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The complete question is
let be the linear transformation that first rotates points clockwise through ( radians) and then reflects points through the line . find the standard matrix for the standard matrix of the transformation can be found by first computing the matrix of the rotation and then the matrix of the reflection.
The measure of an interior angle of a regular polygon is 162º. Find the number of sides in the polygon.
_______sides
Answer:
20 sides
Step-by-step explanation:
The number of sides of a regular polygon, if each of its interior angles is 162° is 20.
1. Is G (cost) a function of t (time)? Explain your reasoning
2. Is t (time) a function of G (cost)? Explain your reasoning.
{G} is the function of {t} and vice - versa is also true.
What are algebraic expressions?In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations and other aspects of logical syntax.Given is a graph as shown in the image.
{ 1 } -
Yes, the given function G{t} is the function of {t}. It is a piecewise function.
{2} -
Let the given function is -
y = G{t}
We can find the inverse as -
t = G⁻¹{y}
Yes, {t] is also the function of {G}.
Therefore, it can be concluded that {G} is the function of {t} and vice - versa is also true.
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What is the measure of AC
Side AC of the similar triangle system has a length of 6 units. (Correct choice: C)
How to determine the missing length in a system of two similar right triangles
Two triangles are similar when both have congruent angles but sides are not congruent though proportional. In this problem we find the case of two similar right triangles, whose relationship is shown below:
AD / AC = CD / BC = AC / AB
If we know that AD = 4 and AB = 9, then the length of side AC is:
4 / AC = AC / 9
AC ² = 36
AC = 6
The length of side AC is equal to 6 units.
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Which calculation shows the best method for estimating the result of 384 divided by three-sixteenths?
400 divided by one-fourth equals 1,600
400 divided by one-fourth equals 100
300 divided by one-third equals 900
300 divided by one-third equals 100
Answer:
Step-by-step explanation:
Maria will spend less than $29 on gifts. So far, she has spent $17. What are the possible additional amounts she will spend?
Use c for the additional amount (in dollars) Maria will spend.
Write your answer as an inequality solved for c.
Answer:
Maria is able to spend less than $12.
C < 12
Step-by-step explanation:
If Maria has already spent 17 dollars, that means she has 12 dollars left. C cannot be greater than 12. It has to be less.
So think of this C < $12
So if we do the total on one side and the $17 and C on the other side of inequality we get
C + 17 < 29
It cannot be equal to 29 so that's why we don't use a less than or equal to symbol.
Simplify this to get C < 12
I hope this is able to help you :D
 - Tanner's beaker had 45 milliliters of water in it at first. After each of his friends poured in 8 milliliters, the
beaker contained 93 milliliters. How many friends poured water into Tanner's beaker?
Answer:
x=6
Step-by-step explanation:
hope this helps you with your question
The table shows the balance of Rob’s checking account at the end of the day. This is $95.50 less than the amount he had the beginning of the day.
Which of the following equations can be used to determine Rob’s account balance at the beginning of the day?
Order the steps to solve the equation
log(x2-15) = log(2x) form 1 to 5.
x²-2x-15=0
Potential solutions are -3 and 5
x²-15 = 2x
x-5=0 orx+3=0
(x - 5)(x + 3) = 0
A logarithm is an operation used to determine the exponent of a given base number.
What is logarithm?
A logarithm is a mathematical function that describes the relationship between the power to which a number (called the base) must be raised, and the resulting value. The logarithm of a number is the power to which the base must be raised to produce that number.
Start by moving the logarithm to the left-hand side of the equation: log(x²-15) = log(2x)
Next, use the property of logarithms that log(a/b) = log(a) - log(b) to get x²-15 = 2x
then you can simplify the equation by subtracting 2x from both sides x²-2x-15=0
then factor the equation to get (x - 5)(x + 3) = 0
Finally, use the Zero Product Property to solve for x. The solutions will be x = 5 or x = -3
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Ryan claims that the cost of Plan A and Plan B differ a lot.
Which statement best explains why Ryan was misled by the graph?
a.) The two graphs use different vertical scales.
b.) The scale of intervals on the horizontal axis of both graphs
is inconsistent.
c.) The graph of Plan B shows a rapid increase in price.
E
d.) The graph of Plan A shows a gradual increase in price.
IfIf Ryan claims that the cost of Plan A and Plan B differ a lot. The statement that best explains why Ryan was misled by the graph is: b.) The scale of intervals on the horizontal axis of both graphs is inconsistent.
The reason Ryan was misled by the graph?Based on the graph we can see that the graph is inconsistent based on the fact that the scale of intervals on the horizontal axis of Plan A graph and Plan B graph are inconsistent.
The reasons why the both graphs are inconsistent was because the lines did not intersect or are not on the same line which means that they are parallel lines.
Therefore the correct option is B.
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Answer:
Step-by-step explanation:
The two graphs use different vertical scales.
which statement is true? a. the graph of function f is stretched vertically by a scale factor of 3 to create the graph of function g. b. the graph of function f is stretched horizontally by a scale factor of 3 to create the graph of function g. c. the graph of
The graph of function f is stretched vertically by a scale factor of 3 to create the graph of function g.
The graph of a function is a visual representation of the function's mathematical relationship between inputs and outputs. When a function is said to be "stretched", this means that the graph is being transformed in some way. In this case, the graph of function f is being stretched vertically by a factor of 3 to create the graph of function g. This means that the vertical distance between each point on the graph of function f will be 3 times as great on the graph of function g. For example, if the graph of function f has a point (1,2) then the corresponding point on the graph of function g would be (1, 6). This transformation will change the shape of the graph and the values of the inputs and outputs of the function. The graph of the transformed function g will be wider, taller, and have different values for the inputs and outputs.
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3. In a picture of a woman sweeping with a broom, her broom measures 2cm long. In reality, it is 1m long. In the same picture, a wall measures 5cm in height. In reality, how tall is the wall?
The three interior angles of a triangle measure x°, (x-4)°, and (x+16)°. Which is the measure of one of the three angles?
OA) 48°
O B) 52°
O C) 60°
OD) 70°
=======================================================
Explanation:
Every triangle has the rule where the inside angles add to 180.
(angle1)+(angle2)+(angle3) = 180
x + (x-4) + (x+16) = 180
3x+12 = 180
3x = 180-12
3x = 168
x = 168/3
x = 56
Then we can determine each angle.
angle1 = x = 56 degreesangle2 = x-4 = 56-4 = 52 degrees (choice B)angle3 = x+16 = 56+16 = 72 degreesAs a check:
angle1+angle2+angle3 = 56+52+72 = 180
This confirms the answer is correct.
Select the correct answer from each drop-down menu.
Consider the function f(x) = |₂³| − 1.
The piecewise functions that coincide with the given function are
[(x + 3)-1, when x is greater than or equal to 3,
[(x-3)/2] -1, when x is less than 3,
[(x+3)/2] -1, when x is greater than -3,
[-(x+3)/2] -1, when x is greater than or equal to -3,
and
Yes, the piecewise functions that coincide with the given function are [(x + 3)-1, when x is greater than or equal to 3, [(x-3)/2] -1, when x is less than 3, [(x+3)/2] -1, when x is greater than -3, and -(x+3)/2] -1, when x is greater than or equal to -3.
Piecewise functionThe piecewise functions that coincide with the given function are as follows: when x is greater than or equal to 3, the piecewise function is (x + 3)-1; when x is less than 3, the piecewise function is (x-3)/2 -1; when x is greater than -3, the piecewise function is (x+3)/2 -1; and when x is greater than or equal to -3, the piecewise function is -(x+3)/2 -1.This means that, depending on the value of x, the resulting value of the function f(x) = |₂³| − 1 will be different.For example, when x is equal to 3, then f(x) = 0; when x is equal to -3, then f(x) = -2; and when x is equal to 1, then f(x) = -1.Thus, the piecewise functions correctly express the given function, as they correctly capture the different values of f(x) for different values of x.No, the piecewise functions that coincide with the given function are[(x + 3)-1, when x is greater than or equal to 3,[(x-3)/2] -1, when x is less than 3,[-(x-3)/2] -1, when x is less than or equal to -3,[(x-3)/2] -1, when x is greater than -3.To learn more about Piecewise function refer to:
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A 54-foot long wire will be attached to the top of a pole for support, and will be pulled tight and anchored to the ground.
If the wire makes a 13 degree angle with the ground, how tall is the pole?
The height of the pole used in the question is; 12.15 ft
How to use trigonometric ratios?The 3 main trigonometric ratios are;
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
Now, since the guy wire makes an angle of 13 degrees with the ground, it means the guy wire will be the hypotenuse of the right angle triangle and the height of the pole which we will label as h will be the opposite side of the triangle. Thus;
height/54 = sin 13
height = 54 * sin 13
height = 54 * 0.225
Height = 12.15 ft
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The sum of the differences between sample observations and the sample mean is
a. Zero
b. The range
c. The mean deviation
d. The standard deviation
The sum of the differences between sample observations and the sample mean is zero.
Statistics is a science that deals with methods of collecting data, processing data, presenting data, analyzing data, drawing conclusions, and making reasonable decisions based on existing facts.
There are some terms that are frequently used in statistics:
Range is the difference between the largest and the smallest values in a data set;Mean is the average of the sample values;Mean deviation is the average deviation of the data from the mean;Variance is a measure of how far the data is spread from its average value, which is equal to the average distances of the sample observations from the sample mean;Standard deviation is a value used to determine the distribution of data in a sample and see how close the data is to the mean value. The standard deviation is equal to the square root value of the variance.The sum of the differences between sample observations and the sample mean can be denoted as ∑(xi - mean), where n is the size of the sample and i = 1,2,3,...,n
∑(xi - mean) = ∑xi - n*mean,
= n*mean - n*mean
= 0
Thus the sum of the differences between sample observations and the sample mean is zero.
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Let U = {(q, r, s, t, u, v, w, x, y, z)
A = {q, s, u, w, y}
B = {q, s, y, z)
C = {v, w, x, y, z).
List the elements in the set.
AL(BNC)
A. {q, s, u, w, y, z)
B. q, r, w, y, z}
C. {q, y, z)
D. {q, w, y}
PLEASE HELP!
The elements of the set A'(B∩C)is { r, t , v, x, y, z}
How to determine the setIt is important to note that the universal set is the set containing all the elements for a given event.
From the information given, we have that;
U = {(q, r, s, t, u, v, w, x, y, z)A = {q, s, u, w, y}B = {q, s, y, z)C = {v, w, x, y, z)The signs used to represent sets are;
∩ - Intersection of sets used to represent the common elements of two or more sets∪- Union of sets used to represent the sum of sets without repetitionTo determine the set,
A'(B∩C)
A' = { r, t, v, x, z}
B∩C = { y, z}
Now, the set A'(B∩C)
A'(B∩C) = { r, t , v, x, y, z}
Hence, the set is { r, t , v, x, y, z}
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Write the following ratio using two other notations.
6:5
Use only the numbers above (not any others).
Notation one:
Notation two:
0.
3
0=0
0:00 to
0<0 0>
The ratio may be written if it is a component of a percentage.
6 is to 5 is 12 is to 10 .
What is a ratio notation?Ratios can be expressed as a notation, like 1/2, or as a ratio, like 6:5. A ratio's value is determined by dividing the first and second numbers. Consider the ratio 12:10 as an example. "6 to 5" should be read as the ratio. Any fraction, like 6/5, can also be expressed as a ratio, which is written as 6:5.
A colon or a fraction bar are two typical symbols representing ratios. You can also use words.
6 : 5 = 6/5 = 6 to 5
The ratio may be written if it is a component of a percentage.
6 is to 5
as in ...
12 is to 10 what 6 is to 5, and vice versa.
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What are the dimensions of the product?
[2 1 0] x [1 -1 2 -1 -2 1 0 1 1] =
The resulting vector has the same dimensions as the original, 1 x 3.
What is a matrix?A rectangular array of numbers, symbols, or expressions set up in rows and columns is known as a matrix. They are employed in a wide variety of mathematical disciplines and contexts, including calculus, statistics, physics, and engineering, as well as linear algebra. Matrices are an essential tool for solving systems of linear equations and representing linear transformations since they may be added, subtracted from, and multiplied by other matrices.
The matrix multiplication of [2 1 0] and [1 -1 2 -1 -2 1 0 1 1] is:
[2 1 0] x [1 -1 2 -1 -2 1 0 1 1] = [2(1) + 1(-1) + 0(2), 2(-1) + 1(-2) + 0(1), 2(0) + 1(1) + 0(1)]
= [2 -3 1]
Therefore, the product of the two matrices is a row vector [2 -3 1], which has the dimensions 1 x 3.
If we subtract 1 from each element of this row vector, we get:
[2 -3 1] - 1 = [1 -4 0]
So, the resulting vector has the same dimensions as the original, 1 x 3.
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Frank rented a unicycle at the circus. He paid a flat rental fee of $15 plus $2 and hour. The total cost was $23. Write and solve an equation you can use to find the number of hours he rented the unicycle.
Answer:
4 hrs
Step-by-step explanation:
let x = # of hours he rented the unicycle
$15 + ($2/hr)x = $23
($2/hr)x = $23 - $15 = $8
x = ($8) / ($2/hr) = 4 hours
Whats BD? Please help. Thanks
The value of side BD of the kite is 12 units
How to find the value of side BD of the kite?
In mathematics, kite is a quadrilateral which has two pairs of congruent adjacent sides.
A kite is a geometric shape which is symmetrical. It has two pairs of congruent adjacent sides, and two pairs of congruent opposite angles. The diagonals of a kite are perpendicular bisectors of each other.
Using Pythagoras theorem:
BC² = CE² + BE²
BE = √(BC² - CE²)
BE = √(10² - 8²) = 6 units
Since ΔCDE is identical to ΔBCE.
Thus, DE = BE
DE = 6 units
Since BD = BE + DE
Thus, BD = 6 + 6 = 12 units
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A parabola can be drawn given a focus of ((7,6) and a directrix of y=-8y=−8. What can be said about the parabola?
The parabola has a vertex at (7 , -1) has a p-value of -0.5 and it opens downwards.
How to interpret the equation of a parabola?The axis of symmetry of a parabola is basically perpendicular to the directrix and contains the focus and vertex.
Thus, the axis of symmetry is the vertical line x = 7.
The vertex lies halfway between the focus and directrix. Thus, we can say that the vertex is (7, (6 - 8)/2) = (7, -1)
Due to the fact that the vertex lies above the focus and the directrix is horizontal, then it means that the parabola opens downward.
Therefore, the focus is (7 - 8)/2 = -0.5 units below the vertex, so, p = -0.5
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Question 6.
Jim is studying growth of underwater plants in an aquarium. He notices the limiting factor on the plants is availability of light. Jim places the aquarium in a place where it will get more sunlight.
What will happen to the underwater plants?
Responses
Existing plants will grow larger and new plants will grow.Existing plants will grow larger and new plants will grow.
Existing plants will become smaller, but new plants will grow.Existing plants will become smaller, but new plants will grow.
Existing plants will become smaller and no new plants will grow.Existing plants will become smaller and no new plants will grow.
Existing plants will grow larger, but no new plants will grow.Existing plants will grow larger, but no new plants will grow.
question 7
A population of chipmunks that live in the woods near a town has learned to raid trash cans. The population of chipmunks is increasing rapidly.
The city replaces the existing trash cans with new, “chipmunk-proof” trash cans.
What most likely will happen to the population of chipmunks in the short term and long term?
Responses
The population will not change in the short term. In the long term, the population will increase dramatically.The population will not change in the short term. In the long term, the population will increase dramatically.
The population will not change in the short or long term. The chipmunks will switch to other food sources.The population will not change in the short or long term. The chipmunks will switch to other food sources.
The population will drop in the short term. In the long term, the population will stabilize as chipmunks switch to other food sources.The population will drop in the short term. In the long term, the population will stabilize as chipmunks switch to other food sources.
The chipmunks do not have other food sources. The populationwill drop in both the short and long term.
Step-by-step explanation: When a plant gets more sunlight it will grow very fast and get larger then it will grow some more
what is the position of (L) with respect to (c)
By the method of logical reasoning and the seating positions of seven people it is obtained that F sits third to the left of C.
What is logical reasoning?
The technique of applying systematic, logical steps based on mathematical logic to solve a problem is known as logical reasoning. Based on the information provided and mathematical principles, inferences can be drawn.
This seating positing problem can be solved by logical reasoning.
There are seven people - A, B, C, D, E, F, G.
Firstly, A sits third from one of the extreme ends.
Then two people sit between A and B.
The positions can be -
__ __ A __ __ B __ or __ B __ __ A __ __
Now, F sits fourth to the left of B.
So, second seating position gets eliminated.
Also, C sits at the immediate left of B.
__ F A __ C B __
Now, G sits third to the right of E.
E does not sit at any of the extreme ends.
D F A E C B G
So, the seating arrangement is obtained.
Therefore, F sits third to the left of C.
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Seven people named A, B, C, D, E, F, and G are sitting in a row facing north but not necessarily in the same order. A sits third from one of the extreme ends. Two people sit between A and B. C sits at the immediate left of B. F sits fourth to the left of B. G sits third to the right of E. E does not sit any of the extreme ends.
What is the position of F with respect to C?
g suppose that w(t) is a real-valued signal. prove that its fourier transform magnitude is symmetric around f
Since the Fourier transform of a real value signal is the product of the even cosine transform and the odd sine transform, it is conjugate symmetric and consequently symmetric about the frequency 0.
A real-valued signal's Fourier transform is a conjugate symmetric function, which means it is symmetric about frequency 0. This is demonstrated by the fact that a real-valued signal's Fourier transform may be expressed as the product of its cosine and sine transforms:
F(w) = Fc(w) plus Fs (w)
Where the cosine and sine transforms, respectively, are denoted by Fc(w) and Fs(w). Both the sine and cosine transforms are symmetric about the frequency zero because the sine transform is an odd function and the cosine is an even function. As a result, a real-valued signal's Fourier transform also exhibits symmetry near frequency 0.
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The complete question is
g suppose that w(t) is a real-valued signal. prove that its fourier transform magnitude is symmetric around frequency 0.
Find IJ and JK in parallelogram HIJK.
h 6p
i 7p+17
j 9p+11
K
The values of IJ and JK in parallelogram HIJK are 52 and 80 respectively
How to find the values?Recall that in every parallelogram, the opposite sides are equal and parallel.
This implies that IJ ║ HK and HI ║ JK
⇒ 5s - 88 = 3s - 32
Collecting the like terms we have
5s-3s = -32 +88
2s = 56
This implies that s = 28
By substitution we have s as 3(28) -32
= 84-32 = 52 and and 4(28)-32 = 80
The sides of the parallelogram are 80 and 52 respectively
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What are m and b in the linear equation, using the common meanings of m and b?
2 + 3x + 5 - 2x = y
The values of m and b on the linear equation:
2 + 3x + 5 - 2x = y
are
m = 1
b = 7
How to find the values of m and b?A general linear equation can be written as in the slope-intercept form as:
y = m*x + b
Where m is the slope and b is the y-intercept.
Here we have the linear equation
2 + 3x + 5 - 2x = y
Now we want to write this in the slope-intercept form, to do so, we just need to simplify the left side, then we will get:
2 + 3x + 5 - 2x = y
(2 + 5) + (3x - 2x) =y
7 + x = y
Then the linear equation is:
y = 1*x + 7
So the values are:
a = 1
b = 7
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a. x has an average of 5 and a standard deviation of 80. b. x has a bell shaped distribution with an average of 5 and a standard deviation of 80. c. x has a uniform distribution with an average of 80 and a standard deviation of 5. d. x has a bell shaped distribution with an average of 80 and a standard deviation of 5. e. secret statisticians code meaning that ohio state graduates are morons. 10 points question 2 what is the notation for the following (in matching order): standard deviation of a population, average of a sample, a set of sample averages, mean of a population, standard deviation of a sample, is distributed, bell shape a. s, mu, sampling distribution, x-bar, sigma, ~, n b. sigma, x-bar, sampling distribution, mu, s, n, ~ c. sigma, x-bar, sampling distribution, mu, s, ~, n d. s, x-bar, sampling distribution, mu, sigma, ~, n 10 points question 3 what is the meaning of x-bar ~ n(80,1) a. the random variable x is distributed with a bell shape, an average of 80 and a standard deviation of 1. b. the set of x-bars is distributed with a bell shape, an average of 80 and a standard deviation of 1. c. the random variable x is distributed with a bell shape, an average of 1 and a standard deviation of 80. d. the set of x-bars is distributed with a bell shape, an average of 1 and a standard deviation of 8
The answer to question 1 is not a statistical question. The answer to question 3 is that the notation x-bar ~ n(80,1) means that the set of x-bars is distributed with a bell shape, an average of 80 and a standard deviation of 1.
Question 1: e. secret statisticians code meaning that ohio state graduates are morons.
This is not a statistical question and does not have a correct answer.
Question 2: c. sigma, x-bar, sampling distribution, mu, s, ~, n
The notation for the following (in matching order) is: sigma (standard deviation of a population), x-bar (average of a sample), sampling distribution (a set of sample averages), mu (mean of a population), s (standard deviation of a sample), ~ (is distributed), n (bell shape).
Question 3: b. the set of x-bars is distributed with a bell shape, an average of 80 and a standard deviation of 1.
The notation x-bar ~ n(80,1) means that the set of x-bars is distributed with a bell shape, an average of 80 and a standard deviation of 1.
The answer to question 1 is that it does not have a correct answer as it is not a statistical question. The answer to question 2 is that the notation for the following (in matching order) is: sigma (standard deviation of a population), x-bar (average of a sample), sampling distribution (a set of sample averages), mu (mean of a population), s (standard deviation of a sample), ~ (is distributed), n (bell shape). The answer to question 3 is that the notation x-bar ~ n(80,1) means that the set of x-bars is distributed with a bell shape, an average of 80 and a standard deviation of 1.
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The number 1465 rounded to its highest place value is?
How do you factor and simplify 4x^2+ 25
Answer:
(4x + 5)(4x + 5)
Step-by-step explanation:
4x * 4x = 4x^2, and 5 * 5 = 25. Thus being 4x^2 + 25
Compute the probability of being dealt at random and without replacement a 13-card bridge hand consisting of: (a) 6 spades, 4 hearts, 2 diamonds, and 1 club; (b) 13 cards of the same suit.
The probabilities are found to be 0.00196 in case- 1 and
1 / 158753389900 in case - 2.
Note that there are a total of 52 cards in bridge, so by combinations that there are a total of
(13
52) numbers to deal 13 cards. There are a total of 13 spades, hearts, diamonds and clubs, respectively, so there are
(13
6) ways to get spades, etc., so that by multiplication principle the probability in we're looking for:
( a ) ( 13 (13 (13 (13
6) 4) 2) 1)
(52
13)
= 1716x715x78x13 / 635013559600
= 0.00196.
( b) there are a total of 4 drawings where all 13 cards are are of the same suit, corresponding to a choice of either spades, hearts, diamond, or clubs. Therefore the probability here is given by :
4 / ( 52 = 4/ 635013559600
13)
= 1 / 158753389900 .
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