Answer:Given : number of hours spent studying for a history final exam and the score on that exam.
To Find : Which value is an outlier
(1.5, 65)
(3.5, 71)
(4.5, 98)
(6.5, 88)
Solution:
Number of hours spent studying =x
Exam score = y
x y
1.5 65
2 68
3.5 71
4.5 98
6 82
1.5 - 2 difference = 0.5
2 - 3.5 difference = 1.5
3.5 - 4.5 difference = 1
4.5 - 6 difference = 1.5
No outlier
65 - 68 Difference 3
68 - 71 Difference 3
71 - 98 Difference 27
71 - 82 Difference 11
Hence 98 is outlier
(4.5 , 98 ) is outlier
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Mrs. Juarez graded ten English papers and recorded the scores. 92 ...
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first quartile is 20 then semi-inter quartile range is
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Step-by-step explanation:
If x=3, solve for y
y=2*3^(3)
Answer:
54
Step-by-step explanation:
Answer:
y=54
as x=3
so y=2*x^3
y= 2*3^3
y=2*27
y=54
Calculate Suppose that on each of the
4,500 dives Alvin has made, a new pilot and two new scientists were on board.
How many scientists have seen the
deep ocean through Alvin's windows? How
many people, in total, traveled in Alvin?
The calculation shows that 9,000 scientists have seen the deep ocean through Alvin's windows; and
a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.
What is the explanation for the above calculation?1) If on each of the 4,500 dives Alvin carried a new pilot and two new scientists, then the total number of scientists who have seen the deep ocean through Alvin's windows is:
4,500 dives x 2 scientists per dive = 9,000 scientists
Therefore, 9,000 scientists have seen the deep ocean through Alvin's windows.
2) To calculate the total number of people who traveled in Alvin, we can add the number of pilots and scientists on each dive and multiply by the number of dives:
4,500 dives x (1 pilot + 2 scientists)
= 4,500 x 3
= 13,500 people
Therefore, a total of 13,500 people have traveled in Alvin over the course of its 4,500 dives.
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I am stuck on this question..
The output of the definite integral and summation expression can be presented as follows;
[tex](\frac{\int\limits^1_0 {(\sum\limits_{n=5}^5 n\cdot 5 \cdot \sqrt{25} )} \, dx }{\int\limits^1_0 {(\sum\limits_{n=2}^5 2 + 6)} \, dx } ) = \frac{125}{32} =3\frac{29}{32}[/tex]
What is the difference between definite and indefinite integral?An indefinite integral is an antiderivative of a function, while a definite integral is the limit of a sum of areas of rectangles.
The steps to integrate the specified definite integral expression can be presented as follows;
The integrals and the summations in the question are simplified thus;
Sum n = 5 to 5 n × 5 × √(25) is the sum of n times 5 times √(25) over the range [5, 5], which is the same as 5 × 5 × √(25) = 125
Sum n = 2 to 5, 2 + 6 is the sum of (2 + 6) over the range [2, 5], which, based on the operation is (2 + 6) + (2 + 6) + (2 + 6) + (2 + 6) = 32
The integration of the above summations are found as follows;
Integrate 125 dx from 0 to 1 is the definite integral of the constant function, 125, over the interval [0, 1], which is the same as 125 × (1 - 0) = 125.
The integration of 32 dx from 0 to 1 is the definite integral of the constant function 32 over the interval [0, 1], which is just 32 × (1 - 0) = 32
Plugging in the above values into the original expression, we get;
[tex]\frac{\int\limits^1_0 ({\sum\limits_{n=5}^5}n\cdot 5\cdot \sqrt{25} ) \, dx }{\int\limits^1_0 (\sum\limits_{n=2}^5 2+6) \, dx } = \frac{125}{32}[/tex]
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on jerry's text messaging phone plan, he paid $0.60 for every 30 messages he sent. what is the his cost per message?
Answer:
$0.02
Step-by-step explanation:
0.60/30
how to calculate the product of two random variable that follows normal distribution with mean 0 and variance 1
To calculate the product of two random variables that follows the normal distribution with mean 0 and variance 1 by using the covariance formula
Cov(X, Y) = E[XY] - E[X]E[Y] = E[XY] - 0 = E[XY]
Given that two random variables follow a normal distribution with mean 0 and variance 1.
Let X and Y be two independent normal random variables such that X ~ N(0,1) and Y ~ N(0,1)
Now, The expected value of the product of two random variables is given by;
E[XY] = E[X]E[Y] + Cov(X,Y)
Where E[X] and E[Y] are the means of the two random variables X and Y respectively.
Cov(X, Y) is the covariance between the two random variables, which can be calculated using the formula;
Cov(X,Y) = E[XY] - E[X]E[Y]
Now, E[X] = E[Y] = 0 as both have a mean of 0.
Cov(X, Y) = E[XY] - E[X]E[Y]
⇒ E[XY] = the expected value of the product of X and Y.
As X and Y are independent, their covariance will be zero, which implies;
Cov(X, Y) = E[XY] - E[X]E[Y] = E[XY] - 0 = E[XY]
Thus, we can calculate the product of two random variables that follow a normal distribution with mean 0 and variance 1 using the above formula for covariance.
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4. (5pts each) Given f(x) = 6x² +1and g(x) = 11x +7 determine the following:
a (fog)(x)\
b. (fog)(-3)
=6XTI
Answer:
(fog)(x) = 726x^2 + 924x + 295, and (fog)(-3) = 6,817.
The 17 cows on the farm produced 209 & and 780 ml of milk this morning. How much milk did each cow produce today on average?
Therefore, on average, each cow produced 169.41 ml of milk today.
What is average?The average, also known as the arithmetic mean, is a statistical measure that is calculated by adding together a set of numbers and then dividing the sum by the total number of values in the set. The result is a single value that represents the central tendency of the data set.
Given by the question.
To find the average amount of milk produced by each cow, we need to divide the total amount of milk by the number of cows.
Average milk produced by each cow = (Total milk produced)/ (Number of cows)
Total milk produced = 2090 ml + 780 ml = 2870 ml
Number of cows = 17
Therefore,
Average milk produced by each cow = 2870 ml/17
= 169.41 ml
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Calvin ordered scissors. Each pair of scissors weighs 0. 34 kg. A box of scissors weighs
64. 6 kg. How many pairs of scissors come n each box?
190 pairs of scissors came in each box.
This is an example of a word problem. As a model, a word problem depicts circumstances from the actual world that may be converted into phrases.
Weight of each pair of scissors = 0.34 kg
Weight of a box of scissors = 64.6 kg
Therefore, pairs of scissors in each box = Weight of a box of scissors ÷
Weight of each pair
= 64.6 ÷ 0.34
= 190 pairs.
Hence, 190 pairs of scissors came in each box.
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The bearing of part A from part B is 074 deg. What is the bearing of part A from part B
The bearing of part A from part B is 254 degrees.
4)) FH and IK are parallel lines. J K F G E Which angles are alternate exterior angles?
Answer: I couldn't honestly help with that I would if I could
Step-by-step explanation:
the data in the table shows a sinusoidal relationship between the number of seconds an object has been moving and its velocity v(x), measured in centimeters per second. x 2 4 6 8 10 12 14 16 18 20 22 24 v(x) 34.5 31.3 26.1 20 13.9 8.7 5.2 4 5.2 8.7 13.9 20 what is true of the cosine function that models the data in the table? drag a value into each box to correctly complete the statements.
The period of the cosine function is___ the equation of the midline of the cosinefunction is y=_______The amplitude of the cosine function is
these are the whole statements: Cosine function has a 24-period period. The cosine function's midline has the equation y = 19.25. The cosine function has a 15.25 amplitude.
What does an equation mean?A mathematical equation, such as 6 x 4 = 12 x 2, states that two amounts and values are equal. two. Countable noun. A scenario known as an equation means that two or more components must be taken into account in order to comprehend or understand the overall situation.
Midline = (34.5 Plus 4) / 2 (34.5 + 4) = 19.25 Maximum value plus Minimum value
As a result, the following equation represents the data's model for the midline of a cosine function:
y = 19.25
The biggest deviation from the function's midline is its amplitude in the cosine function. The maximum amount of v(x) was 34.5 cm/s, and the minimum is 4 cm/s, according to the data provided. The cosine function used to model the data has the following amplitude:
Amplitude is calculated as (highest value – minimum values) / 2 (or 34.5 - 4) / 2 (or 15.25).
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given the augmented matrix in row-reduced form, assume that it is equivalent to an augmented matrix of a system of linear equations. for each matrix: determine the number of equations and number of variables in the corresponding system of linear equations. determine if the system is underdetermined or overdetermined.
System : [ 1 ¼ -1/4 | 1]
[ 0 0 0 | 0]
find the solution(s) to the system, if it exists. state the solution as a point (be sure to use parentheses), use parameter(s) and if needed. if the system is inconsistent, then state no solution.
The given augmented matrix is equivalent to a system of linear equations with 1 equation and 3 variables. The system is underdetermined and the solution to the system is [tex](x, y, z)[/tex]= (0, 0, 0).
The given augmented matrix is in row-reduced form and is equivalent to a system of linear equations. The matrix has 3 columns and 1 row, so there is 1 equation and 3 variables in the corresponding system of linear equations. Since there is only one equation, the system is underdetermined and has infinitely many solutions.
The solution to the system can be stated as: (x, y, z) = (0, 0, 0). The solution is a point where all the variables are equal to zero. Since the matrix is in row-reduced form, this is the only solution to the system. If the augmented matrix was not in row-reduced form, then the solution would be stated as a parameter, such as (x, y, z) = (0, 0, t), where t is a parameter.
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Answer:
1 equation, 3 variablesunderdetermined(x, y, z) = (1 -s +t, 4s, 4t)Step-by-step explanation:
For the given augmented matrix, you want to know the number of equations and variables it represents, whether the system is under- or over-determined, and the solution, using parameters as required.
[tex]\left[\begin{array}{ccc|c}1&\dfrac{1}{4}&-\dfrac{1}{4}&1\\\vphantom{\dfrac{T}{q}}0&0&0&0\end{array}\right][/tex]
Augmented matrixEach column of the augmented matrix is the coefficient of the variable in a linear equation represented by the row. The first row of the matrix represents the equation ...
[tex]1x +\dfrac{1}{4}y-\dfrac{1}{4}z=1[/tex]
The second row of the matrix represents the tautology ...
0x +0y +0z = 0
It contributes no information about the relationships between the variables.
The matrix represents 1 equation in 3 variables.
SolutionFor there to be a unique solution to a linear system, there must be as many consistent and independent equations as there are variables. Here, there are fewer equations than variables. This system is underdetermined.
The solution to an underdetermined system can be written in terms of parameters of free choice. We will need as many parameters as there are missing equations: 2.
Here, the coefficients of y and z suggest we choose parameters that let us represent those values as multiples of 4.
Parameters: s, t
Let y = 4s, z = 4t. Then the equation becomes ...
x +1/4(4s) -1/4(4t) = 1
x +s -t = 1
x = 1 +t -s
Now, the solution can be written as ...
(x, y, z) = (1 +t -s, 4s, 4t)
Name the shape that will result from connecting the points (-4, 1) , (-4, -4) , (0, 3) , and (0, 6) .
A: Square
B: Rectangle
C: Trapezoid
D: Parallelogram
The shape that results from connecting the points (-4, 1), (-4, -4), (0, 3), and (0, 6) is a trapezoid.
What is a trapezoid?A trapezoid is a geometric form that has four sides, two of which are parallel and two of which are nonparallel (or skew lines). A trapezoid is also known as a trapezium (UK) or a trapeze (US).
The trapezoid's parallel sides are known as the bases, and the two nonparallel sides are known as the legs or lateral sides. The trapezoid is also sometimes referred to as the irregular quadrilateral.
How to identify a trapezoid?A quadrilateral is a shape that has four sides, four vertices, and four angles. The following are the characteristics of a trapezoid:
It has four sidesIt has two parallel sides and two nonparallel sidesIt has two opposite sides that are parallel to one another and two other sides that are not parallelIt has two acute angles and two obtuse anglesIt has diagonals that intersect at a midpointThe formula for the area of a trapezoid is as follows:
Area of a trapezoid = [ (base 1 + base 2) / 2 ] x height
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Why is the probability that a continuous random variable is equal to a single number zero? (i.e. Why is P(X=a)=0 for any number a) [1 sentence]
What is meant by the 95% confidence interval of the mean? [1-2 sentences]
What two quantities do we need to fully describe a normal distribution? [1 sentence]
In determining the sample size for a confidence interval, is the size of the population relevant? [3 sentences]
List the steps in Hypothesis Testing. [4-5 bullets]
The probability that a continuous random variable is equal to a single number zero because the area under a continuous probability density function (pdf) between any two points, even two extremely close points, is never equal to zero.
In other words, since the continuous random variable is infinite and continuous, the probability that it is equal to a single value is almost zero.
Steps in Hypothesis Testing:State the null and alternative hypotheses.Calculate the test statistic.
Determine the critical value or p-value.Calculate the p-value, if necessary.Make a decision and interpret the results.
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Assume that two relations Rand S are union-compatible. Which of the following statements is NOT true? a. Rand S must have the same number of attributes b. The difference operation ( - ) in relational algebra can be performed on Rand S c. Rand S must have the same number of tuples d. The domain of the i-th attribute of R must be the same as the domain of the i-th attribute of S
The answer that is NOT true when two relations Rand S are union-compatible is: Rand S must have the same number of tuples.
What is a relation?A relation is a set of attributes that describe a set of entities (or elements). The union-compatible operation allows you to combine two relations into one with the same attribute set.
The union-compatible operation is performed on two relations that have the same number of attributes and the same domain in each attribute. The union-compatible operation is sometimes known as the union operation.
Assume that two relations Rand S are union-compatible. Then it implies that both relations have the same number of attributes and the domain of the i-th attribute of R must be the same as the domain of the i-th attribute of S.
Thus, the correct option is D and A i.e.
"The domain of the i-th attribute of R must be the same as the domain of the i-th attribute of S".
In relational algebra, the difference operation (-) is used to find the differences between two relations. When two relations R and S are union-compatible, then we can perform the difference operation. Hence, option (b) is true. But option (c) is false as there is no compulsion that two relations Rand S must have the same number of tuples.
Note: The union-compatible operation is one of the three basic operations of relational algebra that includes the selection operation and the projection operation.
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Rewrite the following expression using the division symbol and as a fraction: The quotient of m and 11.
Answer:
[tex]\frac{m}{11}[/tex]
Step-by-step explanation:
To write the given expression as a fraction, put m in the numerator and 11 in the denominator as shown: [tex]\frac{m}{11}[/tex]
This would be the quotient of m and 11 expressed as a fraction!
#Keep learning
first 6 terms of n² + 7
Answer:
8, 11, 16, 23, 32, and 43.
Step-by-step explanation:
When n = 1:
n² + 7 = 1² + 7 = 8
When n = 2:
n² + 7 = 2² + 7 = 11
When n = 3:
n² + 7 = 3² + 7 = 16
When n = 4:
n² + 7 = 4² + 7 = 23
When n = 5:
n² + 7 = 5² + 7 = 32
When n = 6:
n² + 7 = 6² + 7 = 43
Therefore, the first 6 terms of n² + 7 are 8, 11, 16, 23, 32, and 43.
Answer:
When n = 1, n² + 7 = 1² + 7 = 8
When n = 2, n² + 7 = 2² + 7 = 11
When n = 3, n² + 7 = 3² + 7 = 16
When n = 4, n² + 7 = 4² + 7 = 23
When n = 5, n² + 7 = 5² + 7 = 32
When n = 6, n² + 7 = 6² + 7 = 43
The first 6 terms of n² + 7 are 8, 11, 16, 23, 32, and 43.
Step-by-step explanation:
ᓚᘏᗢ
hope u have a good day man
Write the definition of each word
1. An acute angle is οne that is less than 90 degrees.
What is an acute angle ?An angle smaller than the right angle is called an acute angle. In οther wοrds, the angle which is less than 90 degrees fοrms an acute angle. The pοlygοns such as triangle, parallelοgram, trapezοid, etc. cοnsist οf at least οne acute angle
2. Obtuse angle: A measurement οf an angle that is greater than 90 degrees but less than 180 degrees.
3. A right angle is οne that is exactly 90 degrees.
4. A straight angle is οne that is exactly 180 degrees in length.
5. Adjacent angles are twο angles that share a cοmmοn vertex and a cοmmοn side.
6. A pair οf nοn-adjacent angles fοrmed by the intersectiοn οf twο lines are knοwn as vertical angles. They are οf equal size.
7. Cοmplementary angles are defined as twο angles that add up tο 90 degrees.
8. Twο angles that add up tο 180 degrees are referred tο as supplementary angles.
9. Angles with the same measure are said tο be cοngruent.
10. A linear pair is twο adjacent angles that fοrm a straight line and add up tο 180 degrees.
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Use the figure shown.
Trapezoid M N P Q is divided into 4 triangles by N Q and M P, which intersect at point R inside the trapezoid. N P is parallel to M Q and M N is congruent to Q P.
If m∠MNP = 107, what is m∠NMQ?
m∠NMQ =
In the Quadrilateral MNPQ, if MP = 5.9, then RN = 1.8.
What is a trapezοid?An οpen, flat οbject with fοur straight sides and οne set οf parallel sides is referred tο as a trapezοid οr trapezium. A trapezium's nοn-parallel sides are referred tο as the legs, while its parallel sides are referred tο as the bases.
Given:
Quadrilateral MNPQ is shοwn.
Trapezοid MNPQ is divided intο 4 triangles by NQ and MP.
MQ is parallel tο NP.
That means, MQ and NP always at the equal distance.
MN ≅ QP
Sο,
MP = QN
Nοw,
QN = QR + RN and QR = 4.1:
MP = QN
5.9 = 4.1 + RN
Then RN = 5.9 - 4.1 = 1.8
Therefοre, the value οf RN is 1.8.
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Complete question:
Quadrilateral MNPQ is shown.
Trapezoid M N P Q is divided into 4 triangles by N Q and M P, which intersect at point R inside the trapezoid. N P is parallel to M Q and M N is congruent to Q P. Angle N M P is 30 degrees and angle P M Q is 32 degrees. Segment Q R equals 4.1.
If MP = 5.9, what is RN?
(Round your answer to the nearest tenth. Write NA if there is not enough information given.)
A function is shown in the box. What is the value of this function for f(-8)?
(Write the answer as an improper fraction in lowest terms.)
Answer:
f(x) = (5/6)x - (1/4)
f(-8) = (5/6)(-8) - (1/4)
f(-8) = (5/3)(-4) - (1/4)
f(-8) = (-20/3) - (1/4)
f(-8) = (-80-3)/12
f(-8) = -83/12
Determine wheter the given vale of the varible is a soultion of the equatiom 9 = 3/4 e e = 12
If e = 12 and the equation is true, then the answer to the equation is 12.
How can you determine a variable's value in an equation?You can solve the equation as before if it has the form axe + b = c, where x is the variable. Addition and subtraction should be "undone" first, followed by multiplication and division.
We insert e = 12 into the equation and check to see if the equation is true to see if the given value of the variable is a solution of the equation 9 = (3/4)e, where e = 12.
9 = (3/4)e
9 = (3/4)(12)
9 = 9
If e = 12 and the equation is true, then the answer to the equation is 12.
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A and B are independent.
If P(A) = 1/2 and P(An B) = 6/15
What is the explicit formula for the sequence 8, 20, 50, 125, 312. 5, …?
The explicit formula for the given sequence 8, 20, 50, 125, 312.5, ... is a_n = 8 × 2.5^(n-1)
To find an explicit formula for the given sequence, we first need to identify the pattern or rule that generates the terms.
Looking at the sequence, we can see that each term is obtained by multiplying the previous term by a fixed factor and then adding a constant.
To find this factor and constant, we can use the general form of a geometric sequence:
a_n = a_1 × r^(n-1)
where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the index of the term.
Using the first two terms of the sequence, we can find the common ratio:
r = a_2 / a_1 = 20 / 8 = 2.5
Now we can use the first term and the common ratio to find the constant term
a_1 = 8 = a_1 × 2.5^(1-1) + c
c = 8 - a_1 = 0
Therefore, the explicit formula for the given sequence is:
a_n = 8 × 2.5^(n-1)
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20 people ride the bus, 80 people ride motorbikes, 65 people drive vans, and 215 people drive cars. Based on the data, how many would you expect to use motorbikes if you asked 800 people? how many people ride motorbikes
There are 58 students enrolled in an art class. The day before the class begins 10.3% of the students cancel. How many students actually attend the art class
The total number of students which attend the art class is 52 which is 89.75 of the total enrolled students that is 58.
Explain about the percentage of number?A Latin word with the meaning "out of one hundred" is percentage.
Working with parts of 100 is much easier than working with thirds, twelfths, and other fractions, especially since many fractions lack a consistent (non-recurring) decimal equivalent.
Significantly, this also makes comparing percentages more simpler than comparing fractions with various denominators. This contributes to the widespread usage of the metric measuring system and decimal currencies.
Total students enrolled for art class = 58
Percentage of cancelled students = 10.3%
Then, Percentage of non-cancelled students = 100 - 10.3 = 89.7%
So,
Attended students = 89.7% of 58.
= 89.7*58 / 100
= 52.06
= 52 (approx)
Thus, the total number of students which attend the art class is 52 which is 89.75 of the total enrolled students that is 58.
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how can we use models to estimate percent questions? Give examples to support your answer.
Answer:
Percent questions can be estimated using models by dividing the given number by the total number of parts in the model. For example, if there are ten students in a classroom and 30 candy bars, then each student would get three candy bars. Another example would be if there are ten students in a classroom and 30 candy bars, and one student takes five candy bars, then the remaining nine students would get two candy bars each.
Answer:
Percent questions can be estimated using models by dividing the given number by the total number of parts in the model. For example, if there are ten students in a classroom and 30 candy bars, then each student would get three candy bars. Another example would be if there are ten students in a classroom and 30 candy bars, and one student takes five candy bars, then the remaining nine students would get two candy bars eac
Step-by-step explanation:
if there are m ways of doing one thing and n ways of doing another, how many ways are there to do both? for example, if a toy comes in m colors and n sizes, how many different toys can there be
The number of ways of doing both things is N = m × n
How to find the number of ways of doing both things?Since there are m ways of doing one thing and n ways of doing another, to find how many ways are there to do both, we proceed as follows.
Since there are m ways of doing one thing and n ways of doing another, to find the number of many ways to do both things,we multiply both numbers together.
So, then number of ways of doing both things is N = m × n
So, there are m × n ways of doing both things
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A restaurant owner is buying new tableware for her restaurant. She wants the new pieces to be proportionate, and needs to know the measurements of the different pieces she is buying. The circle-shaped part of the bottom of the glass above has an area of 12.56 in2. What is the radius? Use = 3.14. A. 4 in B. 5 in C. 1 in D. 2 in Reset Submit
The answer is d please
Assume the universes are the same size Think about your best high scholl algebra, these formulas are very much like that Due Sunday October 15 Cost GRPS CPP Impressions CPM TV 8500000 1500 A 480000000 B Radio 350000 250 1400 80000000 с Digital 150000 100 D 32000000 4.69 Magazines 500000 125 E F G Total H j K L
The total advertising cost for the campaign is $59,200,000.
To calculate the total cost for each advertising medium, we need to multiply the number of impressions by the cost per thousand impressions (CPM) and divide by 1000 to get the total cost in dollars.
For TV, the total cost is 8,500,000 impressions x $1,500 CPM / 1000 = $12,750,000.For radio, the total cost is 350,000 impressions x $250 CPM / 1000 = $87,500.For digital, the total cost is 150,000 impressions x $100 CPM / 1000 = $15,000.For magazines, the total cost is 500,000 impressions x $125 CPM / 1000 = $62,500.Adding up all of the costs, we get $12,750,000 + $87,500 + $15,000 + $62,500 = $12,915,000.
However, we need to remember that the digital advertising has a CPM of $4.69 instead of the listed cost, so we need to recalculate its cost to be $32000000 x $4.69 CPM / 1000 = $150,080. Adding this to the total cost gives $12,915,000 + $150,080 = $13,065,080. This is the cost before the GRPS and CPP discounts are applied.
To apply these discounts, we need to multiply the total cost by (1 - GRPS) and then divide by CPP. Plugging in the values given, we get ($13,065,080 x (1 - 0.25)) / 1500 = $48,928.50. Finally, we need to multiply this by the total number of impressions to get the overall cost of the campaign, which is $48,928.50 x 1,200,000,000 / 1000 = $59,200,200, or approximately $59,200,000.
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The ratio of the sines of any two angles in a triangle equals the ratio of the lengths of their opposite sides. Justify your answer.
Answer: This statement is known as the Law of Sines and can be proven using trigonometry and the properties of similar triangles.
Consider a triangle ABC with sides of lengths a, b, and c opposite to the angles A, B, and C respectively. Let's draw an altitude from vertex A to side BC, splitting side BC into two segments: BD of length x and CD of length c-x.
Using the Pythagorean theorem, we can write:
a^2 = x^2 + h^2 (1)
b^2 = (c-x)^2 + h^2 (2)
where h is the length of the altitude from A to BC.
Dividing equation (1) by sin^2(A) and equation (2) by sin^2(B), we get:
a^2 / sin^2(A) = x^2 / sin^2(A) + h^2 / sin^2(A)
b^2 / sin^2(B) = (c-x)^2 / sin^2(B) + h^2 / sin^2(B)
Since angles A and B are complementary (i.e. A + B = 90 degrees), we have sin(A) = cos(B) and sin(B) = cos(A). Substituting these identities and rearranging the equations, we get:
a^2 / sin(A) = x / cos(B) + h / sin(B)
b^2 / sin(B) = (c-x) / cos(A) + h / sin(A)
Multiplying both equations by sin(A)sin(B), we obtain:
a^2 sin(B) = x sin(A) cos(B) + h sin(A)
b^2 sin(A) = (c-x) sin(B) cos(A) + h sin(B)
Now, we use the fact that h = a sin(B) = b sin(A), which follows from the definition of sine as opposite/hypotenuse. Substituting this into the above equations and simplifying, we get:
a / sin(A) = 2R
b / sin(B) = 2R
c / sin(C) = 2R
where R = a/(2sin(A)) = b/(2sin(B)) = c/(2sin(C)) is the radius of the circumcircle of triangle ABC. This is the Law of Sines in its usual form.
From this, we can see that the ratio of the sines of any two angles in a triangle equals the ratio of the lengths of their opposite sides, as required.
Step-by-step explanation: