The required values of the z-score as per the corresponding blood pressure readings are: z-score = 1.267 when reading is 142, z-score = 0.6 when reading is 130, and z-score = -1.2- when reading is 103.
What is Z -score?A Z-score is defined as the fractional representation of data point to the mean using standard deviations.
Normally distributed with a mean of 121 and a standard deviation of 15.
As per the given question, we have
mean ц = 121
standard deviation σ = 15
For X₁ = 140
z-score = (X₁ - ц )/σ
z-score = (140 - 121)/15
z-score = (19)/15 = 1.267
For X₂ = 130
z-score = (X₂ - ц )/σ
z-score = (130 - 121)/15
z-score = (9)/15 = 0.6
For X₃ = 103
z-score = (X₃ - ц )/σ
z-score = (103- 121)/15
z-score = (19)/15 = - 1.2
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The sin (theta) = -2/5, and theta lies in quadrant IV. Find the exact values of the sine and cosine of 2 theta.
[tex]\displaystyle\\Answer:\ sin(2\theta)=-\frac{4\sqrt{21} }{25} ,\ cos(2\theta)=\frac{17}{25}[/tex]
Step-by-step explanation:
[tex]\displaystyle\\sin(\theta)=-\frac{2}{5} \ \ \ \ \ \ \ \ 270^0 < \theta < 360^0\\\\sin^2(\theta)+cos^2(\theta)=1\\\\cos^2(\theta)=1-sin^2(\theta)\\\\Hence,\\\\cos^2(\theta)=1-(-\frac{2}{5})^2 \\cos^2(\theta)=1-\frac{4}{25} \\\\cos^2(\theta)=\frac{25(1)-4}{25} \\\\cos^2(\theta)=\frac{21}{25} \\\\[/tex]
Extract the square root of both parts of the equation:
[tex]\displaystyle\\cos(\theta)=б\sqrt{\frac{21}{25} } \\\\cos(\theta)=б\frac{\sqrt{21} }{5} \\\\270^0 < \theta < 360^0\\\\Hence,\\\\cos(\theta)=\frac{\sqrt{21} }{5}[/tex]
[tex]\displaystyle\\a)\ sin(2\theta)=2sin(\theta)cos(\theta)\\\\sin(2\theta)=2(-\frac{2}{5})(\frac{\sqrt{21} }{5})\\\\sin(2\theta)=-\frac{4\sqrt{21} }{25}[/tex]
[tex]\displaystyle\\b)\ cos(2\theta)=cos^2(\theta)-sin^2(\theta)\\\\cos(2\theta)=(\frac{\sqrt{21} }{5})^2-(-\frac{2}{5})^2 \\\\cos(2\theta)=\frac{21}{25}-\frac{4}{25} \\\\cos(2\theta)=\frac{17}{25}[/tex]
Choose the equation of the graphed function.
Answer:
h
Step-by-step explanation:
because it shifts the graph to the left by 6 units
the spread of a rumor in a town can be modeled as n equals 500 square root of t, where n is the number of people who have heard of the rumor, and t is time (in days). how long will it take until 2000 people know about the rumor? 12 days 4 days 8 days 16 days
The time it take until 2000 people know about the rumor is 16 days.
The square root of a variety of is described because the value, which offers the variety while it's miles increased with the aid of using itself. The radical symbol √ is used to suggest the square root. Radical is any other call given to the square root symbol. It is likewise called the surds. While Radicand is the variety gift beneath neath the square root symbol.
Find t, when N=200
N=5OO t^1/2
2000=500 t^1/2
t^1/2=4
t=16 days
Thus, the time it take until 2000 people know about the rumor is 16 days.
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Given f (x) = -3x² - 6x +9, find f (-7)
Answer:
f(-7) = - 96---------------------------------
Given function:
f(x) = -3x² - 6x +9Find f(-7) by plugging in the value of x:
f(-7) = - 3(-7)² - 6(-7) +9f(-7) = - 3(49) + 42 +9f(-7) = - 147 + 51f(-7) = - 96Juan claims that y=4x^3+1 is a function, but not a linear function. Select the statement that supports Juan's claim.
A. The function does not contain the point (0,0)
B. The coefficient of x^3 is not 1
C. The graph of the function does not form a straight line
D. The function increases
Answer:
C. The graph of the function does not form a straight line
Solve using a matrix.
2x-6y=22
-5x+y=1
can you please give me something I can copy-paste?
IT is found that the value of x is 4 and value of y is 5.
What is a system of equations?A system of equations is two or more equations that can be solved to get a unique solution. the power of the equation must be in one degree.
The given system of equations are
2x-6y=22
-5x+y=1
The matrix form is
[tex]\left[\begin{array}{ccc}2&-6\\-5&1\end{array}\right] \left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}22\\1\end{array}\right][/tex]
Let as assume
[tex]A = \left[\begin{array}{ccc}2&-6\\-5&1\end{array}\right]\\ \\X = \left[\begin{array}{ccc}x\\y\end{array}\right] \\B = \left[\begin{array}{ccc}22\\1\end{array}\right][/tex]
WE know that AX = B
Then we have;
[tex]\left[\begin{array}{ccc}x\\y\end{array}\right] = \left[\begin{array}{ccc}4\\5\end{array}\right][/tex]
Therefore, the value of x is 4 and value of y is 5.
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The acute angle between the vectors a=i-kj and b=i+jis 60° Calculate the possible values of k
no clue how to reach the answer
Answer:
k = (-55) / 8
k = (-3005) / 8
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 510(0.309016^2))) / 2)^2)) / 2)^2)) / 2
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 1469.59)))))^2)) / 2)
To find the acute angle between two vectors, we can use the dot product formula:
angle = arccos((a * b) / (||a|| * ||b||))
where a and b are the vectors, * is the dot product, and ||a|| and ||b|| are the magnitudes of the vectors a and b, respectively.
In this case, the dot product of a and b is (i - kj) * (i + j) = i^2 - kj * i + kj * i + kj^2 = 2i - k^2j
The magnitudes of the vectors a and b are ||a|| = sqrt(i^2 + (-kj)^2) = sqrt(1 + k^2) and ||b|| = sqrt(i^2 + j^2) = sqrt(2).
Substituting these values into the formula above, we get:
angle = arccos((2i - k^2j) / (sqrt(1 + k^2) * sqrt(2)))
Since the angle is given to be 60 degrees, we can set this equal to 60 degrees and solve for k:
60 = arccos((2i - k^2j) / (sqrt(1 + k^2) * sqrt(2)))
We can use the inverse cosine function to solve for k:
k = sqrt(1 / (cos(60)^2 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
Since cos(60) = 0.5, we can substitute this value in and solve for k:
k = sqrt(1 / (0.5^2 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
k = sqrt(1 / (0.25 - (2i / sqrt(1 + k^2) * sqrt(2))^2))
k = sqrt(1 / (0.25 - (4i^2 / (1 + k^2) * 2)^2))
k = sqrt(1 / (0.25 - (16 / (1 + k^2))^2))
k = sqrt(1 / (0.25 - 256 / (1 + k^2)^2))
k = sqrt((1 + k^2)^2 / (256 - (1 + k^2)^2))
k = sqrt((1 + k^4) / (256 - 1 - 2k^2 - k^4))
k = sqrt((k^4 + 1) / (255 - 2k^2))
We can then solve for the roots of this equation to find the possible values of k:
k = sqrt((k^4 + 1) / (255 - 2k^2))
k^4 - (255 - 2k^2)k^2 + 1 = 0
This is a quartic equation and can be solved using the quartic formula:
k = sqrt((-b +- sqrt(b^2 - 4ac)) / 2a)
where a, b, and c are the coefficients of the polynomial. In this case, a = 1, b = -(255 - 2k^2), and c = 1.
Substituting these values into the quartic formula, we get:
k = sqrt((-(-(255 - 2k^2)) +- sqrt((-(255 - 2k^2))^2 - 4 * 1 * 1)) / 2 * 1)
k = sqrt((255 - 2k^2 +- sqrt((255 - 2k^2)^2 - 4)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(255^2 - 510k^2 + 4k^4)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(255^2 - 510k^2)) / 2)
k = sqrt((255 - 2k^2 +- sqrt(65025 - 510k^2)) / 2)
Solving for the roots of this equation gives us the possible values of k:
k = (-255 + sqrt(65025 - 510k^2)) / 2
k = (-255 - sqrt(65025 - 510k^2)) / 2
The first equation gives us one possible value of k:
k = (-255 + sqrt(65025 - 510k^2)) / 2
Substituting k = (-255 + sqrt(65025 - 510k^2)) / 2 into the second equation gives us the second possible value of k:
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2
Simplifying this expression gives us the final possible value of k:
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2)^2)) / 2
Therefore, the possible values of k are:
k = (-255 + sqrt(65025 - 510k^2)) / 2
k = (-255 - sqrt(65025 - 510((-255 + sqrt(65025 - 510k^2)) / 2)^2)) / 2
solve for k in each
To solve for k in the first equation, we can isolate k by moving everything else to the right side of the equation:
k = (-255 + sqrt(65025 - 510k^2)) / 2
2k = -255 + sqrt(65025 - 510k^2)
2k + 255 = sqrt(65025 - 510k^2)
(2k + 255)^2 = 65025 - 510k^2
4k^2 + 1020k + 65025 = 65025 - 510k^2
4k^2 + 1530k + 65025 = 0
This is a quadratic equation, and we can use the quadratic formula to solve for k:
k = (-b +- sqrt(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the polynomial. In this case, a = 4, b = 1530, and c = 65025.
Substituting these values into the quadratic formula gives us:
k = (-1530 +- sqrt(1530^2 - 4 * 4 * 65025)) / 2 * 4
k = (-1530 +- sqrt(3080400 - 2601000)) / 8
k = (-1530 +- sqrt(477900)) / 8
k = (-1530 +- sqrt(222725)) / 8
k = (-1530 + 1475) / 8
k = (-55) / 8
k = (-1530 - 1475) / 8
k = (-3005) / 8
Therefore, the solutions to the first equation are:
k = (-55) / 8
k = (-3005) / 8
Write a function in any form that would match the graph shown below 
The function that matches the graph is y(x)=(1-x)(x+2)^2
What is function?a function from a set X to the set of Y assigns to each other element of X exactly one element of Y. The set X is the called the domain of to the function of the set Y is called as the condominium of the functions.
From the graph the curve cross the x -axis at X=1
Therefore the expression is (x-1)
And at (-) x axis at X=-2 it is the turning point.
Therefore the expression is
(X+2)^2
And the point on y axis is (0,4)
Therefore the function can be written as
Y(x)=a(x-1)(x+2)^2
4= -4a≈a=-1
Therefore the function is y(x)
(-1)(x-1)(x+2)^2
= y(x)=(1-x)(x+2)^2
Therefore the function that matches the graph is y(x)=(1-x)(x+2)^2
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Select the graph of the solution. Click until the correct graph appears.
|x| + 1 < 3
Answer:
Graph B
Step-by-step explanation:
First, simplify by subtracting 1 on both sides.
|x| + 1 - 1 < 3 - 1
|x| < 2
Using the absolute value definition, we know the inequalities are:
x < 2
-x < 2
Divide both sides by -1.
x < -2
If you multiply or divide both sides of an inequality by a negative number, you must flip the sign.
x > -2
x<2, x>-2
Graph B
Answer:
Graph A
Step-by-step explanation:
if x ≥ 0
x + 1 < 3
x < 3 - 1
x < 2
0≤ x < 2
if x < 0
-x + 1 < 3
-x < 3 - 1
-x < 2
x > -2
-2 < x < 0
Final solution
-2 < x < 2
A number is such that it is as much greater than 112 as it is less than it. find the number.
Answer:
Answer. it is -111 because it is greater than 112 and in positive form it is smaller and in negative it is bigger .
Linear Functions: Model from Two Points-Quiz-Level H
Ready
4) A bathtub has some water in it. Mia turns on the faucet to add more water. The total amount of
water in gallons, y, is a function of the time in minutes since Mia turns on the faucet, a.
4) The graph of the linear function passes through the points (4, 24) and (6, 30).
What is the equation of the function?
?
The equation of function for the given problem is y = 3x + 12.
What is point slope form of the line?
For linear equations, the general form is y - y1 = m(x - x1).
It draws attention to the line's slope and one of the line's points (that is not the y-intercept).
Given:
A bathtub has some water in it. Mia turns on the faucet to add more water.
The total amount of water in gallons, y, is a function of the time in minutes since Mia turns on the faucet, a.
The graph of the linear function passes through the points (4, 24) and
(6, 30).
We have to find the equation of function.
Let the linear function passes through the points (4, 24) and (6, 30).
First to find the slope of equation using given points.
[tex]m = \frac{y_2-y_1}{x_2-x_1} = \frac{30-24}{6-4} = \frac{6}{2} = 3[/tex]
Now to find the equation of function.
Consider the point slope form of the line,
[tex]y-y_1=m(x-x_1)[/tex]
Plug the values of m = 3 and [tex](x_1,, y_1) = (4, 24)[/tex]
⇒
[tex]y-24=3(x-4)\\y-24=3x-12\\y=3x-12+24\\y=3x+12[/tex]
Hence, the equation of function for the given problem is y = 3x + 12.
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Please help me with this
The distance the ball drops in the next 8 second is: 640 m. Using the concept of equation of motion.
What is equation of motion?An item is considered to be at rest when its position doesn't alter throughout time. An item is considered to be in motion if, over time, its location changes.
It is possible to build a relationship using a series of equations between the body's velocity, acceleration, and the distance it travels in a given amount of time when the body is travelling in a straight line with uniform acceleration. The term "motion equations" refers to these equations.
D = kt² where D is the constant of proportionality.
D = 80 m
t = 4 seconds
Putting the values we get -
80 = k(4)²
or, k = 80 / 16
or, k = 5.
So the equation of motion is D = 5t²
When t = 8+4 = 12 seconds
now, D = 5(12)² = 5×144
or, D = 720 m
So, the distance the ball drops in the next 8 seconds = 720 - 80
= 640 m.
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as the sample size increases, the distribution of the sample proportion becomes more normal. this fact is due to
the Central Limit Theorem, which states that as the sample size increases, the sampling distribution of the sample mean tends to become more normal.
The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean tends to become more normal. This means that the mean of the sample will be closer to the mean of the population, and the variability of the sample will be smaller. Since the sample proportion is just the mean of the sample, as the sample size increases, the distribution of the sample proportion will also become more normal. The larger the sample size, the more likely it is that the sample will accurately represent the population, and the more normal the distribution of the sample proportion will be. In other words, the sample proportion will be closer to the true population proportion, and the variability of the sample proportion will be smaller.
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Find the value of k that would make the left side of the equation a perfect square trinomial.
9x² - kx + 4 = 0
Answer: k=±`12
Step-by-step explanation:
[tex]ax^2-bx+c=0\\\\9x^2-kx+4=0\\\\D=b^2-4ac\\\\a=9\ \ \ \ b=-k\ \ \ \ c=4\\\\D=0\\\\Hence,\\\\D=(-k)^ 2-4(9)(4)=0\\\\k^2-144=0\\\\k^2=144\\\\k^2=12^2[/tex]
Extract the square root of both parts of the equation:
[tex]k=б\sqrt{12^2} \\\\k=б12[/tex]
Answer:
k = - 12 or k = 12---------------------------------------------
A perfect square trinomial is:
(a ± b)² = a² ± 2ab + b²Given trinomial:
9x² - kx + 4 = 0Show this as:
(±3x)² - kx + (±2)² = 0Compare and find possible values of k:
k = - 2(±3)(±2) = ± 12Lena's phone is 15 cm long and 8 cm wide. Her tablet is 30 cm long and 22 cm wide.
How much additional area is available on her tablet than on her phone?
Step-by-step explanation:
subtract the area of the phone from that of the phone
Use the marginal tax rate chart to answer the question.
Marginal Tax Rate Chart
Tax Bracket
Marginal Tax Rate
$0-$10,275
10%
$10.276-$41.175
12%
$41.176-$89.075
22%
$89,076-$170.050 24%
$170,051-$215,950 32%
$215,951-$539,900 35%
$539,901
37%
Determine the amount of taxes owed on a taxable income of $49,652.
Answer choices
$4,735.38
$6,600.44
$7,709.92
$10,293.44
The amount of taxes owed on a taxable income of $49,652 is $10,293.44.
What is the percentage?A percentage is a value per hundredth. Percentages can be converted into decimals and fractions by dividing the percentage value by a hundred.
Given, Are the tax percentages for various amounts of income.
Now, From the given chart we conclude that $49,652 falls in the category of $41.176-$89.075 is 22% tax.
So, the Tax owed is 22% of $49,652.
∴ (22/100)×$49,652.
= $10,293.44.
So, an income of $49,652 owed 22% tax of the income which is $10,293.44.
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Answer:THE ANSWER IS B)6,600.44
Step-by-step explanation:
I got it right :)
The figure shows a loading dock and a side view of
an attached ramp, whose run is 12 feet and whose
rise is 39 inches. Joaquin is wondering whether a
long rectangular box can be stored underneath the
ramp, as suggested by the dotted lines. The box is
2 feet tall and 5 feet long. Answer Joaquin's question.
The area of the triangular dock is 19.5 square feet and the area of the rectangular box is 10 feet. The box will fit under the dock.
What is an area?The space occupied by any two-dimensional figure in a plane is called the area. The space occupied by the rectangle in a two-dimensional plane is called the area of the rectangle. The space occupied by the triangle in a two-dimensional plane is called the area of the triangle.
Calculate the area of the triangle first,
Area = 1 / 2 x B x H
Area = 1/2 x 12 x 3.25
Area = 19.5 square feet
Calculate the area of the rectangle,
Area = L x W
Area = 2 x 5
Area = 10 square feet
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11. Define the range of the following:
Answer:
The answer is B.
Step-by-step explanation:
Looking at the points gives you a x-axis and y-axis.
(x , y)
(-2 , 3) ,
(0 , 5) ,
(1 , 5) ,
(2 , 4) ,
(8 , -2)
Focus on the y-axis instead of the x-axis and list them.
{3 , 5 , 4 , -2}
in class a there are 20 students 14 of them are girls in class b there 25 students 15 of them are girls (a) find the percentages of girls in each class
To find the percentage of girls in class A, we need to divide the number of girls by the total number of students and then multiply by 100. This gives us:
14 / 20 * 100 = 70%
To find the percentage of girls in class B, we need to divide the number of girls by the total number of students and then multiply by 100. This gives us:
15 / 25 * 100 = 60%
Thus, the percentage of girls in class A is 70%, and the percentage of girls in class B is 60%.
It takes 73 pounds of seed to completely plant an 11 -acre field. How many pounds of seed are needed per acre?
Step-by-step explanation:
It takes 73 pounds of seed to completely plant an 11 -acre field. How many pounds of seed are needed per acre?
73 / 11 = 6.6363636 pounds per acre
rounded to two decimal places: 6.64 pounds per acre
IN PHOTO⬇️
PLEASE HELP!!!
The simple interest is $2160 and the amount after 30 years is $6160.
Given that, principal =$4000, rate of interest =1.8%, and time period =30 years.
What is the simple interest?Simple interest is a method to calculate the amount of interest charged on a sum at a given rate and for a given period of time.
Simple interest is calculated with the following formula: S.I. = (P × R × T)/100, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years.
a) Now, S.I =(4000×1.8×30)/100
= 40×1.8×30
= $2160
b) Amount = Principal + Interest
= 4000 + 2160
= $6160
Therefore, the simple interest is $2160 and the amount after 30 years is $6160.
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© {2, 4, 6, 8}
(3, 4, 5, 6}
(2, 3), (4, 4), (6, 5), (8, 6)
{2, 3, 4, 5, 6, 8)
X
2
4
6
8
у
3
4
5
6
Is the line perpendicular?
The situation is based on football. One player starts a couple of yards in the endzone while the other starts at the 8-9 yard line. The player in the endzone almost scores when he is tracked down by the guy on the 8-9 yard line. So does the starting position of these players form a perpendicular line?
Yes, the starting position of these players form a perpendicular line.
What is Trigonometry?The area of mathematics that deals with particular angles' functions and how to use those functions in calculations. There are six popular trigonometric functions for an angle. Sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant are their respective names and acronyms (csc).
Given:
Using Trigonometry
sin [tex]\theta[/tex] = 33.33/100
sin [tex]\theta[/tex] = 0.3333
[tex]\theta[/tex] = [tex]sin^{-1}[/tex] (0.3333)
[tex]\theta[/tex] = 19.469
cos 19.469 = B/ 100
0.9428 = B/ 100
B= 94.28
tan [tex]\theta[/tex] = P/B
tan [tex]\theta[/tex] = 94.28/ 33.33
[tex]\theta[/tex] = 70.5
Using Angle Sum property
< 3= 180 - (70.5 + 19.5)
<3 = 180 - 90
<3 = 90
Hence, they form perpendicular line.
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if the value of x is 3 and the value of y is 5, what is displayed as a result of executing the code segment?
The result of executing the code segment is -2
How to determine the result of executing the code segment?The complete question is added at the end of this solution as an attachment
The code in the question is given as
IF X > Y
DISPLAY X + Y
ELSE
DISPLAY X - Y
Given that
X = 3 and Y = 5
When x and y are compared, we have the truth value to be
Y > X
This means that the executed segment is
DISPLAY X - Y
So, we have
DISPLAY 3 - 5
Evaluate
DISPLAY -2
Hence. the displayed result is -2
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how many additional groups would be required to conduct a 3 x 2 x 3 factorial design compared to a 3 x 2 x 2 design?
One additional group would be required to conduct a 3 x 2 x 3 factorial design compared to a 3 x 2 x 2 design with independent variable.
A 3 x 2 x 3 factorial design requires three independent variables (x1, x2, and x3) with three levels each, for a total of 27 conditions. A 3 x 2 x 2 design, however, would only require two independent variables (x1 and x2) with two levels each, for a total of 12 conditions. To conduct a 3 x 2 x 3 factorial design, one additional group would be required, compared to the 3 x 2 x 2 design. This additional group would provide data for the additional 15 conditions that the 3 x 2 x 3 design would require.
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if a line equation 2x +y =4 what is the gradie value
The gradient value of the line 2x + y = 4 is 2. This can be found by rearranging the equation into slope-intercept form, which is y = -2x + 4. The coefficient of the x term, -2, is the slope of the line.
What is the value of y when x = 3 in the equation y = 6x - (9 + 5)
Answer:
y = 4
Step-by-step explanation:
substitute x = 3 into the equation
y = 6(3) - (9 + 5) = 18 - 14 = 4
Answer: y=4
Step-by-step explanation:
y=6(3)-(9+5)
y=18-14
y=4
a company has the following information regarding its forecast performance in the past three periods. icture what is the mean absolute deviation (mad)? question 26 options: 225 -66.7 1200 200
The mean absolute deviation over three period of time is 200
The absolute value of error in period 1 = 300
The absolute value of error in period 2 = 200
The absolute value of error in period 3 = -100
Total absolute value of error = The absolute value of error in period 1 + The absolute value of error in period 2 + The absolute value of error in period 3
Substitute values in the equation
Total absolute value of error = 300 + 200 + 100
= 600
The mean absolute deviation = Total absolute value of error / 3
Substitute the values in the equation
The mean absolute deviation = 600 / 3
= 200
Therefore, the mean absolute deviation is 200
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Let P(n) be the statement that 13 +23 +···+n3 = (n(n+1)/2)2 for the positive integer n.a) What is the statement P(1)?b) Show that P(1) is true, completing the basis step of the proof.c) What is the inductive hypothesis?d) What do you need to prove in the inductive step?e) Complete the inductive step, identifying where you use the inductive hypothesis.f) Explain why these steps show that this formula is true whenever n is a positive integer.
P(n) = 1³ +2³ +···+n³ = (n(n+1)/2)²
P(1) = 1³ = 1
According to statement P(n) = (n(n+1)/2)²
Checking for n = 1,
(n(n+1)/ 2)² = (1(1+1)/ 2)²
P(1) = (1 × 2/ 2)²
P(1) = 1² = 1
P(1) = 1³ = 1
So P(n) holds for n = 1.
The inductive hypothesis is that the statement holds for P(n). To prove this the inductive step under the assumption that the inductive hypothesis is true, we prove it is true for P(n + 1)
Let us assume P(n) = 1³ + 2³ + ... + n³ = (n(n+1)/ 2)²
Then P(n + 1) = 1³ + 2³ + ... + n³ + (n + 1)³ = (n(n+1)/ 2)² + (n + 1)³
P(n + 1) = (n(n+1)/ 2)² + n³ + 3n² + 3n + 1
P(n + 1) = ((n + 1)((n + 1) + 1)/ 2)²
As assuming P(n) holds, it is also true for P(n + 1) so it will be true for all values of n as it is proved already that P(1) is true.
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Answer:
215
Step-by-step explanation:
218+212/2