The composite result function (h o g)(x) in the given functions g(x) = 9 + 4x and h(x) = x + 21/5 is 4x + 66/5.
What is the composite result function (h o g)(x) in the given function?A function is simply a relationship that maps one input to one output.
Given the data in the question;
g(x) = 9 + 4xh(x) = x + 21/5(h o g )(x) = ?To find (h o g)(x), first set up the composite result function h(g(x)).
h(x) = x + 21/5
h( g(x) ) = g(x) + 21/5
Plug g(x) = 9 + 4x
h( g(x) ) = ( 9 + 4x ) + 21/5
Simplify
h( g(x) ) = 9 + 4x + 21/5
h( g(x) ) = 4x + 9 + 21/5
h( g(x) ) = 4x + 66/5
Therefore, the composite function h( g(x) ) is 4x + 66/5.
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A unit of an abstract mathematical system subject to the laws of arithmetic is?
Units are a natural number, a natural number that are negative, or zero. The number zero and any number generated by adding one to it repeatedly. A single integer or the product of two integers (division by zero excluded).
What exactly are integers?
In mathematics, integers are the collection of positive and negative numbers. Integers, like whole numbers, do not include the fractional element. Thus, integers are numbers that can be positive, negative, or zero, but not fractions. On integers, we can do all arithmetic operations such as addition, subtraction, multiplication, and division. Integers include numbers such as 1, 2, 5, 8, -9, -12, and so on. "Z" is the symbol for integers.
The number zero and any number generated by adding one to it repeatedly. A single integer or the product of two integers (division by zero excluded).A natural number, a negative of a natural number, or zero is a unit of an abstract mathematical system according to the principles of arithmetic.
e.g. 1,2,0, 0+1, 0+2+3, 8, 8/2=4, -8/2=-4 etc. all are units of an abstract mathematical system subject to the laws of arithmetic.
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Estimate the local maximum of y=2x3+x2−7x−6.
Answer: (-1.25, 0.41)
Step-by-step explanation:
100 POINTS WILL PLEASEEEEEE HELP
Answer:
The $30,000 in revenue should be entered as a credit in the journal entry.
Step-by-step explanation:
In accounting, revenues are recorded as credits, which means that they increase the credit side of the accounting equation (revenues increase equity, which is a credit account). Debits, on the other hand, are used to record expenses and decreases in assets.Therefore, the journal entry to record $30,000 in revenue for the business on October 1 would be:Date
Account Account Number Debit Credit
1 - Oct Revenue 4050 30,000Since revenue is being credited, there is no debit amount entered in this entry. The credit amount of $30,000 is entered in the credit column, as shown above.
Which of the following is not true of slope of the least-squared regression line?The square of the slope equals the fraction of the variation in the response variable that is explained by the explanatory variable, it is unit less.
The slope of the least-squares regression line is not directly related to the fraction of the variation in the response variable that is explained by the explanatory variable.
The slope of the least-squares regression line is a numerical measure of the rate of change in the response variable for a unit change in the explanatory variable. It does not directly relate to the fraction of the variation in the response variable that is explained by the explanatory variable. This is because the fraction of the variation in the response variable that is explained is determined by the correlation coefficient, which is a measure of how well the regression line fits the data. The slope of the least-squares regression line is unitless, meaning it does not have any specific unit such as meters, seconds, or degrees. The slope of the least-squares regression line is simply a numerical measure of the rate of change in the response variable per unit change in the explanatory variable. This is different from the fraction of the variation in the response variable explained by the explanatory variable, which is determined by the correlation coefficient.
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Solve for the length of the unknown side in the following right triangle. (Side AC is the hypotenuse.) Round your answer to two places, where applicable. Side AB 17, Side BC 22, Side AC ?
Answer:
27.87
Step-by-step explanation:
In a right triangle, the length of the hypotenuse can be found using the Pythagorean theorem, which states that the sum of the squares of the two shorter sides is equal to the square of the length of the hypotenuse.
Using the given sides, we can write the equation:
AB^2 + BC^2 = AC^2
Substituting the given values, we have:
17^2 + 22^2 = AC^2
Simplifying the left side, we get:
289 + 484 = AC^2
773 = AC^2
Taking the square root of both sides, we find:
AC = √773 = 27.87
Rounding to two decimal places, the length of side AC is approximately 27.87.
Solve using elimination. 3x - 2y = 10 3x - y = 17
The first step is to rewrite the equations in standard form by subtracting the constant from each side.
What is standard form ?Standard form is a type of numerical notation used to express large numbers in a concise form. It is based on powers of 10, which means each place value of a number is multiplied by a power of 10. Standard form is typically used in scientific and mathematical applications, and it is also known as scientific notation. To write a number in standard form, the number is written as a coefficient, followed by the base of 10, raised to an exponent.
3x - 2y = 10 becomes 3x - 2y - 10 = 0
3x - y = 17 becomes 3x - y - 17 = 0
Let's multiply the first equation by 2 and the second equation by 3.
2(3x - 2y - 10) = 0
3(3x - y - 17) = 0
6x - 4y - 20 = 0
9x - 3y - 51 = 0
The next step is to add the two equations together.
6x - 4y - 20 + 9x - 3y - 51 = 0
15x - 7y - 71 = 0
The last step is to solve for the variable.
15x - 7y = 71
15x = 71 + 7y
x = (71 + 7y)/15
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55 hours in one week with $15 per hour. What’s gross pay, assuming overtime is time and a half?
Answer:
Step-by-step explanation:
If you work 55 hours in one week and earn $15 per hour, your gross pay before taking into account any overtime can be calculated as follows:
Gross pay = 55 hours x $15/hour = $825
If you work overtime, which is time and a half, the additional pay can be calculated as follows:
Overtime pay = ($15/hour x 1.5) x (55 hours - 40 hours) = ($22.50/hour x 15 hours) = $337.50
So, the total gross pay including the overtime pay is:
Gross pay = $825 + $337.50 = $1,162.50
Therefore, the total gross pay for working 55 hours in one week at $15 per hour, with overtime pay calculated as time and a half, is $1,162.50.
Also when you answer this can you show work please and thank you
Answer:
rectangle: 30 ft²triangle: 12 ft²total: 42 ft²Step-by-step explanation:
You want the area of the composite figure consisting of a rectangle and a triangle.
RectangleThe area of the rectangle is ...
A = LW
A = (5 ft)(6 ft) = 30 ft² . . . . rectangle area
TriangleThe base of the triangle is the difference in length of the horizontal segments:
b = 9 ft -5 ft = 4 ft
The area is given by the formula ...
A = 1/2bh
A = 1/2(4 ft)(6 ft) = 12 ft² . . . . triangle area
Total areaThe area of the figure is the sum of the areas of its parts:
total area = rectangle area + triangle area
total area = 30 ft² +12 ft²
total area = 42 ft²
__
Additional comment
The area of this trapezoid can also be found using the trapezoid formula:
A = 1/2(b1 +b2)h
A = 1/2(9 ft +5 ft)(6 ft) = 42 ft²
The rectangle is shown in the figure as 5 ft wide and 6 ft tall. It does not matter which of these dimensions you call the length and which you call the width. Their product is the same either way.
evaluate the integral using integration by parts with the indicated choices of u and dv. (use c for the constant of integration.) xe5x dx; u
Answer:
[tex]\displaystyle \int {xe^{5x}} \, dx=\frac{1}{5}xe^{5x}-\frac{1}{25}e^{5x}+C[/tex]
Step-by-step explanation:
Given Problem
[tex]\displaystyle \int {xe^{5x}} \, dx[/tex]
Make substitutions for IBP
Let [tex]u=x,\,du=dx[/tex] and [tex]v=\frac{1}{5}e^{5x},\,dv=e^{5x}dx[/tex]
Perform IBP
[tex]\displaystyle \int {u} \,dv=uv-\int {v} \, du\\\\\int {xe^{5x}} \, dx=\frac{1}{5}xe^{5x} -\int {\frac{1}{5}e^{5x}} \, dx\\\\\displaystyle \int {xe^{5x}} \, dx=\frac{1}{5}xe^{5x}-\frac{1}{25}e^{5x}+C[/tex]
Find the perimeter of the triangle whose vertices are (−3,−5)
, (5,−5)
, and (5,1)
. Write the exact answer. Do not round.
From the graph of y = f(x), how do we obtain the graph of y = f(x+1) +3?
Select one:
a. None of the other choices is correct
b. Shift to the right by 1, and then shift up by 3
c. Shift to the left by 3, and then shift up by 1
d. Shift to the left by 1, and then shift up by 3
e. Shift to the right by 3, and then shift up by 1
It is a translation of 3 units up and one unit to the left, then the correct option is D.
How do we obtain the graph of y = f(x+1) +3?For any function y = f(x), we can define:
A vertical translation of N units as:
g(x) =f(x) + N
if N > 0, the translation is up
if N < 0, the translation is down.
A horizontal translation of N units as:
g(x) =f(x + N)
if N > 0, the translation is left
if N < 0, the translation is to the right.
Here we have:
g(x) = f(x + 1) + 3
So we have a translation of 1 unit to the left and 3 units up.
The correct option is d.
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For what value of ‘K’ will the following pair of linear equations are inconsistent 3x+ 4y=1 and (2k-1) x+(k-1)y=2k+1
Answer: k=(1/5)
Step-by-step explanation:
Please help meeeeeee !!??? I begggggggggggggggggg it due tomorrow
Answer:
[tex] \frac{14}{16} \: and \: \frac{13}{16} [/tex]
Step-by-step explanation:
So we have these two fractions
[tex] \frac{7}{8} \: and \: \frac{13}{16} [/tex]
Does 8 go into 16? Yes. It goes in twice because 8 times 2 is 16. That means 16 is our common denominator.
Because of that, since we multiplied 8 by 2, we will do the same to the 7. 7 times 2 is 14.
Because of that we have 14/16 and 13/16 as our answers.
Find the mean for the data items in the given frequency distribution. Score, x 1,2,3,4,5,6,7,8 Frequency, f 1,1,3,6,6,9,8,5,6,3
The mean is -- (Round to 3 decimal places as needed.)
2000 Paul was twice as old as his brother Biko. In 2008 Paul was only four years older than his brother. In what year was Biko born ? Select one: a. 1996 b. 1990 c. 1992 d. 1998 e. 2000
Answer:
a. 1996
Step-by-step explanation:
Let Paul's age in 2000 be P
Let Bilko's age in 2000 be B
Statement: Paul was twice as old as his brother Biko in 2000 means
P = 2B (1)
In 2008 Paul was only four years older than his brother.
Since 2008 is 8 years from 2000, both Paul and Bilko have aged 8 years
Paul's age in 2008 = P + 8
Bilko's age in 2008 = B + 8
Statement: In 2008 Paul was only four years older than his brother means
P + 8 = B + 8 - 4
P = B + 4 (2)
Given these two equations (1) and (2)
Substitute P = 2B from (1) in (2):
2B = B + 4
==> 2B - B = 4
==> B = 4
Therefore in 2000, Paul was 8 years old and Bilko was 4 years old
Bilko's year of birth = 2000 - 4 = 1996
a = [1 0 -4 0 3 -2 -2 6 3] b = [4 1 -4] Denote the columns of A by a1, a2, a3, and let W = Span {a1, a2, a3,}. is b in {a1, a2, a3,}? how many vectors are in {a1, a2, a3,}? is b in W? How many vectors are in W? Show that a1 is in W.[ hint: row operations are unneceassary.]
a. there are three vectors in {a₁, a₂, a₃}.
b. There are infinitely many vectors in W = span {a₁, a₂, a₃} by definition.
c. we have that a₁ is in the W.
What is matrix?A matrix is a set of integers arranged in rows and columns. The rows and columns of integers are arranged in a rectangular matrix. The matrix A, for instance, contains two rows and three columns. The elements of the matrix, also known as the entries, are the integers. In addition to many areas of mathematics, matrices are widely used in the fields of engineering, physics, economics, and statistics.
a. The vector b is not in {a₁ ,a₂, a₃} since b is not equal to any of the column vectors of A. Clearly, there are three vectors in {a₁, a₂, a₃}.
b. Vector b is in W if the augmented matrix [a₁, a₂, a₃ b] is consistent.
The augmented matrix is:
[tex]\left[\begin{array}{cccc}1&0&-4&4\\0&3&-2&1\\-2&6&3&-4\end{array}\right] \sim \left[\begin{array}{cccc}1&0&-4&4\\0&3&-2&1\\0&6&-5&4\end{array}\right] \sim \left[\begin{array}{cccc}1&0&-4&4\\0&3&-2&1\\0&6&-1&2\end{array}\right][/tex]
Thus, since the system is consistent we have that b must be in W.
There are infinitely many vectors in W = span {a₁, a₂, a₃} by definition.
c. since [tex]\left[\begin{array}{c}1\\0\\-2\end{array}\right] = 1. \left[\begin{array}{c}1\\0\\-2\end{array}\right] + 0. \left[\begin{array}{c}1\\0\\-2\end{array}\right] + 0. \left[\begin{array}{c}1\\0\\-2\end{array}\right][/tex] we have that a₁ is in the W.
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a. there are three vectors in {a₁, a₂, a₃}.
b. There are infinitely many vectors in W = span {a₁, a₂, a₃} by definition.
c. we have that a₁ is in the W.
What is matrix?A matrix is a set of integers arranged in rows and columns. The rows and columns of integers are arranged in a rectangular matrix. The matrix A, for instance, contains two rows and three columns. The elements of the matrix, also known as the entries, are the integers. In addition to many areas of mathematics, matrices are widely used in the fields of engineering, physics, economics, and statistics.
a. The vector b is not in {a₁ ,a₂, a₃} since b is not equal to any of the column vectors of A. Clearly, there are three vectors in {a₁, a₂, a₃}.
b. Vector b is in W if the augmented matrix [a₁, a₂, a₃ b] is consistent.
No, b is not in {a1, a2, a3}.
There are 3 vectors in {a1, a2, a3}.
Yes, b is in W.
There are infinitely many vectors in W.
Yes, a1 is in W because it can be expressed as a linear combination of the vectors in W. This is shown by the equation a1 = 1(a1) + 0(a2) + 0(a3). The coefficients of the vectors in the linear combination are all integers and this equation shows that a1 is in the span of the vectors in W.
Thus, since the system is consistent we have that b must be in W.
There are infinitely many vectors in W = span {a₁, a₂, a₃} by definition.
c. since we have that a₁ is in the W.
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n is a whole number. Prove that (n+8)(2n+6) is always an even number.
Answer:
Any odd number multiplied by an even number will make an even number.
the city of alexandria has a high temperature of 18 degrees and a low temperature of -5 degrees on the same day. by how many degrees did the high temperature exceed the low temperature?
Hi there, here's your answer:
Highest temperature = 18 degrees
Least temperature = -5 degrees
The difference between the two is 18 - (-5) = 18 + 5 = 23 degrees
Hope it helps!
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The sugar content of orange juice is decreased from 6% to 4%. How much percent does the sugar content decrease?
Answer:
33.33%
Step-by-step explanation:
amount of sugar at first 6x
now 4x
difference 2x,(6x-4x)
"ratio=2/6"
change in percentage:2/6×100%=33.33%
Which equations represent circles that have a diameter of 12 units and a center that lies on the y-axis? Select two options. x2 + (y – 3)2 = 36 x2 + (y – 5)2 = 6 (x – 4)² + y² = 36 (x + 6)² + y² = 144 x2 + (y + 8)2 = 36
Equations that represent circles that have a diameter of 12 units and a center that lies on the y-axis are
x² + (y - 3)² = 36 and x² + (y + 8)² = 36
What is circle?A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the center, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.
Given equations
⇒ x² + (y - 3)² = 36
comparing with general equation of circle
( x - h )² + ( y - k )² = r²
Center (h , k) = (0 , 3)
Center is on y - axis
Radius r = √36 = 6units
Diameter = 2r = 2×6 = 12units
⇒ x² + (y - 5)² = 6
comparing with general equation of circle
( x - h )² + ( y - k )² = r²
Center (h , k) = (0 , 5)
Center is on y-axis
Radius r = √6 units
Diameter = 2r = 2√6 units
⇒ (x – 4)² + y² = 36
comparing with general equation of circle
( x - h )² + ( y - k )² = r²
Center (h , k) = (4 , 0)
Center is on x-axis
Radius r = √36 = 6units
Diameter = 2r = 2×6 = 12units
⇒ (x + 6)² + y² = 144
comparing with general equation of circle
( x - h )² + ( y - k )² = r²
Center (h , k) = (-6 , 0)
center is on x-axis
Radius r = √144 = 16units
Diameter = 2r = 2×16 = 32units
⇒ x² + (y + 8)² = 36
comparing with general equation of circle
( x - h )² + ( y - k )² = r²
Center (h , k) = (0 , -8)
Center is on y-axis.
Radius r = √36 = 6units
Diameter = 2r = 2×6 = 12units
Hence, equation x² + (y - 3)² = 36 and x² + (y + 8)² = 36 represents the circle with diameter 12 units and a center that lies on y-axis.
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HELP PLEASE ITS DUE TMR
Answer:
[tex]-6[/tex]
Step-by-step explanation:
[tex]-5.f(-2) - 3g(7)[/tex]
The values of [tex]f(-2)[/tex] and [tex]g(7)[/tex] have to be read and deduced from the respective [tex]f(x)[/tex] and [tex]g(x)[/tex] graphs
[tex]f(-2)[/tex]:
For x = [tex]-2[/tex], draw a straight vertical line connecting x = [tex]-2[/tex] and the blue [tex]f(x)[/tex] graph. It can be observed that the the blue [tex]f(x)[/tex] graph is touching the x-axis at x = [tex]-2[/tex]. Therefore, the corresponding y [or [tex]f(x)[/tex]] value is 0.
[tex]g(7)[/tex]:
For x = 7, draw a straight vertical line connecting x = 7 and the red dashed [tex]g(x)[/tex] graph. Then (from this meeting point) draw a straight horizontal line that touches the y-axis. It can be observed that the corresponding y [or [tex]g(x)[/tex]] value is 2
Now substitute the deduced values in the expression:
[tex]-5.[/tex](0) [tex]-3.[/tex](2) = [tex]0 - 6[/tex]
= [tex]-6[/tex]
Taylor invested PhP 100,000 at a rate of 5.8%. If the rate is compounded yearly, how much investment should she have after 3.5 years?
factoriser 30x²-11x-6
Answer:
(3x-2)(10x+3)
Step-by-step explanation:
If y varies directly as x, and y=6 when x=15, find y when x=2.
A.) 4/5
B.) 4
C.) 5
d.) 5/4
Answer:
Option A: 4/5
Step-by-step explanation:
If y varies directly as x written as y ∝ x, this can be written as an equation:
y = kx where k is a constant
Since y = 6 when x = 15:
6 =k · 15
6 = 15k
15k = 6
k = 6/15 = 2/5
The equation beocmes
y = (2/5) x
Therefore when x = 2,
y = (2/5) x 2 = 4/5
Option A: 4/5
Accounting ◆ QUESTION: 2 The information below relates to Winelands Ltd for the year ended 28 February 2021.The company was registered with an authorised share capital of 1 000 000 shares. 600 000 of these shares were in issue on 1 March 2020, the beginning of the current financial year. 13 NSC Grade: 12 (TERM: 1) COMPANIES LEDGER ACCOUNTS REQUIRED: 2.1. Prepare the following accounts in the General Ledger. - Ordinary share capital >> Retained income NOTE: Close off all these accounts correctly at the end of the year. INFORMATION: A. Balances in the ledger on: Ordinary share capital Retained income Shares 28 February 2021 R ? 28 February 2020 R 2 100 000 737 100 - On 31 August 2020 all the unissued shares were issued at 500 cents per share. On 30 November 2020 the directors repurchased 120 000 shares at 620 cents per share from a retired shareholder. This shareholder originally purchased his shares on the JSE at various times and at different prices over the past years.
Showing this information in a ledger is given below":
The Ledger AccountOrdinary Share Capital:
28 February 2021: 1,000,000 authorized shares * 500 cents per share (for the shares issued on 31 August 2020) = R500,000
28 February 2020: 600,000 issued shares * R35 (2,100,000 / 60) = R21,000
Retained Income:
28 February 2021: R500,000 (Ordinary Share Capital) - R21,000 (Ordinary Share Capital) + R120,000 (repurchased shares at 620 cents per share) = R599,000
28 February 2020: R737,100 (Retained Income)
Note: The R35 per share in 28 February 2020 was calculated by dividing the total ordinary share capital of R2,100,000 by the number of shares in issue (600,000).
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What is (-3,2),(4,-3) and (-5,-4) rotated 90° counterclockwise
The coordinates after the rotation of 90° will be (-2, -3), (3, 4), and (4, -5) respectively.
What is geometry rotation?Each point in a figure is transformed into a rotation by rotating it a specific amount of degrees around another point.
Precession, nutation, and intrinsic rotation are the names of these rotations.
Shapes can be rotated by being moved around a fixed point (clockwise or anticlockwise, and by a certain number of degrees).
Although the shape will remain precisely the same, its location within the room will shift.
So, we have:
(-3,2),(4,-3) and (-5,-4)
Now, the graph then as follows.
Now, rotate them to 90°.
(Refer to the graph attached below)
Then, the coordinates after 90° rotation will be:
(-3,2) = (-2, -3)
(4,-3) = (3, 4)
(-5,-4) = (4, -5)
Therefore, the coordinates after the rotation of 90° will be (-2, -3), (3, 4), and (4, -5) respectively.
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Simply the expression 9(1-4n)
Answer: 9 - 36n
Step-by-step explanation: So, first you multiply 9 by 1 and 9 by 4n, which gives you 9 - 36n.
Answer:
n=1/4
Step-by-step explanation:
9(1-4n)
Distribute the 9 to create the equation 9-36n.
Now set it equal to zero.
9-36n = 0
Add 36n to both sides to get 9 = 36n
Now divide by sides by 36 to get 9/36 = n
Now simplify your fraction 9/36 = 1/4 so 1/4=n
100%
At 11:00 A.M. the temperature was 37 °F. At 11:00 P.M. the temperature was -6 °F.
What was the change in temperature between 11:00 A.M. and 11:00 P.M.?
A. 6°F
OB. 31 °F
C. 37 °F
OD. 43 °F
The change in temperature between 11.00am and 11.00pm would be = 43°F. That is option D.
What is temperature?Temperature is defined as the scalar quantity that shows or measure the degree of hotness or coldness of a medium.
The temperature of a medium at 11.00am = 37°F
The temperature of a medium at 11.00 pm = -6°F
The difference between the both given temperature = 37-(-6) = 37+6 = 43°F.
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Fifty percent of the customers who go to ABC Tires for tires buy four tires and 32% buy two tires. Moreover, 13% buy fewer than two tires, with 6% buying none. a. What is the probability that a customer buys three tires? (Round your answer to 2 decimal places.) b. Construct a cumulative probability distribution for the number of tires bought. (Round your answers to 2 decimal places.) x P(X ≤ x) 0 1 2 3 4
The probability of buying 3 tires is 0.05 and the cumulative probability distribution table is created.
What is meant by probability?
The area of mathematics known as probability deals with numerical representations of the likelihood that an event will occur or that a statement is true. An event's probability is a number between 0 and 1, where, roughly speaking, 0 denotes the event's impossibility and 1 denotes certainty.
Given,
Percentage of people who buys 4 tires = 50%
Percentage of people who buys 2 tires = 32%
Percentage of people who buys less than 2 tires = 13%
Percentage of people who buys none = 6%
a) Prob(X=0) = 0.06
Prob(X=2) = 0.32
Prob(X=4) = 0.50
Prob(X < 2) = Prob(X=0) + prob(X=1) = 0.06+ prob(X=1)
prob(X=1) = Prob(X < 2) - 0.06 = 0.13 - 0.06 = 0.07
Therefore prob(X = 3) = 1 - ( 0.5+0.32+0.07+0.06) = 1 - 0.95 = 0.05 = 5%
b) Cumulative probability distribution is as given in the figure.
Therefore the probability of buying 3 tires is 0.05 and the cumulative probability distribution table is created.
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consider the matrix below. are the column vectors of this matrix linearly independent or linearly dependent? show all explanations or computations.| 1 0 5 || -2 1 -6 || 0 2 8 |
The column vectors of this matrix are linearly independent, meaning they are not dependent upon one another, and the determinant is not equal to 0.
The given matrix is:
| 1 0 5 |
| -2 1 -6 |
| 0 2 8 |
We must compute the matrix's determinant in order to determine if the matrix's column vectors are linearly independent or linearly dependent.
A scalar value that can be used to assess linear dependency or independence is the determinant of a matrix.
The determinant of the matrix is:
det(A) = 1(-68) - 0(18) + 5(-2*2) = -6 + 0 - 20 = -26
Since The column vectors of this matrix are linearly independent, meaning they are not dependent upon one another, and the determinant is not equal to 0.
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