Answer: A, B, E
Step-by-step explanation:
For A:
None of the factors have one same factor they can be divided by besides 1.
For B:
Both 20a and 24b can be divisible by 1,2,4.
For E:
Both 20a and 15ab are divisible by 1,5 and a.
Total points in a board game based on the average number of hours played is predicted using the least-square regression line, ŷ = 280. 3 + 85. 7x, where ŷ represents the total points scored and x represents the average number of hours spent playing the game. A player plays the game for an average of 3 hours and has a point total of 550. What is the residual for the player?
As per the given least-square regression line the residual for the player is 12.6.
The term least square regression line in math is defined as the line that makes the vertical distance from the data points to the regression line as small as possible.
Here we have the least square regression line as y = 280. 3 + 85. 7x.
As we all know that the residual is given by the difference between the observed point and the predicted point.
While we looking into the given problem the expected score over x hours is written as,
=> f(x) = 280.3 + 85.7x.
Here we need to find the residual for the player, then we have to apply the value of x as 3, then we get,
=> f(3) = 280.3 + 85.7 x 3 = 537.4.
As here we know that the observed score was of 550, hence the residual is calculated as,
=> R = 550 - 537.4 = 12.6.
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Help with this problem please
The function of u that takes the input u and gives output as v is g(u) = - (8v -2)/ 12.
What are functions?A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.
Given that the equation that takes the input and output value to give the function is:
-12u + 3 = 8v + 1
Isolate the value of u.
Subtract 3 from both sides of the equation:
-12u + 3 - 3= 8v + 1 -3
-12u = 8v -2
Divide both sides of the equation with -12.
-12u / -12 = 8v - 2/ -12
u = - (8v -2)/ 12
Hence, the function of u that takes the input u and gives output as v is g(u) = - (8v -2)/ 12.
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Write 7Y equals 3/8x -1
in standard form
Answer:[tex]-\frac{3}{8} x +7y=-1[/tex]
Step-by-step explanation:
Standard form is ax+by=c, where a,b,c are coefficients.
We are given [tex]7y=\frac{3}{8} x-1[/tex]. All we have to do is move some terms around so that the X is on the left side.
[tex]7y=\frac{3}{8} x-1[/tex] [subtract both sides by 3/8x]
[tex]-\frac{3}{8} x +7y=-1[/tex]
This gives out final answer of [tex]-\frac{3}{8} x +7y=-1[/tex].
A small college has 800 students, 10%, percent of which are left-handed. Suppose they take an SRS of 30 students. Let X= equals the number of left-handed students in the sample.
What is the probability that less than 7 of the 30 students are left-handed?
.
The probability that less than 7 of the 30 students are left-handed is given as follows:
0.9742 = 97.42%.
What is the binomial distribution formula?The mass probability formula, giving the probability of x successes, is of:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are given by:
n is the number of trials of the experiment.p is the probability of a success on a single trial of the experiment.The parameter values for this problem are given as follows:
n = 30, p = 0.1.
The probability that less than 7 of the 30 students are left-handed is given as follows:
P(X < 7) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6).
Using a binomial distribution calculator with the parameter, finding each of the probabilities and adding them, we have that:
P(X < 7) = 0.9742 = 97.42%.
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write the equations described below. a subtraction equation with one variable that has a solution of 2/3.
Answer: SOME1 ANSWER QUICK
Step-by-step explanation:
The sum of 500 consecutive integers is 250.
what is the product of the same 500 numbers?
The required product of the same 500 consecutive integers whose sum is equal to 250 is zero.
Let us consider consecutive integers represented by x, x+ 1, x+ 2 , ..... so on.
Sum of 500 integers = 250
x + x + 1 + x+ 2 + ....+ x + 499 = 250
⇒ 500x + 1 + 2 + ...+499 = 250
⇒ 500x + ( 500 × 499 )/2 = 250
⇒ 500x + 250 × 499 = 250
⇒ 250 ( 2x + 499 ) = 250
⇒ 2x + 499 = 1
⇒x = -249
Required 500 numbers are :
-249 , -248 , -247 , ......, 0, 1, ....... 250
Here zero is also a part of 500 integers.
Product of 499 number with zero is zero.
⇒( -249 ×-248 ×.....× 250 ) × 0 = 0
Therefore, the product of 500 consecutive integers with sum 250 is equal to zero.
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equation of the line that is parallel to x-3y=9 and passes through the point (-10,9)
keeping in mind that parallel lines have exactly the same slope, let's check for the slope of the equation above
[tex]x-3y=9\implies -3y=-x+9\implies y=\cfrac{-x+9}{-3} \\\\\\ y=\cfrac{-x}{-3}+\cfrac{9}{-3}\implies y=\cfrac{1}{3}x-3\qquad \impliedby \begin{array}{|c|ll} \cline{1-1} slope-intercept~form\\ \cline{1-1} \\ y=\underset{y-intercept}{\stackrel{slope\qquad }{\stackrel{\downarrow }{m}x+\underset{\uparrow }{b}}} \\\\ \cline{1-1} \end{array}[/tex]
so we're really looking for the equation of a likne whose slope is 1/3 and it passes through (-10 , 9)
[tex](\stackrel{x_1}{-10}~,~\stackrel{y_1}{9})\hspace{10em} \stackrel{slope}{m} ~=~ \cfrac{1}{3} \\\\\\ \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{9}=\stackrel{m}{ \cfrac{1}{3}}(x-\stackrel{x_1}{(-10)}) \implies y -9= \cfrac{1}{3} (x +10) \\\\\\ y-9=\cfrac{1}{3}x+\cfrac{10}{3}\implies y=\cfrac{1}{3}x+\cfrac{10}{3}+9\implies {\Large \begin{array}{llll} y=\cfrac{1}{3}x+\cfrac{37}{3} \end{array}}[/tex]
Which of the following is the equation of a line parallel to 3y = 6x + 5 that passes through (3, 3)? A. y = 2x - 1 B. y = 2x - 3 C. y + 2x = 1 D. y + 3 = 6x
First, we should find the slope of the line we're starting with.
3y = 6x + 5 can be put into slope-intercept form by dividing both sides by 3.
y = 2x + 5/3
The slope of this line is 2.
A parallel line has to have a slope of 2 as well, so we know we're looking for a line with a slope of 2.
Options A and B have that. Options C and D do not.
Now if (3,3) is a point on the line, then (3,3) must also be a solution for the equation.
Checking Option A:
3 = 2(3) - 1 is not true. 3 ≠ 6 - 1
Checking Option B:
3 = 2(3) - 3 is true. 3 = 6 - 3
Option B is the answer, since it has the right slope and works for the point (3,3).
Answer:
B) y = 2x - 3
Step-by-step explanation:
3y = 6x + 5 To put in the slope intercept form. Divide all the way through by 3
y = 2x + 5/3
When lines are parallel, they have the same slope.
So the slope will be 2. We will use the point to find the y intercept
m = 2
x = 3 This is from the point (3,3)
y = 3 this is from the point (3,3)
y = mx + b
3 = 2(3) + b
3 = 6 + b Subtract 6 from both sides
3-6 = 6- 6 + b
-3 = b
Now that we have the slope (2) and the y intercept (-3) we can write the equation.
y = mx + b
y = 2x -3
Charlie tested three cameras last weekend. The table below shows the number of pictures taken and how much time it took.
Answer: First camera: 5 Pictures per second, Second camera: 6 Pictures per second. Third camera: 7 Pictures per second. The Third camera is the fastest.
Step-by-step explanation: 40/8 = 5, 30/5 = 6, 28/4 = 7
Find the surface area of a square pyramid with side length 2 m and slant height 3 m.
Answer: The surface area of a square pyramid with a side length of 2m and slant height of 3m is 12 square meters.
Step-by-step explanation:
The surface area of a square pyramid can be calculated using the formula:
Surface area = Base area + 4 * (Area of one of the lateral faces)
The base area of the pyramid is the area of the square base, which is the side length squared: 2m * 2m = 4m^2
To find the area of one of the lateral faces, we can use the Pythagorean theorem to find the length of the face, which is the hypotenuse of a right triangle with the slant height and half of the side length as the other two sides.
so, (1/2 * 2m)^2 + 3^2 = x^2
x= (1/2 * 2m)^2 + 3^2
x = 2
Therefore, the surface area of the pyramid is:
Surface area = 4m^2 + 4 * 2m^2 = 4m^2 + 8m^2 = 12m^2
So, the surface area of a square pyramid with a side length of 2m and a slant height of 3m is 12 square meters.
The head of a hammer is located 9 cm to the left of its
center of mass, while the bottom of the handle of the
hammer is 29 cm to the right of the center of mass. Determine the length of the hammer. Justify your
answer using an appropriate geometry principle (definition, theorem, etc.)
The length of the hammer is given as follows:
30.36 cm.
How to obtain the length of the hammer?The Pythagorean Theorem states that for a right triangle, the length of the hypotenuse squared is equals to the sum of the squared lengths of the sides of the triangle.
The sides in this problem are given as follows:
9 cm and 29 cm.
The length of the hammer is the hypotenuse, hence it is obtained as follows:
l² = 9² + 29²
[tex]l = \sqrt{9^2 + 29^2}[/tex]
l = 30.36 cm.
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The total cost of producing a type of car is given by C(x)=21000−90x+0.1x2, where x is the number of cars produced. How many cars should be produced to incur minimum cost? Please explain how you got the answer, THANKS!
Answer: To find the number of cars that should be produced to incur minimum cost, we need to find the minimum value of the cost function C(x). Since C(x) is a quadratic function, it will have a minimum value, which is either a relative minimum or an absolute minimum. To find the number of cars that should be produced to incur the minimum cost, we will find the derivative of C(x) and set it equal to zero.
To find the derivative of C(x), we can use the power rule, which states that the derivative of x^n is n*x^(n-1).
The derivative of C(x) = 21000 - 90x + 0.1x^2 is
C'(x) = -90 + 0.2x
To find the minimum cost, we set C'(x) equal to zero and solve for x:
-90 + 0.2x = 0
0.2x = 90
x = 450
So, producing 450 cars will incur the minimum cost.
To verify that this is indeed a minimum, we can find the second derivative of C(x) which is 0.2, since the second derivative is positive, we can confirm that x = 450 is a relative minimum.
Alternatively, we can substitute x = 450 into the cost function and find the minimum value, which is C(450) = 21000 - 90(450) + 0.1(450)^2, which is less than any other value of C(x) for x≠450.
Therefore, to incur the minimum cost, 450 cars should be produced.
Step-by-step explanation:
An air traffic controller is taking two planes. To start plane a is at an altitude of 2457 feet and plane B is just taking off. Plane eight is gaining altitude at 25.25 ft./s in plane, B is gaining altitude at 70.75 ft./s, how many seconds will pass before the planes are at the same altitude and what will they are out of to be when they’re at that same altitude
Answer: To find the number of seconds it takes for the planes to reach the same altitude, we can use the following formula:
time = (difference in altitude) / (rate of plane B - rate of plane A)
In this case, the difference in altitude is the altitude of plane A (2457 feet) and plane B is just taking off so its altitude is 0 feet.
So, substituting these values into the formula:
time = (2457) / (70.75 - 25.25)
time = 2457 / 45.5
time = 53.7 seconds
So it takes 53.7 seconds for the planes to reach the same altitude. To find out what altitude they will be at when they reach the same altitude, we can use the formula:
Altitude = (rate of plane A x time) + altitude of plane A
Substituting the values we have:
Altitude = (25.25 x 53.7) + 2457
Altitude = 1351.6 + 2457
Altitude = 3809.6 ft
So, both planes will be at 3809.6 ft. when they reach the same altitude.
Step-by-step explanation:
a certain family has 6 children, consisting of 3 boys and 3 girls. assuming that all birth orders are equally likely, what is the probability that the 3 eldest children are the 3 girls?
The probability that the 3 eldest children are the 3 girls is 0.05.
Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.
To determine the probability, we can use the permutation.
A permutation is an arrangement of a set of objects in a particular order. It is a way of counting the number of ways that a group of items can be arranged in a specific order.
There are a total of 6! ways in which the 6 children can be ordered. If the first 3 children are girls, there are 3! ways to arrange the 3 girls, and 3! ways to arrange the 3 boys. Therefore, the probability that the 3 eldest children are the 3 girls is
(3! x 3!) / 6! = (6 x 6) / 720 = 0.05
Therefore, there is a 0.05 chance or probability that the 3 eldest children are the 3 girls.
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25 POINTS!! ... Guessers shall be reported!
Q: Write a set of all natural numbers which divide 24 in tabular and set-builder form.
Set of all natural numbers which divide 24 in tabular form {1,2,3,4,6,8,12,24} and set-builder form is {x ∈ ℕ | x ∣ 24} ℕ denotes the set of natural numbers and x ∣ 24 means x divides 24.
What do math natural numbers mean?Normative Data We utilize the numerals 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11 to count or arrange objects in a particular order. Complete numbers are those that fall between the zero and artificial number range. not a decimal or fraction.
Integers make up all natural numbers, right?Decimal numbers and fractions are not integers. Although all natural whole and numbers are integers (as well as all whole numbers), not all integer are whole or natural numbers. 5 is an integer even though it is not a whole number or a natural number.
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a hotel chain, with 3 locations, wants to survey customers who stayed at their facilities over the last year to determine customer satisfaction profiles. they plan to use a sample of customers who visited the hotel chain over the last year. which of the below sampling methods describes a stratified sampling method?
It describes the Cluster Sampling method.
It is a cluster sampling method because customers are divided into clusters at random and no personal judgment is involved. Thus, each cluster has an equal probability of selection, and sampling is done randomly.
Cluster sampling involves splitting the population into subgroups, but each subgroup should have similar properties to the whole sample.
Cluster sampling is a probability sampling method in which we divide a population into clusters, such as districts or schools, and then randomly select some of these clusters as our sample.
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The graphs of lines k1 and k2 are shown on the grid. Which system of equations is best represented by this graph?
F. 3x-y=2
4x+9y=36
G. 3x-y=6
4x+9y=4
H. x-3y=-18
9x+4y=9
J. x+y=10
9x+4y=13
3x-y=2 and 4x+9y=36 make up the graph equation in option F. next we receive [tex]x=\frac{2+y}{3}$$[/tex] and [tex]x=\frac{36-9 y}{4}$$[/tex] .
Which is then graph equation ?solve for [tex]$x, 3 x-y=2 \quad: \quad x=\frac{2+y}{3}$[/tex]
Steps
3 x-y=2
Turn y to the right.
3 x=2+y
Divide both sides by 3
[tex]\frac{3 x}{3}=\frac{2}{3}+\frac{y}{3}$$[/tex]
Simplify
[tex]x=\frac{2+y}{3}$$[/tex]
solve for x, 4 x+9 y=36: x=[tex]\frac{36-9 y}{4}[/tex]
Steps
4 x+9 y=36
Turn 9 y to the right.
4 x=36-9 y
By dividing both sides by 4,
[tex]\frac{4 x}{4}=\frac{36}{4}-\frac{9 y}{4}$$[/tex]
Simplify
[tex]x=\frac{36-9 y}{4}$$[/tex]
Therefore the correct answer is option F ) 3x-y=2 , 4x+9y=36 .
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Nicole is buying dog nuts for the bus ride to cheer competition. Plain glazed donuts are $4 per box and chocolate cake donuts cost $5 per box. She spends a total of $91 for 20 boxes. How many boxes of each kind of donut did she buy?
Answer: We can start solving this problem by using a system of equations. Let x be the number of plain glazed donuts Nicole buys, and y be the number of chocolate cake donuts she buys. We know that:
x + y = 20 (The total number of boxes of donuts she buys is 20)
4x + 5y = 91 (The total amount of money she spends is $91)
We can use the first equation to solve for one variable in terms of the other. For example, we can substitute x = 20 - y into the second equation:
4(20 - y) + 5y = 91
80 - 4y + 5y = 91
-4y + 5y = 91 - 80
y = 11
So Nicole bought 11 boxes of chocolate cake donuts. We can substitute that back into the first equation to find the number of plain glazed donuts she bought:
x + 11 = 20
x = 9
So Nicole bought 9 boxes of plain glazed donuts.
We can check our answer by substituting our values of x and y back into the original equations:
x + y = 9 + 11 = 20 (This is the total number of boxes of donuts she bought, which is 20)
4x + 5y = 4(9) + 5(11) = 91 (This is the total amount of money she spent, which is $91)
Both equations are true, so our solution is correct.
Step-by-step explanation:
2.4-1.2x=20.5-(3x+10)
Answer:
x=4.5
Step-by-step explanation:
Isolate the variable by dividing by each side by factors that don't contain the variable.
The highest point of a pool slide is 10 feet. The lowest point of the deep end of the pool is 8 feet below water level. A height of 0 is the water level.
(A) Give an expression involving absolute value and addition that represents the vertical distance between the top of the slide and the bottom of the deep end of the pool.
(B) Find the value of the expression in (a)
10-8 is the expression involving absolute value and addition that represents the vertical distance between the top of the slide and the bottom of the deep end of the pool.
What is Expression?An expression is combination of variables, numbers and operators.
Given that the highest point of a pool slide is 10 feet.
The lowest point of the deep end of the pool is 8 feet below water level.
A height of 0 is the water level.
10-8 is the expression involving absolute value and addition that represents the vertical distance between the top of the slide and the bottom of the deep end of the pool.
10-8=2, 2 is the value of the expression in (a)
Hence, 10-8 is the expression involving absolute value and addition that represents the vertical distance between the top of the slide and the bottom of the deep end of the pool.
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The function f(x)=3x+1 is shown above. Complete each of the following statements by dragging the best choice from the list at the bottom of the page. Some answers will remain after you have completed each statement.
Each of the statements should be completed as follows;
The base value for this function is 3.The function f(x) = 3^x + 1 is a growth function. The point (0, 2) is the y-intercept of the graph. The purple line on the graph is a horizontal asymptote and has an equation of 1. The domain of the function is all real numbers. The range of the function is all the real numbers >1.What is a horizontal asymptote?In Mathematics, a horizontal asymptote simply refers to a horizontal line (y = b) where the graph of a function approaches the line as the input values approach negative infinity (-∞) to positive infinity (∞).
In this context, the purple line on the graph of this exponential growth function represents a horizontal asymptote and it has an equation of y = 1.
Additionally, the base value of this exponential growth function f(x) = 3^x + 1 is equal to 1 and its domain includes all real numbers while its range includes all real numbers that are greater than 1 i.e 1 < y < ∞.
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Complete each statement about angles by choosing from the drop-down menus:
The terminal side of a 130° angle in standard position lies in the ... quadrant
Two angles that are coterminal to 130° are ... and ...
The terminal side of a 130° angle in standard position lies in the second quadrant. Two angles that are co-terminal to 130° are 490° and -230°.
What is an angle measure?When two lines or rays intersect at a single point, an angle is created. The vertex is the term for the shared point. An angle measure in geometry is the length of the angle created by two rays or arms meeting at a common vertex.
Given angle is 130°.
The first quadrant, second quadrant, third quadrant, and fourth quadrant make up the whole graph plane.
Angles in the first quadrant vary from 0° to 90°.
Angles in the second quadrant vary from 90° to 180°.
The angles in the third quadrant can be found between 180° and 270°.
Angles in the fourth quadrant can be found between 270° and 360°.
Therefore, the plot's beginning side is at zero degrees and its terminal side is at 130° in the second quadrant, since the specified angle is 130°.
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Answer: 2nd, -230, 490
Step-by-step explanation:
Edge 2023
PLSSSS IF YOU TURLY KNOW THISSS
For given linear equation x will be 5.
What is a linear equation ?
A linear equation is an equation in which the highest power of the variable is always 1. It is also known as a one-degree equation. The standard form of a linear equation in one variable is of the form
Ax + B = 0
e.g. x-10=0. Here, x is a variable, A is a coefficient and B is constant.
The standard form of a linear equation in two variables is of the form
Ax + By = C
e.g. 2x-4y=10. Here, x and y are variables, A and B are coefficients and C is a constant.
Now,
As given linear equation is
15-x=10
x=15-10
x=5
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How many groups of 1/8 are in 15/4
Answer:
30 groups
Step-by-step explanation:
We can first set 15/4 to have the same denominator as 1/8, which will be 30/8.
Since the question asks for the number of groups of 1/8, 30/8 will consist of 30 groups of 1/8.
a person weighing 180 pounds stands on snowshoes if the total area of the 2 snowshoes is 900 square inches, what is the total pressure on the snow
The pressure on the snow on the boy is P=1379.83 N.
A person weighing 180 pounds stands on snowshoes.
If the total area of the 2 snowshoes is 900 square inches
The mass of person on the snowshoes is given by,
m=180 pounds=180×0.454=81.6455≅81.65 kg
Force acting on a body due to gravity is given by, f = mg
Where f is the force acting on the body, g is the acceleration due to gravity, m is mass of the body.The weight force of the person is given by,
W=F=mg=81.65×9.81=800.99N.The area of the snowshoes is given by,
⇒A=900 square inches
⇒1inch=0.025 m
Thus, Area in [tex]m^{2}[/tex] given by A=900×[tex]0.0254^{2}[/tex]
=900×0.000645
=0.5805[tex]m^{2}[/tex]
The expression of the pressure in the fluid containing tube is given by FA
; where F is the force exerted by the liquid and A is the area of the tube.
From the expression of pressure, the resulting pressure value is given by FA.
=800.900.5805
=19.82≈19.8pa
The pressure on the snow on the boy is =1379.83 N.
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a license plate begins with two letters. if the possible letters are A, B, C, D, and E, how many different ways of these letters can be made of if no letter is used more than once?
Answer:
20
Step-by-step explanation:
A, B, C, D, E is a set of 5 letters.
A letter can only be used once.
You have 5 choices for the first letter.
5
Since the letter that was selected to go first cannot be used again, now you have 4 choices for the second letter.
5 × 4
5 × 4 = 20
Answer: 20
A school district signs a contract to purchase 50 of the electric typewriters (normally selling for $129. 95) from ABC at a quantity discount of 18%. What price does the school district pay for each typewriter?
The price paid by school district for each typewriter after applying 18% quantify discount is equal to $106.56.
Regular selling price of each typewriter = $129.95
Number of typewriter required by school district = 50
Total cost of 50 type writers = $129.95 × 50
= $6497.5
Discount percent on whole quantity = 18%
18% of $6497.5
= ( 18 / 100 ) × $6497.5
= $1169.55
Total price paid by school district for 50 typewriters
= $6497.5 - $1169.55
= $5327.95
Price paid by school district for 1 typewriters
= $5327.95 / 50
= $106.559
= $106.56
Therefore, the price paid school district for each typewriter is equal to
$106.56.
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You are buying an old "go to work" pickup with a price tag of $1,600. You apply a down payment of $200. 0. Sales tax is 6. 5%. You will have to borrow $1,504 at 8% interest for one year to complete the purchase. What will your full purchase price be with principal, tax and interest
For the given case the full purchase price will be $1,628.92,Principal will be $1,600, Tax will be $101.60,Interest will be $27.32 and Total is $1,728.92.
To calculate the full purchase price, first, you must calculate the sales tax. To do this, multiply the price of the pickup ($1,600) by the sales tax rate (6.5%), which equals $101.60. Next, you must calculate the interest. To do this, multiply the amount borrowed ($1,504) by the interest rate (8%) and the number of years (1), which equals $27.32.
Finally, you must add the principal ($1,600), the sales tax ($101.60), and the interest ($27.32) to get the full purchase price of $1,728.92. Once you subtract the down payment of $200, the total purchase price is $1,628.92.
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need answer asap ! thank you to anyone who helps <3 !
AB = BA, the first pair of matrices does not satisfy [tex]AB \neq BA[/tex].
Is AB BA correct for all matrices?AB = BA in general, even if A and B are both square. We say that A and B commute if AB = BA. We cannot argue that AB = AC provides B = C for a generic matrix A. (However, since A is invertible, we may multiply both sides of the equation AB = AC to the left by A1 to yield B = C.)
For matrices, A and B, the product AB is not necessarily equal to BA. In fact, matrix multiplication is not commutative in general. Therefore, we need to check each pair of matrices to see if their products are equal in order to determine which pair satisfies [tex]$AB \neq BA$[/tex].
[tex]$$\begin{align*}AB &= \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] \\\&= \left[\begin{array}{cc}1 \cdot 7 + 0 \cdot 3 & 1 \cdot 0 + 0 \cdot 4 \ 3 \cdot 7 - 2 \cdot 3 & 3 \cdot 0 - 2 \cdot 4\end{array}\right] \\\&= \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]\end{align*}[/tex]
[tex]$$\begin{align*}BA &= \left[\begin{array}{cc}7 & 0 \ 3 & 4\end{array}\right] \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \\\&= \left[\begin{array}{cc}7 \cdot 1 + 0 \cdot 3 & 7 \cdot 0 + 0 \cdot (-2) \ 3 \cdot 1 + 4 \cdot 3 & 3 \cdot 0 + 4 \cdot (-2)\end{array}\right] \\\&= \left[\begin{array}{cc}7 & 0 \ 15 & -8\end{array}\right]\end{align*}[/tex]
Since AB = BA, the first pair of matrices does not satisfy [tex]AB \neq BA[/tex]. We can proceed in the same manner to check the other pairs of matrices:
[tex]$$\begin{align*}\\AB &= \left[\begin{array}{cc}1 & 0 \ 3 & -2\end{array}\right] \left[\begin{array}{cc}8 & 0 \ 11 & -3\end{array}\right] \\\&= \left[\begin{array}{cc}1 \cdot 8 + 0 \cdot 11 & 1 \cdot 0 + 0 \cdot (-3) \ 3 \cdot 8 - 2 \cdot 11 & 3 \cdot 0 - 2 \cdot (-3)\end{array}\right] \\\&= \left[\begin{array}{cc}8 & 0 \ 2 & 6\end{array}\right]\end{align*}[/tex]
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How do I evaluate the integral by interpreting it in terms of areas?
The process of evaluate the integral by interpreting it in terms of areas is by apply the summation rule on it.
The term called integral is defined as a mathematical object that can be interpreted as an area or a generalization of area.
Here we need to find the way to evaluate the integral by interpreting it in terms of areas.
As we all know that the integration symbol ∫ is an elongated S, suggesting sigma or summation.
Here on a definite integral, we have to use the above and below the summation symbol are the boundaries of the interval, [a, b].
Where the numbers a and b are x-values and are called the limits of integration and it specifically a is the lower limit and b is the upper limit.
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