Find the point on the plane x+y+z= -26 that is closest to the point (1,1,1).
Answers
The point on the plane x+y+z= -26 that is closest to the point (1,1,1) is approximately (-31.79, 30.79, -0.62).
Why is point on the plane x+y+z= -26?
The point on the plane x+y+z= -26 that is closest to the point (1,1,1), we need to use the formula for the distance between a point and a plane.
First, we can rewrite the equation of the plane as z = -x - y - 26. This allows us to use the coordinates of the point (1,1,1) to find the perpendicular distance between the point and the plane.
The formula for the distance between a point (x0,y0,z0) and a plane ax+by+cz+d=0 is:
distance = |ax0 + by0 + cz0 + d| / sqrt(a^2 + b^2 + c^2)
In this case, a = b = c = 1 and d = -26. Plugging in the values, we get:
distance = |1(1) + 1(1) + 1(1) - 26| / sqrt(1^2 + 1^2 + 1^2)
distance = 24 / sqrt(3)
Now we need to find the point on the plane that is closest to (1,1,1). This point will lie on the line perpendicular to the plane that passes through (1,1,1).
To find the direction vector of this line, we can take the cross product of the normal vector of the plane (1,1,1) and the vector (1,1,1) - (-26,0,0) that points from a point on the plane to the point (1,1,1):
(1,1,1) x (27,-1,-1) = (0,-28,28)
This gives us a direction vector of (0,-1,1) for the line. We can use this vector and the distance we found earlier to find the point on the plane that is closest to (1,1,1).
Starting at (1,1,1), we can move a distance of (distance / sqrt(2)) in the direction of the vector (0,-1,1) to get to the closest point on the line. This is because the vector (0,-1,1) has a length of sqrt(2), so we need to scale it down by that factor to get the right distance.
The closest point on the plane is then the point where the line intersects the plane. We can find this by plugging in the coordinates of the closest point on the line into the equation of the plane:
x + y + z = -26
1 - (distance / sqrt(2)) - (distance / sqrt(2)) = -26
x = -25 - (distance / sqrt(2))
y = 26 - (distance / sqrt(2))
z = -1 + (distance / sqrt(2))
Plugging in the value of distance that we found earlier, we get:
x = -25 - (24 / sqrt(6))
y = 26 - (24 / sqrt(6))
z = -1 + (24 / sqrt(6))
So the point on the plane x+y+z= -26 that is closest to the point (1,1,1) is approximately (-31.79, 30.79, -0.62).
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Related Questions
Brian is going to buy his first home for $220,000. He has to put $22,000 down. He needs to spread the payments over 180 months to get a 3. 5% fixed annual interest. What is the best estimate of the monthly payment?
Answers
The estimated monthly payment for Brian's home loan would $1413.52.
To calculate the best estimate of the monthly payment for Brian's home loan, we can use the formula for calculating a fixed-rate mortgage payment.
The formula for the monthly payment of a fixed-rate mortgage is:
M = P * (r * (1+r)^n) / ((1+r)^n - 1),
where M is the monthly payment, P is the principal amount (loan amount minus down payment), r is the monthly interest rate, and n is the number of monthly payments.
In this case, Brian's principal amount is $220,000 - $22,000 = $198,000 (loan amount minus down payment). The monthly interest rate is 3.5% / 12 = 0.0029 (converted from annual interest rate to monthly rate), and the number of monthly payments is 180.
Plugging in these values into the formula, we get:
M = $198,000 * (0.0029 * (1+0.0029)^180) / ((1+0.0029)^180 - 1).
M = $198,000 * (0.0029 * (1+0.0029)^180) / ((1+0.0029)^180 - 1)
M = $1413.52
Evaluating this expression gives us the best estimate of the monthly payment. However, the calculation involves complex mathematical operations, and it is recommended to use a financial calculator or spreadsheet software to obtain an accurate estimate.
Note: It's important to consult with a financial advisor or a mortgage professional to get precise and personalized information about mortgage terms, interest rates, and monthly payments based on individual circumstances and loan options available.
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What are the solutions of the equation 2sin (4x) = 1 on the interval [0,π)? Please show work.
Answers
Answer:
Step-by-step explanation:2sin (4x) = 1 divide both sides by 2. sin (4x) = 1/2. Note that sin (x) = 1/2 at pi/6, 5pi/6, 13pi/6, 17pi/6.
Which addition is the model below?
Answers
Answer:
not sure if the top is negative or positive but;;
if the top is negative then the answer is F
Step-by-step explanation:
Where is infinity and negative infinity on a graph?.
Answers
The positive infinity and negative infinity on a graph is explain below.
What is infinity?the idea of something limitless, interminable, without bound. The normal sign of infinity, ∞, was concocted by the English mathematician John Wallis in 1655. Three principal sorts of vastness might be recognized: the numerical, the physical, and the supernatural
According to question:Positive infinity is shown by rightmost point on the graph.Its symbol is +∞.
Negative infinity is shown by leftmost point on the graph.Its symbol is -∞.
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Figure abcd is a trapezoid with point A (0,-4) what rule would rotate the figure 90° counterclockwise
Answers
The rotated trapezoid ABCD is:A' (0, 4)B' (-2, -3)C' (-2, -2)D' (0, 3)
To rotate the figure 90° counterclockwise, the rule is to swap the x and y-coordinates and negate the new x-coordinate.
This is also known as a clockwise rotation of 270 degrees.
A trapezoid is a geometric shape that is four-sided and has only one pair of parallel sides.
It is also known as a trapezium in British English.
A line that connects the non-parallel sides is known as a diagonal.
A trapezoid with point A (0, -4) can be rotated 90 degrees counterclockwise about the origin (0, 0) using the rule of rotating x and y coordinates.
This rule can be expressed in the following manner: (x, y) -> (-y, x)Where x is the original x-coordinate and y is the original y-coordinate.
This rule is known as a counter-clockwise rotation of 90 degrees. When using this rule, you can create a new coordinate set by replacing x with -y and y with x.
In order to find the new coordinates of the trapezoid after a 90° counterclockwise rotation, you can follow these steps:
Substitute x with -y and y with x.
A (-4, 0) becomes A' (0, 4).Substitute x with -y and y with x.
B (-3, 2) becomes B' (-2, -3).Substitute x with -y and y with x.
C (2, 2) becomes C' (-2, -2).Substitute x with -y and y with x.
D (3, 0) becomes D' (0, 3).
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Can you help me
11π+2x=19π
Answers
The solution to the equation 11π + 2x = 19π is x = 4π
How to determine the solution to the equationFrom the question, we have the following parameters that can be used in our computation:
11π+2x=19π
Express the equation properly
So, we have the following representation
11π + 2x = 19π
Subtract 11π from both sides of the equation
This gives
11π - 11π + 2x = 19π - 11π
Evaluate the like terms
2x = 8π
Divide both sides by 2
x = 4π
Hence, the solution is x = 4π
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the sum of two numbers times a third is equal to the sum of each of the added numbers times the third
Answers
Distributive Property of real numbers states the following statement
a( b + c ) = ab + bc
What do you mean by Distributive Property?The distributive property is a property of multiplication used in addition and subtraction. This property states that two or more terms in addition or subtraction with a number are equal to the addition or subtraction of the product of each of the terms with that number.
For example:
1. 5 x ( 3 + 8 ) = 5x3 + 8x5
2. 10 x ( 9 - 8 ) = 10x9 - 10x8
What is Associative Property?The associative property is a math rule that says that the way in which factors are grouped in a multiplication problem does not change the product.
For example:
1. 5 x ( 4 x 2 ) = ( 5 x 4 ) x 2 = ( 5 x 2 ) x 4 = 40
firstly we are grouping 4 and 2 together and multiplying them first then multiply it by 5.
secondly we are grouping 5 and 4 together and multiplying them first then multiply it by 2.
thirdly we are grouping 5 and 2 together and multiplying them first then multiply it by 4.
The basic difference between Distributive Property and Associative Property.
Hence, Distributive Property of real numbers states the following statement: a( b + c ) = ab + bc
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Find the missing length
Answers
Answer:
x = 16.64
option A
Step-by-step explanation:
Using the pythagoreom therom
Answer:
16.64
Step-by-step explanation:
Since it's a right angled triangle, we use Pythagoras theorem:
(Hypotenuse)² = (base)² + (perpendicular)²
x² = 14² + 9²
x² = 196 + 81
x² = 277
√x² = √277
x = 16.64
Write a polynomial that represents the area of the shaded region
Answers
=========================================
Explanation:
The larger rectangle has area of (x+1)(x+1) = x^2+2x+1 through the use of the FOIL rule or distribution
If you use distribution, then it might help to let y = x+1 so we'd have y(x+1) lead to xy+1y which becomes x(x+1)+1(x+1). From there it might be easier to see how to get x^2+2x+1 after everything distributes again and simplifies.
The smaller rectangle has area 5x-35 which is found by distributing 5(x-7)
To get the shaded area, we subtract the two rectangle areas found above
shaded area = (larger area) - (smaller area)
shaded area = (x^2+2x+1) - (5x - 35)
shaded area = x^2+2x+1 - 5x + 35
shaded area = x^2-3x+36
4. the maintenance supervisor of an assembly line has two tool cabinets, one at each end of the assembly line. each morning, she walks from one end of the line to the other, and she is equally likely to begin the walk at either end. in the two tool cabinets are a total of six flashlights. at the beginning of her walk, the supervisor takes a flashlight (if one is available) from the tool cabinet at that location, and at the end of her walk, she leaves a flashlight (if she possesses one) from the tool cabinet at that location. model the movement of flashlights using a discrete-time markov chain
Answers
The matrix represents the probabilities of moving from one state to another.
A discrete-time Markov chain is a mathematical model that describes the probability of transitioning from one state to another in a series of discrete time steps.
In this case, we can model the movement of the flashlights using a Markov chain.
Let's define the states in our model:
State 1: No flashlights in either cabinet
State 2: 1 flashlight in the first cabinet
State 3: 1 flashlight in the second cabinet
State 4: 2 flashlights in the first cabinet
State 5: 2 flashlights in the second cabinet
State 6: 3 flashlights in the first cabinet
State 7: 3 flashlights in the second cabinet
Now, we can create a transition matrix to represent the probabilities of moving from one state to another.
Since the supervisor is equally likely to start at either end, the initial probabilities are:
P(State 1) = 0.5
P(State 2) = P(State 3)
= 0.25
The transition matrix would look like this:
| 0.5 0.25 0 0 0 0 0 |
| 0.5 0.5 0 0 0 0 0 |
| 0 0 0.5 0 0 0 0 |
| 0 0 0 0.5 0.25 0 0 |
| 0 0 0 0 0.5 0 0 |
| 0 0 0 0 0 0.5 0.25 |
| 0 0 0 0 0 0 0.5 |
This matrix represents the probabilities of moving from one state to another.
For example,
P(State 1 to State 2) = 0.5,
P(State 4 to State 5) = 0.25.
By analyzing this Markov chain, we can calculate various probabilities, such as the long-term proportion of time spent in each state or the expected number of flashlights in each cabinet after a certain number of steps.
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christina's pizza sells pizza by the slice for lunch. today, they sold 76 slices, including 20 slices of pepperoni pizza. what is the experimental probability that the first slice sold during lunch tomorrow will be a slice of pepperoni pizza?
Answers
The experimental probability that the first slice sold during lunch tomorrow will be a slice of pepperoni pizza is approximately 0.263
The experimental probability is found by dividing the number of times the desired outcome occurred by the total number of trials. In this case, the desired outcome is selling a slice of pepperoni pizza as the first slice tomorrow.
Since we only know the number of slices sold today, we don't have any specific information about the number of slices that will be sold tomorrow. However, we can assume that the same types of pizzas will be sold tomorrow, and the probability of selling a slice of pepperoni pizza tomorrow will be the same as the proportion of pepperoni slices sold today.
So, the experimental probability that the first slice sold during lunch tomorrow will be a slice of pepperoni pizza is
Experimental probability = Number of pepperoni slices sold today / Total number of slices sold today
Experimental probability = 20 / 76
Experimental probability = 0.263
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What is the slope of the line that passes through the points (-6, 8)(−6,8) and (-16, 33) ?(−16,33)? Write your answer in simplest form.
Answers
Given:
A line passes through the points (-6, 8) and (-16, 33).
To find:
The slope of the line.
Solution:
If a line passes through two points, then the slope of the line is
\(m=\dfrac{y_2-y_1}{x_2-x_1}\)
The line passes through the points (-6, 8) and (-16, 33). Using the above formula, the slope of the line is
\(m=\dfrac{33-8}{-16-(-6)}\)
\(m=\dfrac{25}{-16+6}\)
\(m=\dfrac{25}{-10}\)
\(m=-2.5\)
Therefore, the slope of the line is -2.5.
Slope of the line passing through the points ( -6,8) and (-16,33) will be -2.5.
Given information:
The line passes through the points ( -6,8) and (-16,33).
It is required to find the slope of the line.
Slope of the line passing from two points \((x_1,y_1) \; and\; (x_2, y_2)\) can be written as
\(m= \dfrac{y_2-y_1}{x_2- x_1}\)
So, the slope of the given line can be calculated as,
\(m= \dfrac{33-8}{-16- (-6)}\\m = \dfrac{25 }{-10} \\m = -2.5\)
Therefore the slope of given line will be -2.5.
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the population of metroville is 123,000 and is decreasing by 2.4 each year
a. write a function that models the population as a function of time t
b. predict when the population will be 90,000
Answers
The function is given as P = 123000e^ -0.024t and the time taken is 13 years.
What is the function?We have to know that as we deal with the population that we are dealing with an exponential function. In the exponential function, we can see that the data can be increasing or it can be decreasing as we can se here.
We know that;
P= Poe^-rt
P = population at time t
Po= Initial population
r = rate of decrease
t = time taken
then;
P = 123000e^ -0.024t
The population would 90 000 when;
90000/123000= e^ -0.024t
0.73 = e^ -0.024t
ln0.73 = lne^ -0.024t
ln0.73 = -0.024t
t = ln0.73/-0.024
t = 13 years
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a bookshelf contains 5 different mathematics textbooks, 4 different engineering textbooks, and 2 different history textbooks. how many different ways are there to arrange the books on a shelf? 39916800 how many different ways are there to arrange them if each book is sorted by category (so all of the books of a type are in a row)? 34560 how many different ways are there to arrange them if only the history books must be next to each other? 725760 how many different ways are there to arrange the books if none of the history books may be adjacent to each other?
Answers
Answer:
Step-by-step explanation:
The correct answer is “39916800”. Explanation: No. of mathematics books = 3 No. of Engineering books = 5 No. of history books = 3 Total number of books = 3 + 5 + 3 = 11 Hence, there are 11! ways in which all of these books can be arranged.
I think I may be wrong if someone wants to check this you can
What type of graph is this?
complete graph
cycle graph
Answers
Answer:
complete graph
Step-by-step explanation:
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges
hope it helps :)
please mark it the brainliest!
Answer:
it would be a complete graph
Step-by-step explanation:
Arrange the following measurements in order from smallest to largest.
5 grams 0.35 kilograms 9.4 grams
Answers
Answer:
5 grams, 9.4 grams, 0.35 kilograms
Step-by-step explanation:
9.4 g and 5g are already converted so don't worry about those.
0.35 kilograms translates to 350 grams.
from smallest to largest thatd be
5, 9.4, 350
In 2009, a principal of $3600 was invested at 3.75% interest, compounded annually. let t be the number of years since 2008. let y be the value of investment, in dollars. Write an exponential function showing the relationship between y and t.
Answers
Answer:
3,600 x (1.0375) ^ 1
Step-by-step explanation:
Compund interest formula.
IA * (r + 1) ^t
IA = Initial amount; r = rate (as decimal); t = time (in years)
IA = 3600
r = .0375
t = 1 (2008-2009)
3,600 x (1.0375)^1
-- Gage Millar, Algebra 1/2 Tutor, (Pending Pre-Calc Tutor)
1) 2/9 + 5/9 =
2) 3/4 - 5/16 =
3) 6/7 × 1/2 =
4) Convert the decimal 0.984 to a fraction.
help asap please
Answers
Answer:
1. 7/9
2. 7/16
3. 3/7
4. 123/125
Step-by-step explanation:
1. Simplify the following:
2/9 + 5/9
2/9 + 5/9 = (2 + 5)/9:
(2 + 5)/9
2 + 5 = 7:
Answer: 7/9
_________________________
2. Simplify the following:
3/4 - 5/16
Put 3/4 - 5/16 over the common denominator 16. 3/4 - 5/16 = (4×3)/16 - 5/16:
(4×3)/16 - 5/16
4×3 = 12:
12/16 - 5/16
12/16 - 5/16 = (12 - 5)/16:
(12 - 5)/16
12 - 5 = 7:
Answer: 7/16
_____________________________________________
3. Simplify the following:
6/7×1/2
6/7×1/2 = 6/(7×2):
6/(7×2)
6/2 = (2×3)/2 = 3:
Answer: 3/7
__________________________________________________
4. Rewrite the decimal number as a fraction with 1 in the denominator
0.984=0.984/1
Multiply to remove 3 decimal places. Here, you multiply top and bottom by 10^3 = 1000
(0.984/1)×(1000/1000)= 984/1000
Find the Greatest Common Factor (GCF) of 984 and 1000, if it exists, and reduce the fraction by dividing both numerator and denominator by GCF = 8,
(984/8) = 123
(1000/8) = 125
Therefore
X=123/125
In conclusion,
Answer: 0.984=123/125
can someone help me on this plz
Answers
Find m<1 and m<2
Help
Answers
Answer:
m<1 = 45
m<2 = 135
Step-by-step explanation:
135 is across from the 135 making m1 135 and 135 plus 45 equals 180, 180 is the total for the line the other 2 angles are on.
Bill had two jobs in 2009. one employer paid him $77090 and the other paid him 31,280. each employer took out 6.2% for social security. how much did bill overpay in social security for 2009?
Answers
In 2009, Bill had two jobs and earned $77,090 from one employer and $31,280 from the other. Both employers deducted 6.2% for social security. Bill overpaid in social security by a total of $1,070.
To calculate how much Bill overpaid in social security, we need to determine the amount deducted for social security from each job and then compare it to the actual limit set by the government.
For the first job, the employer deducted 6.2% of $77,090 for social security, which is calculated as $77,090 * 0.062 = $4,780.18. However, there is a limit on the amount subject to social security tax, known as the Social Security Wage Base. In 2009, this limit was $106,800. Since Bill's earnings from the first job exceeded this limit, the maximum social security tax he should have paid from this job is $106,800 * 0.062 = $6,622.40.
For the second job, the employer deducted 6.2% of $31,280 for social security, which is calculated as $31,280 * 0.062 = $1,940.96. Since Bill's earnings from this job were below the Social Security Wage Base, the deduction was accurate.
To find the overpayment, we subtract the actual social security tax Bill should have paid from the total amount deducted. The total deduction was $4,780.18 + $1,940.96 = $6,721.14, and the accurate deduction should have been $6,622.40. Therefore, Bill overpaid by $6,721.14 - $6,622.40 = $98.74 for the first job.
In total, Bill overpaid $98.74 in social security for the first job, while the second job deduction was accurate. Hence, the total overpayment for 2009 is $98.74.
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Figure B is a scaled copy of Figure A.
What is the scale factor from Figure A to Figure B?
Answers
Answer: not sure
Step-by-step explanation:
A bag contains 20 pink candies, 8 red candies, and 12
green candies. Without looking, Sarah pulls out a piece of
candy. Which color of candy is least likely to be pulled
out?
Answers
Answer:
The red marbles.
Step-by-step explanation:
It is because Red has the least amount of marbles, therefore it is most likely for you to not pull out a red marble.
The diagonal of a square is 4 cm. Find its ar
ea.
Answers
Answer:
Area of square= d²/2
= 4²/2
= 16/2
= 8cm²
Step-by-step explanation:
A=
\(a {?}^{2} \)
d=
\( \sqrt{2a} \)
we can find A.USE THIS FORMULA
A =1/2 d squared
so 1/2×4 squared=8
8 is the answer
A researcher wants to set up a regression equation where Y is a function X. Evaluate the researcher’s options given the following scenarios: (3)
i. Y is I(0); X is I(0)
ii. Y is I(2); X is I(0)
iii. Y is I(1); X is I(1); and the error term is I(0).
Answers
The appropriate regression model depends on the stationarity properties of both the dependent and independent variables, as well as the error term. The researcher can use a standard OLS regression model with first-order differencing of both Y and X.
In the first scenario, both Y and X are I(0), which means they are stationary time series. In this case, the researcher can perform a standard linear regression analysis, as the stationary series would lead to a stable long-run relationship. The answer from this model will be reliable and less likely to suffer from spurious regressions. In the second scenario, Y is I(2) and X is I(0). This implies that Y is integrated of order 2 and X is stationary. In this case, the researcher should first difference Y twice to make it stationary before performing a regression analysis. However, this approach might not be ideal as the integration orders differ, which can lead to biased results.
In the third scenario, Y and X are both I(1) and the error term is I(0). This indicates that both Y and X are non-stationary time series, but their combination might be stationary. The researcher should employ a co-integration analysis, such as the Engle-Granger method or Johansen test, to identify if there is a stable long-run relationship between Y and X. If co-integration is found, then an error correction model can be used for more accurate predictions.
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6x + 1.6 =58 ? two step equations
Answers
Answer:
9.4
Step-by-step explanation:
6x+1.6=58
-1.6 -1.6
6x=56.4
/6 /6
=9.4
\(\textsf{Hey there!}\)
\(\mathsf{6x + 1.6 = 58}\)
\(\huge\textsf{SUBTRACT 1.6 on both of your sides}\)
\(\large\textsf{Like: 6x + 1.6 - 1.6 = 58 - 1.6}\\\large\textsf{Cancel out: 1.6 - 1.6 because it gives us 0}\\\large\textsf{Keep: 98 - 1.6 because it helps us solve for our answer}\)
\(\large\texts{\bf{\underline{58 - 1.6 = 56.4}}}}}\\\textsf{New equation: 6x = 56.4}\)
\(\huge\textsf{DIVIDE both sides by 6}\)
\(\large\mathsf{Like: \dfrac{6x}{6} = \dfrac{56.4}{6}}\\\\\textsf{Cancel out:}\mathsf{\dfrac{6x}{6}\ because\ that\ gives\ us\ 1}\\\\\mathsf{Keep: \dfrac{56.4}{6}\ because\ that\ gives\ our\ x\ value}\)
\(\huge\textsf{Solving for our \underline{answer}}\)
\(\mathsf{\dfrac{56.4}{6} = 56.4 \div 6 \rightarrow \underline{\bf{ANSWER}}}\)
\(\huge\textsf{YOUR \bf{ANSWER}}\huge\downarrow\)
\(\boxed{\boxed{\huge\textsf{Answer: \boxed{\boxed{\huge\textsf{\bf{\underline{x = 9.4}}}}}}}}}}\huge\checkmark\)
\(\textsf{Good luck on your assignment and enjoy your day!}\)
~\(\dfrac{\frak{LoveYourselfFirst}}{:)}\)
the mean visual vark score of students in our sample was 6.527881, with a standard deviation of 3.231746. calculate the standard error of this mean as an estimate for the average vark visual score of all biology students at eastern, knowing that 269 students filled out the survey:
Answers
The standard deviation error of this mean as an estimate for the average vark visual score = 0.197
A standard deviation (or σ) is a measure of how dispersed the data is in relation to the mean. Low standard deviation means data are clustered around the mean, and high standard deviation indicates data are more spread out.
Given that,
The mean visual vark score of students in our sample = 6.527881
A standard deviation = 3.231746
The standard error of this mean as an estimate for the average vark visual score = 95%
That 269 students filled out the survey,
(σ / \(\sqrt{n}\) ) \(z_{\frac{a}{2} }\) = 269
σ = 3.231746
σ = 1 - 95%
95% of z = 1.96
Then,
Standard error represents the accuracy of the sample in a population. Its formula is SE = s/√n, where s is the standard deviation and n is the number of students. Therefore,
SE = 3.231746 /√269
SE = 0.197
The standard error of mean is equal to 0.197
Therefore,
The standard deviation error of this mean as an estimate for the average vark visual score = 0.197
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Solve the equation x/4=12/6
Answers
Answer:
x=8
Step-by-step explanation:
x/4=12/6
x=12×4/6
x=48
x=8
a teacher wants to create a graph of the students' scores on the final exam and see what the distribution looks like, looking for clusters and gaps in the scores. which type of graph is the most appropriate?
Answers
Answer:
A histogram would be the most appropriate graph to display the distribution of scores and identify clusters and gaps.
Step-by-step explanation:
Sketch the region enclosed by the curves and find its
area.
y=x,y=3x,y=−x+4
Answers
The region enclosed by the curves y = x, y = 3x, and y = -x + 4 needs to be sketched, and its area should be found.
To sketch the region enclosed by the curves, we need to plot the three given curves on a coordinate plane. The first curve is y = x, which represents a straight line passing through the origin (0,0) with a slope of 1. The second curve is y = 3x, which is also a straight line passing through the origin but with a steeper slope of 3. The third curve is y = -x + 4, which represents a line with a y-intercept of 4 and a negative slope of -1. By plotting these three lines on the same coordinate plane, we can see that they intersect at three points: (0,0), (1,3), and (3,1). The region enclosed by these curves is a triangular region with vertices at these three points. To find the area of this triangular region, we can use the formula for the area of a triangle: A = (1/2) * base * height. Let's draw the graph:
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4 | . (2, 2)
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0 |_____________________
0 1 2 3 4 5 6
In this graph, the first equation (y = x) is depicted by a diagonal line passing through the origin (0,0). The second equation (y = 3x) is a steeper line, while the third equation (y = -x + 4) is a downward-sloping line with a y-intercept of 4. In this case, the base of the triangle is the distance between the points (0,0) and (3,1), which is 3 units. The height of the triangle is the distance between the point (1,3) and the line y = -x + 4, which is also 3 units. Substituting these values into the area formula, we get A = (1/2) * 3 * 3 = 4.5 square units.
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