Let L be the line through A = (1, 2,−1) and B = (5,−3, 4).
Answer the following questions below:
1. Find a direction vector of L
2. Find a parameterization of L.
3. Find the two points on L that are at distance 2 away from A.
4. Find a unit speed parameterization of L.
Answers
Answer:
(4, -5, 5)L = (1+4t, 2-5t, -1+5t)(1+4√66/33, 2-5√66/33, -1+5√66/33), (1-4√66/33, 2+5√66/33, -1-5√66/33)(1-2t√66/33, 2+5√66/66, -1-5√66/66)Step-by-step explanation:
Given L is the line through A(1, 2, -1) and B(5, -3, 4), you want its direction vector, a parameterization of L, 2 points on L that are 2 units from A, and a unit speed parametrization of L.
1. Direction vectorA direction vector for L will be the vector AB.
[tex]\overrightarrow{AB}=B-A=(5,-3,4)-(1,2,-1)\\\\\boxed{\overrightarrow{AB}=(4, -5,5)}[/tex]
2. Parameterized lineThe parameterized equation will be ...
[tex]L=A+t\cdot\overrightarrow{AB}\\\\\boxed{L=(1+4t,2-5t,-1+5t)}[/tex]
3. Distance 2The direction vector has length √66, so we can find two points on L that are 2 units from A by using t = ±2/√66 in the parameterized equation.
Here, they are:
P1 = L(2/√66) = (1 +8/√66, 2 -10/√66, -1 +10/√66)
P2 = L(-2/√66) = (1 -8/√66, 2 +10/√66, -1 -10/√66)
In the Answer section above, these are written with rationalized denominators.
4. Unit speedThe unit speed parametrization uses the unit direction vector instead of the unnormalized direction vector in the parameterized equation. Effectively, t is scaled by a factor of 1/√66.
L = (1 +4t/√66, 2 -5t/√66, -1 +5t/√66)
In the Answer section above, this is written with rationalized denominators.
You might notice here that it is easier to answer part 3 if you have this equation. You need only substitute t=±2 to get the points 2 units from A.
Related Questions
Need it ASAP, please!!!
Answers
Answer:
27y - 45 + 7z
S Janine is considering buying a water filter and a reusable water bottle rather than buying bottled water. Will doing so save her money? First, determine what information you need to answer this question, then click here to display that info (along with other info). • How much water does Janine drink in a day? She normally drinks 6 bottles a day, each 16.9 ounces. How much does a bottle of water cost? She buys 24-packs of 16.9 ounce bottles for $3.99. • How much does a reusable water bottle cost? About $10. • How long does a reusable water bottle last? Basically forever (or until you lose it). . • How much does a water filter cost? How much water will they filter? o A faucet-mounted filter costs about $28 and includes one filter cartridge. Refill filters cost about $33 for a 3-pack. The box says each filter will filter up to 100 gallons (378 liters) o A water filter pitcher costs about $22 and includes one filter cartridge. Refill filters cost about $20 for a 4-pack. The box says each filter lasts for 40 gallons or 4 months • An under-sink filter costs $130 and includes one filter cartridge. Refill filters cost about $60 each. The filter lasts for 500 gallons. Which option is cheapest over one year (365 days)? faucet-mount filter The cheapest option saves her $ Give your answer to the nearest cent. Pro-rate the costs of additional filters (so if you only use part of a filter, only count the corresponding fraction of the filter cost) and pro-rate the cost of the bottled water (so if you only use part of a case, only count the cost for the portion of the case used). over a year.
Answers
Janine currently spends about $150.11 on bottled water every year.The total sum of the cost for the pitcher filter is $42. The sink filter costs $130
The first step in finding how much money Janine will save is to calculate how much water Janine drinks and how much bottled water costs her in one year.
To find how much money Janine currently spends on water, calculate the number of cases of water she buys each year (365 days / (24 bottles in a case/ 3 bottle per day) and multiply this number by the price per case ($3.29) to find that Janine currently spends about $150.11 on bottled water every year.To calculate the amount of water Janine drinks each year we multiply the amount she drinks in one day (3 bottles x 16.9oz per bottle) by 365 and then since there are 128 ounces in a gallon, we divide this number by 128 to find that she drinks 144.6 gallons of water each yearThe next step in this problem is to calculate the cost of each water filter in the first year. To find this, add the cost of the reusable water bottle, the cost of the filter, and the cost of any additional filters needed.
The faucet filter costs $28 dollars and includes a filter, plus each additional filter costs $11. Since the filter will last for 100 gallons, we can subtract that amount from the total water she drinks in a year to find the cost of filtering the remainder of the water in the year (144.6 gallons- 100gallons /$11). The total sum of the cost of the faucet filter and reusable water bottle in the first year is $42.91.The pitcher filter costs $22 and each additional filter costs $5. Since the included filter lasts 4 months, Janine will need two additional replacement filters for the other 8 months of the first year. The total sum of the cost for the pitcher filter is $42.The sink filter costs $130 and since the included filter will work for 500 gallons of water, Janine will not need to purchase additional filters in the first year. The total cost of the sink filter is $140.From these calculations, you can see that the pitcher filter is the cheapest option and will save Janine $108.11 in her first year of using this filter.Lastly the question asks you to pro-rate the cost of each filter
Since the faucet filters cost $11 each and filter 100 gallons of water, this filter costs 11 cents per gallon and $15.91 per year. She will have to begin replacing these filters approximately 8 months after the initial purchase of the filter.Filtering 40 gallons and costing $5 each the pitcher filter costs 12.5 cents per gallon and $18.08 per year. She will have to replace these filters every 4 months.The sink filter filters 500 gallons of water and costs $60, so it costs 12 cents per gallon and $17.35 per year. She will not have to replace this filter for about 3 and half years after the initial purchase.Learn more about cost at:
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Mr Paul has 4 bars of chocolate. He distributes 1/3
of each bar to 4 children.
a) Write the fraction of chocolate distributed.
b) Write the fraction of cholates left with Mr. Paul
Answers
a) The fraction of chocolate distributed is given as follows: 4/3.
b) The fraction of chocolates left with Paul is given as follows: 8/3.
How to obtain the fractions?The fractions are obtained applying the proportions used in the context of this problem.
He distributes 1/3 of each bar to 4 children, hence the total number of bars distributed is given as follows:
4 x 1/3 = 4/3.
(multiplication of fractions, multiply the numerators and the denominators, keeping the quotient).
The total number of bars that Paul had initially is given as follows:
4 bars.
Then the fraction of chocolate remaining with Paul is given by the subtraction of the total amount of bars by the amount given to the children, as follows:
4 - 4/3 = 12/3 - 4/3 = 8/3 bars.
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Which problem situation matches the
equation $95.25 = 15p?
A Cassie is paid $95.25 for 15 hours
of work. Find p, the amount she
was paid per hour.
B Kris is paid $6.35 per hour for
mowing the lawn. Find p, the
amount he is paid for 15 hours of
mowing.
CA iPad was on sale for $95.25. Find
p, the amount Noah needs to buy
the iPad if he already has $15.
D Fifteen friends spend $95.25 each
for tickets to a Cowboy's game.
Find p, the amount they paid total.
Answers
Answer:
Step-by-step explanation:
Which problem situation matches the
equation $95.25 = 15p?
So We Can Just Cross B Cause $6.35 Wasnt Even In The equation
Lets look at A.
So Im pretty sure its A, But Lets look at the rest.
.
D Wouldnt Work Cause It Says The Spend $95.25 EACH
C dont work so its A.
what is the area of 12 1/2 feet long and 11 3/4 feet wide
Answers
now, we're assuming this is some rectangular area, and thus is simply the product of both quantities, let's change all mixed fractions to improper fractions firstly.
[tex]\stackrel{mixed}{12\frac{1}{2}}\implies \cfrac{12\cdot 2+1}{2}\implies \stackrel{improper}{\cfrac{25}{2}} ~\hfill \stackrel{mixed}{11\frac{3}{4}} \implies \cfrac{11\cdot 4+3}{4} \implies \stackrel{improper}{\cfrac{47}{4}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{25}{2}\cdot \cfrac{47}{4}\implies \cfrac{1175}{8}\implies 146\frac{7}{8}~ft^2[/tex]
Photo attached. Someone please help
Thanks
Answers
By using the graph, a solution the given system of equations include the following:
x = 0, y = 3
x = 3, y = 6
What are quadratic functions?In Mathematics, the graph of every quadratic functions (equations) is either an upward or downward parabola, which is typically a u-shaped curve. This ultimately implies that, the graph of a quadratic function (equation) would always form a parabolic curve.
In this scenario, we would use an online graphing calculator to plot the graph of the given system of equations and then determine the point of intersection.
y = x + 3
y = x² - 2x + 3
By critically observing the graph of the given system of equations, the solutions are as follows;
x = 0, y = 3.
x = 3, y = 6.
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What is 10/83 simplified
Answers
Answer:
5/19
Step-by-step explanation:
To identify if a fraction is further divisible or not, we find a common number that the numerator and denominator both can be divided by. In this case, 10/83 cannot be commonly divided any further than it is. Therefore, the fraction is in its simplest form.
-20 POINTS- only answer If you know dont guess again.
Answers
Using the Pythagoras theorem, the value of y is [tex]6\sqrt{7}[/tex], and the value that goes in the green box is 6.
What is Pythagoras Theorem?
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side.
Mark the intersections as A, B, C, and D.
It is given that BC=12 and CD=9.
The relationship between the height on the hypotenuse of a right triangle and the two line segments it generates on the hypotenuse is described by the right triangle altitude theorem, also known as the geometric mean theorem. It claims that the altitude is equal to the geometric mean of the two segments.
Apply the geometric mean theorem -
(AC)^2 = BC x CD
(AC)^2 = 12 x 9
(AC)^2 = 108
AC = [tex]\sqrt{108}[/tex]
AC = [tex]6\sqrt{3}[/tex]
Triangle ABC is a right-angled triangle, so apply Pythagoras theorem -
(AB)^2=(BC)^2+(AC)^2
(AB)^2=(12)^2+([tex]6\sqrt{3}[/tex])^2
(AB)^2=144+108
(AB)^2=252
AB=[tex]\sqrt{252}[/tex]
AB=[tex]6\sqrt{7}[/tex]
Therefore, the value of y is obtained as [tex]6\sqrt{7}[/tex].
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Help answer this pls if really appreciate it!! Also if you could explain a little I’d appreciate it extra since I’m a little lost. Thank you!!!
Answers
According to the question the angle between right angle triangle hence the M∠1 = [tex]90^{0}[/tex] .
What does a triangle's right angle look like?90 degrees or a straight angle is always one angle. The hypotenuse is the side that has an angle of 90° on it. Always, the hypotenuse is the longest side. The remaining two inner angles add up to a total of 90°.
Which three angles make up a right triangle?Right triangles are triangles with one of the internal angles at a right angle of 90 degrees. Since a triangle's three interior angles sum up to 180 degrees, a right triangle's three interior angles must always add up to 90 degrees since one of them is always 90 degrees (they are complementary).
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The angle of elevation to the top of a Building in New York is found to be 4 degrees from the ground at a
distance of 1.75 miles from the base of the building. Using this information, find the height of the building.
Round to the nearest whole number.
Your answer is
Submit Question
feet.
Answers
The height of the building can be 649.44 ft
What is angle of elevation ?
angle of elevation states that the angle which is formed between line of sight and horizontal line.
Given, the base of the triangle is 1.75 mile. The angle opposite of the tall side, x, is 4 degrees.
height of the building is an unknown variable (x) .
tan(angle) = opposite length /adjacent length.
Opposite length is x, adjacent length is 1.75 mile, angle is 4 degrees.
tan(4º) = x / 1.75
rearrange for x
x = 1.75*tan(4º) = ~1.75*0.0699 miles.
x = ~0.123
Now to get feet, just multiply by 5,280 ft in one mile
Height of the building is 0.123 miles * 5280 ft/mile = 649.44 ft
Hence , The height of the building can be 649.44 ft .
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Taylor's band is recording their first studio album at Midnight Studios. The studio's
day rate is $75 per day with a fee of $50 per hour for their sound engineers. The
band has $375 and they think they can earn up an additional $200 off merch sales.
What is the range of hours the band can record at the studio within their budget?
Answers
Number of Hours the band can record there music is 10hrs
What are Word Problems ?A word problem is a type of mathematics exercise where the majority of the issue's context is supplied in spoken language as opposed to mathematical notation.
A math word problem is a question that is phrased as one or more sentences and asks students to use their mathematical understanding to solve an issue from "real world." In order for kids to understand the word problem, they must be familiar with the terminology that goes along with the mathematical symbols that they are used to.
Word issues, also known as narrative problems, may be found in both daily life and the Common Core State Standards for every school level, from K to 12. For everything from counting pennies to calculating a gratuity, you can utilize the three primary forms of word problems: part-part whole, separate and join, and multiply and divide.
Total money the band has after selling the merch is = $375 + 200 = $575.
Rate for studio is $75 per day.
Total money left = $575 - 75 = $ 500
So number of Hours the band can record there music is= 500/50 = 10hrs.
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Suppose a researcher compiled a data set of the following variables fora sample of 100 retired men
Answers
Answer:
The researcher compiled a data set consisting of variables such as age, income, and marital status for a sample of 100 retired men
Step-by-step explanation:
A discrete variable is a type of variable that can only take on specific, distinct values that can be counted. Examples of discrete variables include the number of countries visited, shoe size, and number of pets living in a household.
On the other hand, a continuous variable is a type of variable that can take on any value within a certain range through measurement. Examples of continuous variables include pounds of chocolate consumed in the past and height in inches.
6+√-18 simplified for edmentum
Answers
Answer:
= 18√2
decimal =25.45584412
Malcom wrote the equivalent ratios shown below. Find the scale factor he used to write the equivalent ratios
9/40 and 54/240
Answers
In this question, we're looking for the number Malcom used to scale the two numbers. We can see there's a pattern, 9 is 1/6 of 54 and 40 is 1/6 of 240!
(54 divided by 9 is 6 and 240 divided by 40 is 6!)
therefore, the scale factor must be 6
a line has a slope of 5. it passes through the points (-5, -1) and (x, 9). what is the value of x?
Answers
A line has a slope of 5. it passes through the points (-5, -1) and (x,9) than the value of x is -3
We are going to use the fact that when a line passes through two points : [tex](x1,y1) and (x2,y2)[/tex]
the slope is given by the following formula:
[tex]slope=y2-y1/x2-x1[/tex]
In the present case, the slope is 5 and the points are (-5, -1) and (x, 9)
Hence,
[tex]5=9-(-1)/x-(-5)[/tex]
[tex]5=10/x+5\\x+5=10/5\\x+5=2\\x=-3[/tex]
Thus, the value of x is -3 when it travels between the coordinates (-5, -1) and (x, 9).
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Please help me with this question
Answers
Answer:
cos(x -π/3) = (1/2)cos(x) +(√3/2)sin(x)
Step-by-step explanation:
You want cos(x -π/3) written in terms of sine and cosine.
Angle differenceThe formula for the cosine of the difference between two angles is ...
cos(a -b) = cos(a)cos(b) +sin(a)sin(b)
ApplicationWe have a=x, b=π/3, so the desired function is ...
cos(x -π/3) = cos(x)cos(π/3) +sin(x)sin(π/3)
We know that ...
cos(π/3) = 1/2sin(π/3) = √3/2so the rewrite is ...
cos(x -π/3) = (1/2)cos(x) +(√3/2)sin(x)
Write an equation and solve for x using the diagram
below. Also give the angle measure. YOU MUST
WRITE THE EQUATION IN ORDER TO GET
Answers
Answer:
x = 45
Step-by-step explanation:
2x + 6 and 96 are vertically opposite angles and are congruent , then equation is
2x + 6 = 96 ( subtract 6 from both sides )
2x = 90 ( divide both sides by 2 )
x = 45
Reduce 17/5 to It lowest term
Answers
Answer:
17/5
It is already simplified to lowest terms.
This number, as a mixed fraction, would look like this:
3 2/5.
How many real solutions does this system of equations have? x²+y²=36
3x-y+1=0
A.0
B. 2
C. 3
D. 1
Answers
This system of equations have 2 real solution.
Do fractions actually solve problems?As a result, all of these numbers—rational and otherwise—as well as fractions are regarded as real numbers. Due to the decimal's ability to float within the numbers, real numbers with decimal points are known as floating point numbers. No whole number or integer can be a floating point number.
y = 3x + 1 is a solution to 3x - y + 1 = 0.
This line's y-intercept is represented by (0, 1). Plot this point inside the circle with radius 6 that was previously mentioned, and then draw a straight line through it with a slope of 3.
Two points where this line crosses the circle denote actual solutions.
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Simplify the expression
-4|2.5-6|
Answers
Answer: -14
Step-by-step explanation:
g(x) = (2 (x+3))^3 + 4
Answers
.....................................
(5 x 7) + (n x 4) = 5 x (7+ 4) write each missing number
Answers
The missing number in the expression (5 x 7) + (n x 4) = 5 x (7+ 4) is 5.
What are GCF and distributive property?The GCF of two or more than two numbers is the highest number that divides the given two numbers completely.
We also know that distributive property states a(b + c) = ab + ac.
Given, An expression (5 × 7) + (n × 4) = 5 × (7 + 4).
Now, If we expand the RHS we have,
5×(7 + 4).
= (5 × 7) + (5 × 4).
Now, Writing the obtained RHS with LHS we have,
(5 × 7) + (n × 4) = (5 × 7) + (5 × 4).
So, The missing number is 5.
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Here is a graph of the function h.
Use the graph to find the following
If there is more than one answer, separate them with commas
(a)
All local minimum values of
h
(b) All values at which h has a local minimum:
Answers
The graph also shows two other local maxima, at x=0.5 and x=1.5.
The local minimum(a) -4, -2, 0, 2 (b) The local minimum values of h occur at x = -4, -2, 0, and 2.These values can be seen on the graph as points at which the graph of h changes from decreasing to increasing.At each of these points, the value of h is at a minimum compared to the values of h on either side of the point.All values at which h has a local maximum: 0.5, 1.5All local maximum values of h: 2.5, 4.5.The graph of the function h is a parabola that opens upwards, with a turning point at x=1.This indicates that the function has a local maximum at this point.The local maximum values of h at these points can be determined by looking at the y-values at these points, which are 2.5 and 4.5 respectively.To learn more about All local minimum values refer to:
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Which equations are true for x = –2 and x = 2? Select two options x2 – 4 = 0 x2 = –4 3x2 + 12 = 0 4x2 = 16 2(x – 2)2 = 0
Answers
4x^2 = 16
The first equation is true when x = -2 or x = 2, because substituting either of these values into the equation results in a true statement. The second equation is also true when x = -2 or x = 2, because substituting either of these values into the equation results in a true statement. The other three equations are not true when x = -2 or x = 2, because substituting these values into the equations results in false statements.
Suppose that R, S and T are digits and that N is the four-digit positive integer
8RST. That is, N has thousands digit 8, hundreds digit R, tens digits S, and ones
(units) digit T, which means that N = 8000 + 100R + 10S + T. Suppose that the
following conditions are all true:
• The two-digit integer 8R is divisible by 3.
• The three-digit integer 8RS is divisible by 4.
The four-digit integer 8RST is divisible by 5.
The digits of N are not necessarily all different.
The number of possible values for the integer N is
(A) 8
(B) 16
(C) 12
(D) 10
87
(E) 14
Answers
Answer:
8RST. That is, N has thousands digit 8, hundreds digit R, tens digits S, and ones
(units) digit T, which means that N = 8000 + 100R + 10S + T. Suppose that the
following conditions are all true:
• The two-digit integer 8R is divisible by 3.
• The three-digit integer 8RS is divisible by 4.
The four-digit integer 8RST is divisible by 5.
The digits of N are not necessarily all different.
The number of possible values for the integer N is
(A) 8
(B) 16
(C) 12
(D) 10
87
(E) 14
Since 8R is divisible by 3, R must be a multiple of 3. The possible values for R are 3, 6, and 9.
Since 8RS is divisible by 4, S must be a multiple of 2. The possible values for S are 0, 2, 4, 6, and 8.
Since 8RST is divisible by 5, T must be 0 or 5.
The possible values for N are therefore:
8000 + 300 + 00 + 0 = 8300
8000 + 600 + 00 + 0 = 8600
8000 + 900 + 00 + 0 = 8900
8000 + 300 + 20 + 0 = 8320
8000 + 600 + 20 + 0 = 8620
8000 + 900 + 20 + 0 = 8920
8000 + 300 + 40 + 0 = 8340
8000 + 600 + 40 + 0 = 8640
8000 + 900 + 40 + 0 = 8940
8000 + 300 + 60 + 0 = 8360
8000 + 600 + 60 + 0 = 8660
8000 + 900 + 60 + 0 = 8960
8000 + 300 + 80 + 0 = 8380
8000 + 600 + 80 + 0 = 8680
8000 + 900 + 80 + 0 = 8980
8000 + 300 + 00 + 5 = 8305
8000 + 600 + 00 + 5 = 8605
8000 + 900 + 00 + 5 = 8905
8000 + 300 + 20 + 5 = 8325
8000 + 600 + 20 + 5 = 8625
8000 + 900 + 20 + 5 = 8925
8000 + 300 + 40 + 5 = 8345
8000 + 600 + 40 + 5 = 8645
8000 + 900 + 40 + 5 = 8945
8000 + 300 + 60 + 5 = 8365
8000 + 600 + 60 + 5 = 8665
8000 + 900 + 60 + 5 = 8965
8000 + 300 + 80 + 5 = 8385
8000 + 600 + 80 + 5 = 8685
8000 + 900 + 80 + 5 = 8985
There are a total of 30 possible values for N. The answer is therefore (E) 14.
Step-by-step explanation:
Determine whether the statement is true or false. If false, give a counterexample.
The constant of proportionality in an equation can never be 0.
Answers
Answer:
Step-by-step explanation:
42. Let f be a function from the set A to the set B. Let S and T be subsets of A. Show that
a) f(S ∪ T)=f(S) ∪ f(T).
b) f(S ∩ T) ⊆ f(S) ∩ f(T).
Answers
Start with any random elements of sets and compute LHS and RHS individually.
a) The proof of f(S ∪ T) = f(S) ∪ f(T) is explained below.
b) The proof of f(S ∩ T) ⊆ f(S) ∩ f(T) is expliane dbelow.
What are sets and subsets?A set is a collection of well-defined objects.
A subset contains all the elements or a few elements of the given set.
The improper subset is when it contains all the elements of the given set and the proper subset is when it doesn't contain all the elements of the given set.
Let, The set A = {1, 2, 3, 4, 5} and B = {2, 4, 5}.
Therefore, S = {1, 2, 3} and T = {1, 4, 5}.
Now, f(S ∪ T) = {1, 2, 3, 4, 5} and f(S) ∪ f(T) = {1, 2, 3} ∪ {1, 4, 5}.
Hence, f(S ∪ T)=f(S) ∪ f(T).
Now, f(S ∩ T) = {1} and f(S) ∩ f(T) = {1, 2, 3} ∩ {1, 4, 5} = {1}.
Therefore, f(S ∩ T) ⊆ f(S) ∩ f(T). (⊆ implies improper subset)
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An algebraic expression 2x² + 2xy is trinomial algebraic expression true or false
Answers
Answer:
False
Step-by-step explanation:
right!!
there are no good answers on brainily can someone pls answer this
Answers
Answer:
D
Step-by-step explanation:
first simplify everything
2/9v + 1/3v - 5/9
A and B is wrong because it is - 5/9 not + 5/9
C is wrong because it is -1/3v not 1/3, it is supposed to be - 5/9
You can solve this problem without much solving
hope this helps :)
Calculate the sample mean and sample variance for the following distribution of heights in centimeters for an 8 year old boy
Answers
Answer: To calculate the sample mean of a distribution, you add up all the values and divide by the number of values in the sample. The sample mean is also known as the average.
The formula for the sample mean is:
X-bar = (X1 + X2 + ... + Xn) / n
To calculate the sample variance of a distribution, you first need to calculate the sample mean. Then, for each value, you subtract the sample mean and square the result. You add up all these squared differences, and divide the sum by the number of values in the sample.
The formula for the sample variance is:
s^2 = ( (X1 - X-bar)^2 + (X2 - X-bar)^2 + ... + (Xn - X-bar)^2 ) / (n - 1)
It would be helpful to have the sample data, however I cannot complete this task without it.
Please provide me with the data.
Step-by-step explanation: