So, the solution is x = 1, y = -1, z = 3.
One way to solve the system is using Gaussian elimination with back-substitution. The steps are:
1. Rewrite the system in matrix form, called the augmented matrix:
[3 3 6 | 9]
[1 1 2 | 3]
[2 5 10| 15]
[-1 2 4 | 6]
2. Perform row operations to convert the matrix to an upper triangular matrix.
[1 1 2 | 3]
[0 2 4 | 3]
[0 2 6 | 6]
[0 0 -2 | -6]
3. Use back-substitution to find the values of the variables. Starting with the last equation:
-2z = -6 => z = 3
4y + 6(3) = 6 => y = -1
x + 2(-1) + 2(3) = 3 => x = 1
So, the solution is x = 1, y = -1, z = 3.
Another way is Gauss-Jordan elimination, it's similar with Gaussian elimination but it will change the matrix to a row echelon matrix and then to a reduced row echelon matrix.
So, the solution is x = 1, y = -1, z = 3.
Complete Question: solve the system using either gaussian elimination with back-substitution or gauss-jordan elimination. (if there is no solution, enter no solution. if the system has an infinite number of solutions, express x, y, and z in terms of the parameter t.) 3x + 3y + 6z = 9
x + y + 2z = 3
2x + 5y + 10z = 15
−x + 2y + 4z = 6
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The lengths of the sides of triangle XYZ are written in terms of the variable m, where m ≥ 6.
Triangle X Y Z is shown. The length of side X Y is m + 8, the length of side Y Z is 2 m + 3, the length of side Z X is m minus 3.
Which is correct regarding the angles of the triangle?
mAngleX < mAngleZ < mAngleY
mAngleY < mAngleZ < mAngleX
mAngleY < mAngleX < mAngleZ
mAngleZ < mAngleY < mAngleX
The correct arrangement regarding the angles of the triangle is;
m∠Y < m∠Z < m∠X
What are the missing angles of the triangle?In triangle XYZ, the lengths of sides are;
XY = m + 8
YZ = 2m + 3
ZX = m - 3
If m ≥ 6, then;
m + 8 ≥ 14
2m + 3 ≥ 15
m - 3 ≥ 3
Thus;
m - 3 ≤ m + 8 ≤ 2m + 3
The greatest angle is opposite to the greatest side, the smallest angle is opposite to the smallest side, so;
the greatest side is YZ = 2m + 3- the greatest angle is ∠X
the smallest side is ZX = m - 3 - the smallest angle is ∠Y
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Left F(x) be the number, in thousands, of vehicles a manufacturer estimates will be sold when x million dollars are spent on advertising. If f(1.8)=240, and f(2.5)=325, compute the average rate of change for 1.8 < x< 2.5
The average rate of change for 1.8 < x < 2.5 is given as follows:
121.43.
How to obtain the average rate of change?The average rate of change of a function is given by the change in the output divided by the change in the input.
The change in the output for this problem is given as follows:
325 - 240 = 85.
The change in the input for this problem is given as follows:
2.5 - 1.8 = 0.7.
Hence the average rate of change is given as follows:
85/0.7 = 121.43.
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enter the correct 4 digit code
Answer:
DCBA
Step-by-step explanation:
It's just reverse look closely and you'll notice
enter a formula in cell b3 using the vlookup function to find the meaning for the medical abbreviation listed in cell a3. use the name abbreviation for the lookup table. the item names are located in column 2 of the lookup table. be sure to require an exact match.
The formula in cell B3 using the VLOOKUP function to find the meaning for the medical abbreviation listed in cell A3, using the "abbreviation" lookup table, would be: =VLOOKUP(A3, "abbreviation", 2, FALSE).
A3 is the cell that contains the medical abbreviation we want to find the meaning for.
"abbreviation" is the name of the lookup table that contains the item names and meanings.
2 is the column number of the lookup table where the meanings are located.
FALSE is used to require an exact match for the abbreviation in cell A3, so that the correct meaning is returned.
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I NEED HELP ASAP WHOEVER ANSWERS GETS BRAINLIST
The side length of the cube is 1/6 and the volume is 1/216
How to determine the side length of the cubeFrom the question, we have the following parameters that can be used in our computation:
Face area = 1/36
This means that
Side length = √(1/36)
Evaluate
Side length = 1/6
How to determine the volue of the cubeFrom the question, we have the following parameters that can be used in our computation:
Side length = 1/6
This means that
Volume = (1/6)^3
Evaluate
Volume = 1/216
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For each of the following subsets of R3 determine whether it is a vector subspace of R3 or not. Justify your answer.
So, b) and e) are subspaces of R3, and a), c), and d) are not.
a) {(x,y,z) ∈R3 : x = 0} is not a subspace of R3, because it doesn't contain the zero vector (0,0,0) which is a requirement for a subspace. It's also not closed under scalar multiplication.
b) {(x,y,z) ∈R3: x + y = 0} is a subspace of R3, because it's closed under scalar multiplication and addition and it contains the zero vector (0,0,0).
c) {(x,y,z) ∈R3 : xz = 0} is not a subspace of R3, because it's not closed under scalar multiplication and it's not closed under addition
d) {(x,y,z) ∈ R3: y ≥0} is not a subspace of R3, because it's not closed under addition, it doesn't contain the zero vector (0,0,0) and it's not closed under scalar multiplication.
e) {(x,y,z) ∈ R3: x = y = z} is a subspace of R3 because it satisfies all the properties of a subspace, it is closed under addition, scalar multiplication, it contains the zero vector (0,0,0) and the negative of any vector in the set is also in the set. This set contains all the vectors that are multiples of (1,1,1) which forms a subspace.
So, b) and e) are subspaces of R3, and a), c), and d) are not.
Complete Question: For each of the following subsets of R3 determine whether it is a vector subspace of R3 or not. Justify your answer.
a) {(x,y,z) ∈R3 : x = 0}
b) {(x,y,z) ∈R3: x + y = 0}
c) {(x,y,z) ∈R3 : xz = 0}
d) {(x,y,z) ∈ R3: y ≥0}
e) {(x,y,z) ∈ R3: x = y = z}
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Problem 1)
a) What is the possible number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails?
b) For a series of 1000 tosses, what is the total number of microstates associated with 50% heads and 50% tails?
c) How less probable is the outcome that the coin will land 40% heads and 60% tails?
a) The number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails is given by the binomial coefficient "N choose H" (or "N choose T", as H+T=N), which is equal to N!/(H! * T!)
b) For a series of 1000 tosses, with 50% heads and 50% tails, there are a total of "1000 choose 500" microstates. This is approximately 2.07*10²⁵ microstates.
c) The probability of getting a specific outcome when tossing a coin is given by the binomial probability mass function. The probability of getting 40% heads and 60% tails in 1000 tosses is given by (1000 choose 400)(0.4⁴⁰⁰)(0.6⁶⁰⁰) which is very less probable.
Coin Toss Microstates Probabilitya) The number of microstates associated with a coin flip can be determined using combinatorics. The coin flip can have two possible outcomes: heads or tails. To calculate the number of microstates for N coin flips with H heads and T tails, we use the binomial coefficient formula "N choose H" (or "N choose T", as H+T=N), which is equal to N!/(H! * T!) where ! denotes factorial. This formula gives the number of ways to choose H heads out of N coin flips.
b) To calculate the total number of microstates for a series of 1000 tosses with 50% heads and 50% tails, we use the binomial coefficient formula "1000 choose 500". This is equivalent to (1000!/(500!500!)) which is approximately 2.0710²⁵ microstates.
c) The probability of getting a specific outcome when tossing a coin is given by the binomial probability mass function. The probability of getting 40% heads and 60% tails in 1000 tosses is given by (1000 choose 400)(0.4⁴⁰⁰)(0.6⁶⁰⁰) which is very less probable. We can calculate this using the binomial probability mass function which is ( n choose k ) * p^k * (1-p)^(n-k) where n is the number of trials, k is the number of successful outcomes, p is the probability of success in one trial, 1-p is the probability of failure in one trial.
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Benita buys 2 shirts for the same price s the sales tax rate is 6.5
Write an expression to represent the total cost of the shirt in two different ways .
The required total cost of the two shirts after 6.5% of the tax is given as 2.13s.
What are equation models?The equation model is defined as the model of the given situation in the form of an equation using variables and constants.
Here,
Benita buys 2 shirts for the same price s the sales tax rate is 6.5%
1 way, paid the cost of 2 shirts jointly
The total cost of the two shirts becomes,
Total cost = 2[ s + 0.065s]
Total cost = 2.13s
2nd way paid the cost of two shirts individually,
total cost = s + 0.065s + s + 0.065s
= 2.13s
Thus, the required total cost of the two shirts after 6.5% of the tax is given as 2.13s.
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From a 15-inch by 15-inch piece of metal, squares are cut out of the four corners so that the sides can be folded up to make a box. Let x represent the length of the sides of the squares, in inches, that are to be cut out. Express the volume of the box as a function of x.
By applying the formula of cuboid volume, the volume of the box can be expressed as 225x - 60x² + 4x³.
Cuboid is a 3-D shape formed by three pairs of rectangles (6 sides), 12 edges, and 8 vertices.
The volume of a cuboid can be calculated as follows:
Volume = length * width * height
Initially, we have a 15 x 15-inch piece of metal, then we cut out the four corners so that the sides can be folded up to make a box.
Now we have a shape as the attached image, where x represents the length of the sides of the squares that are to be cut out.
Thus, after the cutting, the length of the new side is 15 - 2x.
This value also represents the length and width of the box, whilst the height is x.
Now we can calculate the volume of the box:
Volume = length * width * height
= (15 - 2x) * (15 - 2x) * x
= (225 - 60x + 4x²) * x
= 225x - 60x² + 4x³
Thus, the volume of the box can be expressed as 225x - 60x² + 4x³.
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according to the law of large numbers
Answer:
In probability and statistics, the law of big numbers holds that as a sample size increases, its mean gets closer to the average of the entire population. This is because as the sample size increases, the sample becomes more representative of the population.
According to the law of large numbers, a large entity that is expanding quickly cannot continue doing so indefinitely. Often used as instances of this phenomenon are the largest blue chips, with market values in the hundreds of billions.
Two distinct subjects can be covered by the law of huge numbers. First, there are many topics to which the law of large numbers can be applied in statistical analysis. In order to acquire the necessary amount of data, it may not be practical to survey every member of a given population, but every additional data point has the potential to raise the probability that the result is an accurate representation of the mean.
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Problem 3: A city ballot includes a local initiative that would legalize gambling. The issue is hotly contested, and two groups decide to conduct polls to predict the outcome. The local newspaper finds that 53% of 1200 randomly selected voters plan to vote "yes," while a college Statistics class finds 54% of 450 randomly selected voters in support. Both groups will create 95% confidence intervals.
a) Without finding the confidence intervals, explain which one will have the larger margin of error.
b) Find both confidence intervals.
c) Which group concludes that the outcome is "too close to call"? Explain.
a. Without finding the confidence intervals, the larger margin of error will be the statistics class.
b. The confidence interval is (50.5%, 55.5%)
c. The group that concludes that the outcome is "too close to call is the statistics class.
.
How to calculate the interval?It should be noted that the Without finding the confidence intervals, the larger margin of error will be the statistics class. This is because of the smaller sample size.
The group that concludes that the outcome is "too close to call is the statistics class. This is because the intervals that are obtained contains 50%.
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At Lauren's school library, students are allowed to have a balance of no less than -$12 and still check out books. Five days ago, Lauren's balance was $0, but now she owes for her books that are five days late. If the total charge on her checked out books is $2 per day and she turns them in today, can she still check out books? No Yes
Answer:
No
Step-by-step explanation:
2 * 5 = 10
12 does not equal 10
The figure shows a rectangular box in three-dimensional space that contains several vectors. (The vector c is in the xz-plane, and the vector e is in the xy-plane.)
Are the following statements true or false?
1. e = a - b
2. g = f + a
3. a = d
4. d = g - c
5. c = f
6. a = -b
The statement 2( g = f + a) and 5(c = f) are true where as statement 3(a = d), 4(d = g - c), and 6(a = -b) are false.
The statement a = d is FALSE because they have opposite directions, a is in the direction of positive x, while d is in the direction of negative x axis.The statement c= f is TRUE because they have the same directions, in the direction of positive z axis and same length, the height of the rectangular box.The statement d=g−c is FALSE because by the triangle rule c=d+g=> d=−g+c≠g−cThe statement g=f+a is TRUE because the reference vector of f (given by the vector with the initial point at the origin and direction in the positive direction of z), added with the vector a satisfies the triangle rule to obtain the vector g:g=f+aThe statement a=− b is FALSE because their directions are not parallel, they are perpendicular.To learn more about the triangle, visit:
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suppose that you have a photograph and you want to crop it to cut off 2 inches from the right side of the picture and then resize it using various scale factors (for our purposes here we will focus on a scale factor of 1.5). the transition from the dimensions of the original photograph to the dimensions of any scaled version is a two-step process.
Yes, cropping a picture and then using a scale factor to resize it is a two-step process.
The image is first cropped by subtracting 2 inches from the right side, altering the image's proportions.
The image is then enlarged using a scale factor of 1.5, which increases the image's dimensions by 1.5.
The final measurements of the image would be its original measurements less 2 inches (for cropping), then multiplied by 1.5. (for the resizing).
It's vital to keep in mind that both picture cropping and image scaling change the size of the final product.
Yes, cropping a photo, then resizing it with a scale factor, is a two-step process.
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lian's truck has a 25 gallon tank and uses 0.05 gallon of gas for every mile driven.when creating a table for the function y=-0.05x + 25 ,lian argues that he can only use positive rational numbers for the input .is lian correct? justify your answer
Since it is possible to drive a fraction of a mile, so it will be appropriate to use fractions; however, it is not possible to drive a negative number of miles, so only positive rational numbers are appropriate.
How to interpret Linear equations?A rational number is said to be positive if its numerator and denominator are both positive.
Now, we are told that lian's truck uses 0.05 gallon of gas for every mile driven. Thus;
Rate of consumption of gas = 0.05 gallons per mile
The function for number of gallons is;
y = -0.05x + 25
where;
x is number of miles
y is number of gallons
We can use a fraction for the number of miles because it is possible but then the number of miles must be positive rational numbers because number of miles cannot be negative.
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A student is standing 20 ft away from the base of a tree. He looks up to the top of the tree at a 50° angle of elevation. His eyes are 5ft above the ground. What is the height of the tree to the nearest foot?
The height of the tree is 29 feet.
What is an angle of elevation?A measure of an angle formed which an object is viewed above the horizontal plane is termed angle of elevation.
Let the distance from the top of the tree to the student's eyes be represented by x. Thus, applying the required trigonometric function;
Tan θ = opposite/ adjacent
Tan 50 = x/ 20
x = 20*Tan 50
= 23.8351
x = 24 ft
So that;
the height of the tree = 24 + 5
= 29 feet
Therefore, the height of the tree is 29 feet.
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write the measure in degrees 11pi/6
well, we know that π is 180°, so what's 11π/6?
[tex]\begin{array}{ccll} radians°rees\\ \cline{1-2} \pi &180\\\\ \frac{11\pi }{6}&x \end{array}\implies \cfrac{\pi }{ ~~ \frac{11\pi }{6} ~~ }~~ = ~~\cfrac{180}{x}\implies \cfrac{ ~~ \frac{\pi}{1} ~~ }{\frac{11\pi }{6}}~~ = ~~\cfrac{180}{x} \\\\\\ \cfrac{6\pi }{11\pi }=\cfrac{180}{x}\implies \cfrac{6 }{11 }=\cfrac{180}{x}\implies 6x=1980\implies x=\cfrac{1980}{6}\implies x=330[/tex]
The path of a firework can be represented by F9t=16^+92t+160
What is the maximum height of the firework?
How long did it take to reach its maximum height?
a) The maximum height of a firework is 292.25 feet.
b) The time required by a firework to reach its maximum height is 2.875 second.
How to determine the maximum height reached by a firework and its time needed
Herein we find the case of a firework being launched under free fall motion, that is, a vertical uniformly accelerated motion due to gravity, represented by following quadratic equation:
f(t) = - 16 · t² + 92 · t + 160
Where:
f(t) - Height, in feett - Time, in secondsPart A - The maximum height can be found by modifying the quadratic equation into vertex form. This can be done by completing the square:
f(t) = - 16 · t² + 92 · t + 160
f(t) = - 16 · [t² - (23 / 4) · t - 10]
f(t) - 16 · (529 / 64) = - 16 · [t² - (23 / 4) · t + 529 / 64] + 160
f(t) - 292.25 = 16 · (t - 2.875)²
The firework reaches a maximum height of 292.25 feet.
Part B - The firework takes a time of 2.875 seconds to reach its maximum height.
RemarkFree fall equation has several mistakes and was corrected according to principles of free fall motion.
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Given that 5 x : 9 = 8 : 3 Calculate the value of x . Give your answer in its simplest form.
James got a 75/100 and James teacher wants to convert that mark over 100 into ?/40 how much will James mark be now out of 40
Events A and B are mutually exclusive. Events B and C are non-mutually exclusive. which Venn diagram could represent this description?
The Venn diagram that represents the description on Events A and B would be the A. First Diagram.
What does mutually exclusive mean ?A statistical term used to describe events that cannot occur concurrently is "mutually exclusive." It is frequently used to refer to circumstances in which the occurrence of one event takes precedence over the other.
This means that Events A and B cannot happen at the same time and therefore cannot share circles on the Venn diagram.
Events B and C are non - mutually exclusive however and so will share a circle on the Venn Diagram as shown below.
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Answer: D
last one
Step-by-step explanation: just did the test
Modeling Problems with Equations Dean ran 2.3 fewer kilometers than Sam. If Dean ran 6.8 km, how far did Sam run? A 2-column table with 4 rows. Column 1 is labeled Situation with entries increasing, difference, finding part of a total, sharing or grouping. Column 2 is labeled Operation with entries +, minus, times, divided by. Select all that apply. You know the difference in the distances the boys ran, so this is a subtraction problem. You are finding the total distance the boys ran, so this is an addition problem. Dean ran part of the distance Sam ran, so this is a multiplication problem. The correct equation is s + 2.3 = 6.8. The correct equation is s – 2.3 = 6.8. The correct equation is 2.3s = 6.8.
Since Dean ran 2.3 fewer kilometers than Sam. If Dean ran 6.8 km, the true statements about the distance which Sam ran are:
A. You know the difference in the distances the boys ran, so this is a subtraction problem.
E. The correct equation is s – 2.3 = 6.8.
What is distance?In Mathematics, distance can be defined as the total amount of ground that is covered or travelled by a physical object (body) over a particular period of time and speed, irrespective of its direction, starting point or ending point.
Since Dean ran 6.8 km and 2.3 fewer kilometers than Sam, we can logically deduce that this is a subtraction problem. Assuming the variable s represent the distance which Sam ran, Sam's distance can be calculated as follows;
s – 2.3 = 6.8
s = 6.8 + 2.3
s = 9.1 kilometers.
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A coin is flipped 200 times. The table shows the frequency of each event.
Outcome Frequency
Heads 96
Tails 104
Determine the experimental probability of landing on tails.
50%
52%
96%
104%
Answer:
52%
Step-by-step explanation:
Since the question asks about tails, we don't need the information about heads.
So out of 200 flips, 104 were tails. If we put 104 out of 200 in fraction form, you get;
104
-----
200
Now we just divide;
104/200 = 0.52, also known as 52%
The venn diagram shows sports played by 10 students let event A= The student play basketball let event B= the student plays soccer
The value of P(A|B) from the given data is 1/6 ≅ 0.17
Conditional Probability:The term P(A/B) is known as a conditional probability which indicates the probability of an event A that depends on another event B.
The probability of an event or outcome occurring due to the existence of a prior event or outcome is known as conditional probability.
P(A/B) is used to find this conditional probability.
The formula for P(A/B) is given by
P(B/A) = P(A∩B) / P(A)Here we have
A ven diagram which shows sports played by 10 students
From given figure
Number of students who play basketball, p(A) = 3
Number of students who play soccer, p(B) = 2
Number of students who play both games, A∩B = 3
As we know, P(A/B) = P(A∩B)/P(B)
= 1/3 [2]
= 1/6 ≅ 0.17
Therefore,
The value of P(A/B) from the given data is 1/6 ≅ 0.17
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de matics class into equal groups 2,5,6 without leaving any learner out what is the smallest possible answer in the class?
the smallest possible answer in the class is 30.
What are mathematical operations?The term "operation" in mathematics refers to the process of computing a value utilizing operands and a math operator. For the specified operands or integers, the math operator's symbol has predetermined rules that must be followed. In mathematics, there are five basic operations: addition, subtraction, multiplication, division, and modular forms.
Given, Mr. Abraham’s is able to divide the learners in his Mathematics class into equal groups of 2, 5 or 6 members without leaving any learners out.
You just need to find the smallest number that can be divided by 2 or 5 or 6 evenly
The Least Common Multiple
there are various ways to do this
start with 5
multiples of 5 are 5 10 15 20 25 30 35 etc.
see any number in there that is also divisible by 2 and 6
Therefore, 30 is the smallest possible answer in the class.
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Complete question:
Mr. Abraham’s is able to divide the learners in his Mathematics class into equal groups of 2, 5 or 6 members without leaving any learners out. What is the smallest possible number of learners in the class?
knewton.com a
First up, let's review the assignment's learning
objectives
Get familiar with this topic by reviewing instruction and answering a couple of
questions.
Provide your answer below:
LTE 36%
Question
Montana has campgrounds over 41 counties. Nevada has 637 campgrounds over 18 counties. If
Montana was proportional to Nevada in the number of campgrounds to counties, how many
campgrounds would Montana expect to have? Round to the nearest whole number.
Answer: In order to find the number of campgrounds that Montana would expect to have if it was proportional to Nevada in the number of campgrounds to counties, we need to determine the ratio of campgrounds to counties in Nevada and then apply that same ratio to Montana.
First, we'll find the ratio of campgrounds to counties in Nevada by dividing the number of campgrounds (637) by the number of counties (18): 637 / 18 = 35.389
Now we'll apply that same ratio to Montana by multiplying the number of counties in Montana (41) by the ratio of campgrounds to counties in Nevada: 41 x 35.389 = 1458.369
Rounding to the nearest whole number, we get 1458. So if Montana was proportional to Nevada in the number of campgrounds to counties, we would expect to find 1458 campgrounds in Montana.
Step-by-step explanation:
write an equation of the line that passes through (-6,0) and (0,-24)
Answer:
y=-4x-24
Step-by-step explanation:
draw the line on a grid paper so that you know the slope must be negative. then plug in the numbers.
Sam has 9 1/3 cups of cereal. Each serving is 2/4 cup. How many servings does Sam have?
The number of servings that Sam has would be = 18.67 servings.
What is a fraction?A fraction is part of a whole value that is being represented in a form of numerator and denominator.
The number of cups that Sam has = 9 1/3 = 28/3
The number of cups used for each serving = 2/4
That is ;
1 serving = 2/4 cups
X servings = 28/3 cups
Make X the subject of formula;
X servings = 28/3 ÷ 2/4
= 28/3 × 4/2
= 112/ 6
= 18.67 servings.
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Recall that the side length of the square was 14 units long. What is the base area of the prism?
Diameter of cylinder base - 19.8
base area of cylinder - 307.8
volume of cylinder - 9.234
Answer:
56
Step-by-step explanation:
The base area will be 4 times the side length
56 units
Find the exact value of sin(x + y), given sin(x) = Negative one-half, cos(y) = Negative one-third, x is an angle in quadrant III, and y is an angle in quadrant II.
StartFraction 2 minus StartRoot 5 EndRoot Over 6 EndFraction
StartFraction 1 minus 2 StartRoot 6 EndRoot Over 6 EndFraction
StartFraction 6 minus StartRoot 5 EndRoot Over 6 EndFraction
StartFraction StartRoot 3 EndRoot minus StartRoot 5 EndRoot Over 6 EndFraction
The exact value of sin(x + y) is (1 - 2√6) /6. The 2nd option is the answer.
How to find the exact value of sin(x + y)?We have sin(x) = -1/2 and cos(y) = -1/3
x is an angle in quadrant III, and y is an angle in quadrant II
For x:
Using trigonometric ratio
sin(x) = -1/2 (sin = opposite/hypotenuse)
adjacent = √(2²-(-1)²) = √3
cos(x) = (-√3)/2 (cos = adjacent/hypotenuse)
Note: adjacent in quadrant III is negative
For y:
Using trigonometric ratio
cos(y) = -1/3 (cos = adjacent/hypotenuse)
opposite = √(3²-(-1)²) = √8 = 2√2
sin(y) = (2√2)/3 (sin = opposite/hypotenuse)
Using trigonometric identity:
sin (x + y) = sin (x)·cos(y) + cos (x)·sin (y)
sin (x + y) = -1/2·(-1/3) + [(-√3)/2]·[(2√2)/3]
sin (x + y) = 1/6 - (√6)/3
sin (x + y) = (1 - 2√6) /6
Thus, the exact value of sin(x + y) is (1 - 2√6) /6
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