how do i do part (iii). the answer is 2 but i got the answer based on guessing. how do i do this w/o guessing? thank you.
Answers
Answer:
(i) 2² · 3 · 7
(ii) 3024
(iii) k = 16
(iv) 12 cm
Step-by-step explanation:
Part (i)The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, ...
To find which prime numbers multiply together to make 84, start by dividing 84 by the first prime number, 2:
⇒ 84 ÷ 2 = 42
As 42 is not a prime number, we need to divide again:
⇒ 42 ÷ 2 = 21
As 21 is not a prime number, we need to divide again.
21 is not divisible by 2, so let's try dividing by the next prime number, 3:
⇒ 21 ÷ 3 = 7
As 7 is a prime number, we can stop.
Therefore, 84 is the product of:
⇒ 84 = 2 · 2 · 3 · 7
As 2 appears two times, we can write this using exponents:
⇒ 84 = 2² · 3 · 7
Part (ii)Lowest Common Multiple (LCM): The lowest multiple shared by two or more numbers.
Prime Factors Method of finding LCM
Step 1
Find the prime factorization of each number.
From part (i): 84 = 2² · 3 · 7
Given in part (ii): 432 = 2⁴ · 3³
Step 2
Write each number as a product of primes, matching primes vertically where possible:
[tex]\begin{array}{ r c c c c c c c ccccccccc}84&=& 2 & \cdot & 2 & \cdot & & & & & 3& \cdot & & & && 7\\\\432 &=& 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3\\\\\cline{1-17} \\\sf LCM & = & 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3 &\cdot & 7\end{array}[/tex]
Step 3
Bring down the primes in each column then multiply the factors to get the LCM:
⇒ LCM = 2 · 2 · 2 · 2 · 3 · 3 · 3 · 7
⇒ LCM = 2⁴ · 3³ · 7
⇒ LCM = 3024
Part (iii)From part (ii) we know that the prime factorization of 432 is:
432 = 2⁴ · 3³
A perfect cube is a number multiplied by itself three times.
One of the factors of 432 is a perfect cube → 3³.
Therefore,
[tex]\begin{aligned}432 & = 2^4 \cdot 3^3\\\\\implies \dfrac{432}{2^4} & = 3^3\\\\\dfrac{432}{16} & = 3^3\end{aligned}[/tex]
So we need to divide 432 by 2⁴ to get the perfect cube 3³.
Therefore,
⇒ k = 2⁴
⇒ k = 2 · 2 · 2 · 2
⇒ k = 16
Part (iv)To find the largest possible length of the side of the square, we need to find the greatest common factor of 432 and 84.
Prime Factors Method of finding GCF
Step 1
Find the prime factorization of each number.
From part (i): 84 = 2² · 3 · 7
Given in part (ii): 432 = 2⁴ · 3³
Step 2
Write each number as a product of primes, matching primes vertically where possible:
[tex]\begin{array}{ r c c c c c c c ccccccccc}84&=& 2 & \cdot & 2 & \cdot & & & & & 3& \cdot & & & && 7\\\\432 &=& 2 & \cdot & 2 &\cdot & 2& \cdot & 2& \cdot & 3& \cdot & 3& \cdot & 3\\\\\cline{1-17} \\\sf GCF & = & 2 & \cdot & 2 & \cdot & & & & & 3& & & & && \end{array}[/tex]
Step 3
Bring down the primes in the columns where both numbers have the factor, then multiply the factors to get the GCF:
⇒ GCF = 2 · 2 · 3
⇒ GCF = 12
Therefore, the largest possible length of the side of the square is 12 cm.
Related Questions
A recipe uses both carrots and onions. The recipe uses 4 times more carrots than onions. If the recipe uses a total of 25 ounces of chopped carrots and onions, how many ounces of onions does the recipe use?
Answers
The ounces of onions is 5 ounces.
How to calculate the value?Let the onions be represented by x.
The recipe uses 4 times more carrots than onions. The carrots will be 4 × x = 4x.
Therefore, the equation will be:
4x + x = 25
5x = 25
Divide by 5
x = 25/5
x = 5
Therefore, the ounces of onions is 5 ounces.
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Answer:
Step-by-step explanation:
The ounces of onions is 5 ounces.
The recipe uses 4 times more carrots than onions. The carrots will be 4 × x = 4x.
Therefore, the equation will be:
4x + x = 25
5x = 25
Divide by 5
x = 25/5
x = 5
Therefore, the ounces of onions are 5 ounces.
PLEASE HURRY UP
Lamar needs to buy plastic forks. Brand A has a box of 42 forks for $2.17. Brand B has a box of 48 forks for $3.97.
Find the unit price for each brand. Then state which brand is the better buy based on the unit price.
Round your answers to the nearest cent.
Answers
Answer:
A = .05 and B = .08 so A is the better buy
Step-by-step explanation:
In each case, divide total cost by number of units to get unit cost.
The number has 2 digits the number has 13 as a factor the sum of the digits is 11
Answers
Answer:
65
Step-by-step explanation:
- 2 digits ✅
- Has 13 as a factor ✅
- Sum of digits is 11 ✅
13. There are 4,585 students who
rode the bus to school today.
There were 3,369 students who
came to school another way.
How many students were there
in all at the school?
Answers
The number of students in all at the school is 7,954 students.
How many students are in school?In order to determine the number of students in all at the school, you would add the number of students that rode the bus with the number of students that came to school another way.
Addition is the mathematical operation that is used to determine the sum of two or more numbers. The sign used to denote addition is +. Other mathematical operations are subtraction, division and multiplication.
Number of students in school = number of students that rode the bus + number of students that came another way
4,585 + 3,369 = 7,954 students
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Simplify completely:
(c^-6 d^4)^1/2
Answers
Answer:
d^2/c^3
Step-by-step explanation:
write problem with negative exponent rule
(1/c^6 d^4)^1/2
combine 1/c^6 & d^4
(d^4/c^6)^1/2
use exponent rules to finish simplifying
d^2/c^3
Evan invested $2500 in the stock market.
Four months later, the value of the
investment dropped to $2248. Find the
average monthly change in the value of
the investment.
Answers
Answer:
63
Step-by-step explanation:
Value dropped = 2500 - 2248 = 252
divide 252 across 4 months
252/4 = 63
The average monthly change in the value of the investment is $63.
Given that, Evan invested $2500 in the stock market. Four months later, the value of the investment dropped to $2248.
What is the investment?Investment definition is an asset acquired or invested in to build wealth and save money from the hard earned income or appreciation. Investment meaning is primarily to obtain an additional source of income or gain profit from the investment over a specific period of time.
Now, change in investment is
2500-2248 = 252
The average monthly change
= 252/4
= 63
Hence, the average monthly change in the value of the investment is $63.
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(4x+1)(4x+1)
im so confused
Answers
Answer:
16x² + 8x + 1
Step-by-step explanation:
(4x + 1)(4x + 1)
each term in the second factor is multiplied by each term in the first factor, that is
4x(4x + 1) + 1(4x + 1) ← distribute both parenthesis
= 16x² + 4x + 4x + 1 ← collect like terms
= 16x² + 8x + 1
Answer:
16 x 2 + 8x + 1
Step-by-step explanation:
Using the foil method we expanded the polynomial.
solve the equation. (find only the real solutions. enter your answers as a comma-separated list.) 5x2 10x 4
Answers
After changing the form the equation becomes 5x (x + 2) = 4.
What is an equation?An is a mathematical statement composed of two expressions joined by an equal sign. An equation is 3x - 5 = 16. By solving this equation, we obtain the value of the variable x as x = 7.We cannot solve the equation as the value of x is not given but we can change its form.
Given equation is 5x² + 10x - 4.
Then,
5x² + 10x - 4 = 05x² + 10x = 45x (x + 2) = 4 (Take 5x as common)Therefore, after changing the form the equation becomes 5x (x + 2) = 4.
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The correct question is given below:
solve the equation. (find only the real solutions. enter your answers as a comma-separated list.)
5x² + 10x - 4
Can someone help me on this
Answers
Answer:
one solution
Step-by-step explanation:
-7 works
7 does not
In+a+recent+poll,+64%+of+the+respondents+supported+stricter+gun+laws.+is+this+an+example+of+statistic+or+a+parameter?
Answers
In a recent poll, 64% of the respondents supported stricter gun laws is statistic.
What is an example of a parameter?
To describe the entire population under study, a parameter is utilized. For instance, we are interested in learning the typical length of a butterfly. This information about the total butterfly population makes it a parameter.What is an example of a population parameter?
A parameter is a descriptive measure applied to a whole population. Numerous population parameters exist. For instance, the population size (N) is one parameter, while other parameters that relate to continuous variables are the population's mean diastolic blood pressure or mean body weight.Learn more about population parameter
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In a certain school, there were 180 students in year 7. 110 study french, 88 study german, 65 study indonesian, 40 study both french and german while 38 study german and german only. find the number of students who study all the three languages. also, find the number of students who study indonesian only
Answers
The number of students who study Indonesian only = 10
Total pupils= 110
Study French = 88
Study Indonesian = 65
Study French and German = 40
Study German and Indonesian = 38
Study French and Indonesian = 26
Study German only = 19
Students who study German only = G - F∩G - G∩I + F∩G∩I
19= 88-40-38 + F∩G∩I
F∩G∩I = 9
Students who study all three languages = 9
So the students who study only Indonesian = I - F∩I - G∩I + F∩G∩I
= 65- 38-26+ 9
= 10
Therefore the number of students studying Indonesian only = 10.
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The complete question is:
In a certain school, there are 180 pupils in Year 7. One hundred and
ten pupils study French, 88 study German and 65 study Indonesian.
Forty pupils study both French and German, 38 study German and
Indonesian, and 26 study both French and Indonesian, while 19 study
German only. Find the number of pupils who study Indonesian only.
What is your monthly take home pay if your take home pay is $897.84 every monday
Answers
|2x-5| < 7
How to solve for x?
Answers
Answer:
-1 < x < 6
Step-by-step explanation:
|2x - 5| < 7
-7 < 2x - 5 < 7
-2 < 2x < 12
-1 < x < 6
need help with this! show work please!
Answers
Answer:
(a) 1/2
(b) $1,080,000
(c) 72°
Step-by-step explanation:
We have to find the ratio in which the money is being distributed
To do this first add up the individual values in the ratio 5:3:2 which represents the ratio C:D:E for apportioning savings
5 + 3 + 2 = 10
Chris's share = 5/10 = 1/2
Danny's share = 3/10
Evelyn's share = 2/10 = 1/5
Answers
(a) Fraction of Danny's share = 1/2
(b) If Danny is to receive $540,000 and his share is 1/2 of total savings then the total savings = 2 x 540,000 = $1,080,000
(c) The ratio of the angles in a pie chart will be the same as the ratio of the division. A pie chart shows the relative shares as part of a circle. Since a circle contains 360° , the angle of the sector that represents Evelyn's share will be 1/5 x 360 = 72°
The data below shows the number of people visiting a local clinic per day to be vaccinated against Measles.
5 12 19 29 35 23 15 33 37 21 26 18 23 18 13 21 18 22 20
1.1 Determine the mean of the given data. (2)
1.2 Calculate the standard deviation of the data.(2)
1.3 Determine the number of people vaccinated against Measles that lies within ONE standard deviation of the mean.(2)
1.4 Determine the interquartile range for the data.(3)
1.5 Draw a box and whiskers diagram to represent the data.(3)
1.6 Identify any outliers in the data set. Substantiate your answer.(2)
Answers
Answer:
1.3
2.5
3.7
4.95.0
5.38
6.67
Step-by-step explanation:
Answer:
Step-by-step explanation:
34
45
11
9
78
56
If the sum of squares for a sample containing 51 items equals 400, what is the standard deviation? 8.00 7.84 4.47 2.83
Answers
The standard deviation is 2.83 if the sum of squares for a sample containing 51 items equals 400 option (D) is correct.
What is the standard deviation?It is defined as the measure of data disbursement, It gives an idea about how much is the data spread out.
[tex]\rm SD = \sqrt{\dfrac{ \sum (x_i-X)^2}{n-1}[/tex]
SD is the standard deviation
xi is each value from the data set
X is the mean of the data set
n is the number of observations in the data set.
∑(x(i) - X)² = 400
n = 51
SD = √(400/51-1)
SD = √8
SD = 2.828 ≈ 2.83
Thus, the standard deviation is 2.83 if the sum of squares for a sample containing 51 items equals 400 option (D) is correct.
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14) Out of the following options, pick the one that
can be expressed as a rational number.
a) 1.16321632...
b) 0.00120345...
c) 5.672101234 ...
d) 2.75757575...
i have no idea what this means
Answers
0.7+y=−1.34 Please solve it I tried 0.64 and it said it was incorrect.
Answers
Answer:
-2.04
Step-by-step explanation:
it'll be
-1.34 - 0.7 = y
- 2.04 = y
compute 9600 divided by 30
Answers
Answer:
9600 divided by 30 is 320
Please Mark as brainliest.
Better taxi service charges $3.00 for the first mile plus $0.20 for each additional mile. Best
taxi service charges $6.00 for the first two miles plus $0.10 for each additional mile. How
many miles will a person travel in order to spend an equal amount of money using either
service?
PLEASE AN EXPERT THIS IS WORTH 50 POINTS!!!!!!!!!
Answers
An angle is defined as the space between two rays that emanate from a common point. True or false
Answers
True, there are some undefined terms like line, point angle, distance between two points .
Which definition of angle is the best?
The figure in which two rays intersect at the vertex, or common point, is referred to as an angle. The symbol "∠" is used to represent the angle.How does Euclid define angle?
A plane angle is the inclination toward one another of two intersecting, non-straight lines in the plane.
True, there are some undefined terms like line, point angle, distance between two points .
we intuitively know about the angle but can not give mathematical definition .
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R, S, and T are collinear. S is between R and T. RS =2w + 1, ST = w - 1, and RT = 18. Use the Segment Addition Postulate to slove for w. Then determine the lenght of RS.
Answers
R, S, and T are collinear . Then 13 is the length of RS.
What do collinear points mean?
Collinear points are those that are situated along a single or shared straight line. In Euclidean geometry, two or more points are said to be collinear if they are located on a line either near or far from one another.The question to know
RS + ST = RT
RS = w + 1 , ST = w - 1
RT = 18
2w + 1 + w - 1 = 18
3w = 18
w = 18/3
w = 6
RS = 2 * 6 + 1 = 13
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Triangle PQR is transformed to triangle P′Q′R′. Triangle PQR has vertices P(3, −6), Q(0, 9), and R(−3, 0). Triangle P′Q′R′ has vertices P′(1, −2), Q′(0, 3), and R′(−1, 0).
Plot triangles PQR and P′Q′R′ on your own coordinate grid.
Part A: What is the scale factor of the dilation that transforms triangle PQR to triangle P′Q′R′? Explain your answer. (4 points)
Part B: Write the coordinates of triangle P′′Q′′R′′ obtained after P′Q′R′ is reflected about the y-axis. (4 points)
Part C: Are the two triangles PQR and P′'Q′'R′' congruent? Explain your answer. (2 points)
Answers
Answer:
A: sf 2 or 1/2
Step-by-step explanation:
Answer:
Part A: The scale factor of the dilation that transforms Triangle PQR to Triangle P'Q'R' is 1/4
Part B:
P''(-1, 0)
Q''(0, -1)
R''(2, -1)
Part C:
The two Triangles PQR and P''Q''R'' are not congruent.
Step-by-step explanation:
Noah is trying to bisect angle BAC. He draws circles of the same radius with centers B and C and then uses one of the points of intersection for his ray. What mistake has Noah made in his construction
Answers
The mistake done by Noah in his construction is that; length of the line segment AB and AC should be equal.
How to bisect an angle?
The steps in bisecting an angle say PQR is as follows;
Start with angle PQR that we will bisect.
1. Place the compasses' point on the angle's vertex Q.
2. Adjust the compasses to a medium wide setting. The exact width is not important.
3. Without changing the compasses' width, draw an arc across each leg of the angle.
4. The compasses' width can be changed here if desired. Recommended: leave it the same.
5. Place the compasses on the point where one arc crosses a leg and draw an arc in the interior of the angle.
6. Without changing the compasses setting repeat for the other leg so that the two arcs cross.
7. Using a straightedge or ruler, draw a line from the vertex to the point where the arcs cross.
8. This is the bisector of the angle ∠PQR.
Therefore, the mistake done by Noah is that length of the line segment AB and AC should be equal in order to bisect it.
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A particle moving uniformly along the x axis is located at 16.5 m at 1.82 s and at 4.97 m at 3.81 s. find its displacement during this time interval. answer in units of m\
Answers
For a particle moving uniformly along the x axis is located at 16.5 m at 1.82 s and at 4.97 m at 3.81 s, the displacement is -11.53 m
For given question,
A particle moving uniformly along the x axis is located at 16.5 m at 1.82 s and at 4.97 m at 3.81 s.
We can observe that, for t1 = 1.82 s and t2 = 3.81 s
object is moving from right to left along the x axis.
We know that, the displacement of a particle moving in a straight line is the change in its position.
so, d = 4.97 - 16.5
d = -11.53
The negative sign indicates that the object is moving from right to left along the x axis for t1 = 1.82 s and t2 = 3.81 s
Therefore, for a particle moving uniformly along the x axis is located at 16.5 m at 1.82 s and at 4.97 m at 3.81 s, the displacement is -11.53m.
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I need help 10-25 please help asap step by step
Answers
Find the 14th term of the arthmetic sequence x-1, 7x -7 , 13x-13,....
Answers
Since they're in arithmetic progression, there is a constant difference between consecutive terms of
[tex](7x - 7) - (x - 1) = 6x - 6[/tex]
Recursively, the sequence is given by
[tex]\begin{cases} a_1 = x-1 \\ a_n = a_{n-1} + 6x - 6 & \text{for }n\ge2\end{cases}[/tex]
By substitution, we have
[tex]a_2 = a_1 + 6x-6[/tex]
[tex]a_3 = a_2 + 6x-6 = a_1 + 2(6x-6)[/tex]
[tex]a_4 = a_3 + 6x-6 = a_1 + 3(6x-6)[/tex]
and so on, up to the 14th term
[tex]a_{14} = a_1 + 13(6x-6)[/tex]
[tex]a_{14} = (x-1) + 78 (x - 1)[/tex]
[tex]a_{17} = 79 (x - 1) = \boxed{79x-79}[/tex]
What is 0.2302 to the thousands place
Answers
Read the problem below and find the solution. Draw a diagram on your own
paper to help solve it.
A group of 29 friends gets together to play a sport. First, people must be
divided into teams. Each team has to have exactly 4 players, and no one can
be on more than one team. How many teams can they make? (It is possible
that not everyone can be on a team.)
(Do not include units in your answer.)
Answer here
Answers
Answer:
yes
Step-by-step explanation:
28/4 = 7 teams
1 is left without a team
PLEASE HELP ASAP!!
Answers
The equation of the graph is (x-3)(x+1), the y-intercept of y=[tex]x^{2} -5x+4[/tex] is "4" and the factors of y=[tex]x^{2} -5x+4[/tex] are (x-4) ad (x-1)
Given that a graph touch x-axis at 2 points i.e. 2 roots are there for given graph.The graph touches the x-axis at (3,0) and (-1,0) it implies x=3 and x=-1 are the solutions the equation,(x-3) and (x+1) are factors of equation
Therefore,the equation of graph is (x-3)(x+1).
Given the equation y=[tex]x^{2} -5x+4[/tex]
The y-intercept is defined as the graph passing a fixed point at x=0 line
i.e.(0,c) pass through it . c is the y-intercept
c=0-0+4
c=4
Therefore,The y-intercept of the equation y=[tex]x^{2} -5x+4[/tex] is 4
Factorising y=[tex]x^{2} -5x+4[/tex]
⇒ y=[tex]x^{2} -5x+4[/tex]
⇒y=[tex]x^{2} -x-4x+4[/tex]
⇒y=x(x-1)-4(x-1)
⇒y=(x-4)(x-1)
Therefore, the factors of y=[tex]x^{2} -5x+4[/tex] are (x-4) and(x-1)
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