Given:-
A small cup has height of 6in. and a large cup has a height of 8in.Volume of coffee in large cup is 18oz .To find:-
The volume of coffee small cup .Answer:-
Here we can assume the shape of the cup to be cylindrical . And the volume of a cylinder is given by;
[tex]\implies V = \pi r^2 h \\[/tex]
where ,
[tex]r[/tex] is the radius of the base of the cylinder.[tex]h[/tex] is the height of the cylinder.Assuimg that the cups differ only in height and not in width (diameter) , we can say that ;
[tex]\implies V \propto h \\[/tex]
So , we can write;
[tex]\implies \dfrac{V_s}{V_l}=\dfrac{h_s}{h_l}\\[/tex]
[tex]\implies \dfrac{V_s}{18\ oz}=\dfrac{6\ in.}{8 \ in.}\\[/tex]
[tex]\implies V_s = \dfrac{6(18)}{8} \\[/tex]
[tex]\implies \underline{\underline{ V_s =13.5 \ oz}} \\[/tex]
Hence the volume of the smaller cup is 13.5 oz .
and we are done!
Directions: Work in your small group to complete this worksheet Submit the completed Wor full credit. Canvas by the beginning of the next class. All answers must be justified and work Trigonometric functions are periodic, meaning they repeat the same pattern as x goes toward positive and negative infinity. Hence, trigonometric functions fail to be one to one. Recall, one to one functions must pass BOTH vertical and horizontal line tests. Therefore, to define the trigonometric inverse functions, we must restrict the domain of each trigonometric function to a place where the function IS one to one. 1. Let g(x) be the function whose domain is [0,π] and whose outputs are determined by cos(x) on this interval. Hence, g(x) is the solid line part of the graph pictured to the right. Note: g(x) is defined in terms of cos(x), however g(x)=cos(x) since it has a different domain. a) Does y=cos(x) have an inverse function? Why or why not? b) Explain why g(x) has an inverse function, g−1(x)=cos−1(x). c) State the domain and range of g(x) in interval notation. Domain: Range: d) State the domain and range of g−1(x)=cos−1(x) in interval notation. Domain: Range:
a)Does not have an inverse function
b) g(x) has an inverse function
c)Domain of g(x) is [0, π],range is [-1, 1]
d)domain of g−1(x)=cos−1(x) is [-1, 1], range is [0, π]
a) No, y=cos(x) does not have an inverse function because it fails the vertical line test.
b) g(x) has an inverse function, g−1(x)=cos−1(x), because it passes the vertical and horizontal line tests when its domain is restricted to the interval [0, π].
c) The domain of g(x) is [0, π], and the range is [-1, 1] in interval notation.
d) The domain of g−1(x)=cos−1(x) is [-1, 1], and the range is [0, π] in interval notation.
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The equation of the passing through the points
(-1,-4) and (3,0)
The equation of the line passing through the points (-1, -4) and (3, 0) is y = x - 3.
To find the equation of the line passing through two points, we need to use the point-slope form of the equation of a line.
Point-slope form: y - y₁ = m(x - x₁)
where:
(x₁, y₁) is one of the given points on the line
m is the slope of the line
To find the slope of the line passing through (-1, -4) and (3, 0), we can use the formula:
m = (y₂ - y₁)/(x₂ - x₁)
where:
(x₁, y₁) = (-1, -4)
(x₂, y₂) = (3, 0)
m = (0 - (-4))/(3 - (-1))
m = 4/4
m = 1
Now we can use either of the two points and the slope to write the equation of the line.
Let's use point (-1, -4):
y - (-4) = 1(x - (-1))
y + 4 = x + 1
y = x - 3
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Answer the question below:
Fill in the Box and give the frequency for A,B,C and D.
find the equation of a straight line passing through a point (-2, 1) and perpendicular to the line -3x +5y +2=0
Answer:
y= -5/3x - 7/3
Step-by-step explanation:
1. solve the line
-3x +5y + 2 = 0
+3x +3x
5y + 2 = +3x
-2 -2
5y = +3x - 2
/5 /5
y= 3/5x -2/5
2. find slope
perpendicular lines have an opposite reciprocal slope to original line so your slope is now -5/3
3. find b ( y - intercept )
point ( -2 , 1 )
y - y1 = m ( x - x1 )
y - 1 = -5/3 ( x + 2 )
y -1 = -5/3x - 10/3
+1 + 1
y= -5/3x - 7/3
in pawnee, indiana, the price of a pound of bacon (x) varies from day to day according to a normal distribution with mean of $4.10 and a standard deviation of $0.16. the price of a dozen eggs (y) also varies from day to day according to a normal distribution with a mean of $1.90 and a standard deviation $0.06. assume the prices for a pound of bacon and a dozen eggs are independent. (a) find the probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs. that is, find p(x > 2y). (b) ron wants to cook breakfast for himself, so he buys 10 pounds of bacon and 6 dozen eggs. find the probability that he paid more than $50 for the bacon and eggs.
The probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs is 0.0001. The probability that Ron paid more than $50 for the bacon and eggs is 0.0028.
(a) To find the probability that on a given day, the price of a pound of bacon is more than twice as expensive as a dozen eggs, we need to use the normal distribution formula. This is given by P(x > 2y) = P(X > 2µy + 2σy) = P(Z > (2µy + 2σy - µx) / σx) = P(Z > (2(1.90) + 2(0.06) - 4.10) / 0.16) = P(Z > -4.24) = 1 - P(Z < -4.24) = 1 - 0.9999 = 0.0001
(b) To find the probability that Ron paid more than $50 for the bacon and eggs, we need to calculate the probability that the 10 pounds of bacon and 6 dozen eggs cost more than $50. This is given by P(10x + 6y > 50) = P(10X + 6µy + 6σy > 50) = P(Z > (50 - 10µx - 6µy - 6σy) / (√10σx + √6σy)) = P(Z > (50 - 10(4.10) - 6(1.90) - 6(0.06)) / (√10(0.16) + √6(0.06)) = P(Z > -2.87) = 1 - P(Z < -2.87) = 1 - 0.9972 = 0.0028.
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Polygon ABCD with vertices at A(−4, 6), B(−2, 2), C(4, −2), D(4, 4) is dilated using a scale factor of 3/4 to create polygon A′B′C′D′. Determine the vertices of polygon A′B′C′D′.
A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3)
A′(−12, 18), B′(−6, 6), C′(12, −6), D′(12, 12)
A′(3, −4.5), B′(1.5, −1.5), C′(−3, 1.5), D′(−3, −3)
A′(4.5, −3), B′(1.5, −1.5), C′(−1.5, 3), D′(3, 3)
The vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).
What is polygon?
A polygon is a closed plane figure that is formed by three or more straight sides, which are connected end-to-end to create a closed shape.
To find the vertices of the dilated polygon, we need to apply the scale factor of 3/4 to each of the original vertices. This will result in a new set of coordinates for each vertex.
The coordinates of point A' can be found by multiplying the x-coordinate and y-coordinate of A by 3/4:
x-coordinate of A' = -4 × 3/4 = -3
y-coordinate of A' = 6 × 3/4 = 4.5
Therefore, A' has coordinates (-3, 4.5).
Using the same method, we can find the coordinates of the other vertices:
B':
x-coordinate of B' = -2 × 3/4 = -1.5
y-coordinate of B' = 2 × 3/4 = 1.5
Therefore, B' has coordinates (-1.5, 1.5).
C':
x-coordinate of C' = 4 × 3/4 = 3
y-coordinate of C' = -2 × 3/4 = -1.5
Therefore, C' has coordinates (3, -1.5).
D':
x-coordinate of D' = 4 × 3/4 = 3
y-coordinate of D' = 4 × 3/4 = 3
Therefore, D' has coordinates (3, 3).
Thus, the vertices of the dilated polygon A'B'C'D' are A'(-3, 4.5), B'(-1.5, 1.5), C'(3, -1.5), and D'(3, 3). Therefore, the correct option is (A) A′(−3, 4.5), B′(−1.5, 1.5), C′(3, −1.5), D′(3, 3).
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How many solutions does the nonlinear system of equations graphed below
have?
O A Zero
• B. One
O C. Four
O D. Two
Answer:
D. Two.
Step-by-step explanation:
The two graphs touch each other at two different points, meaning they have two solutions.
QUESTION 6 Any square matrix multiplied by its inverse will be that size identity matrix.
Given statement: "Any square matrix multiplied by its inverse will be that size identity matrix."
The statement is correct because for a square matrix A, if it is multiplied by its inverse, we get the identity matrix of the same order. The product of the matrix with its inverse is commutative.
The identity matrix is a square matrix with 1's along the diagonal and 0's elsewhere. For example, the 3x3 identity matrix is shown below.
1 0 0
0 1 0
0 0 1
The inverse of a square matrix A is a matrix A^-1, such that AA^-1 = A^-1A = I, where I is the identity matrix.
For example, let us consider a 2x2 matrix A. A = (a, b, c, d)
The inverse of the matrix A is given by the formula. A^-1 = 1/(ad-bc) * (d, -b, -c, a)
Thus, AA^-1 = A^-1A = I, where I is a 2x2 identity matrix. Therefore, any square matrix multiplied by its inverse will be that size identity matrix.
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Dima has twice as many books as Ramiro. If dima gives 12 books to Ramiro, they will have the same number of books.How many books do they have in total
Answer:
They have 36 books in total.
Step-by-step explanation:
Let d = The number of Dima's books
Let r = The number of Ramiro's books.
d = 2r
d - 12 = r Substitute 2r for d in the second equation and solve for r
2r - 12 = r Subtract 2r from both sides
2r - 2r - 12 = r - 2r
-12 = -r Multiply both sides by -1
12 = r
Ramiro has 12 books.
d = 2r substitute 12 for r and solve for d
d = 2(12)
d = 24
Dima has 24 books.
Together they have 36 books (12 + 24)
Helping in the name of Jesus.
Answer:36
Step-by-step explanation:
In a standard normal distribution, an area of \( 0.3974 \) on the right corresponds to a z-score of:
A. -0.26 B. -0.92
C. 2.60 D. 0.26
Option A -026 is the correct answer
The standard normal distribution represents a normal distribution with a mean of 0 and a standard deviation of 1. It is also known as the z-distribution.What is the z-score formula?The z-score formula is: z = (x - μ) / σ where z is the z-score, x is the raw score, μ is the mean, and σ is the standard deviation.In a standard normal distribution, an area of 0.3974 on the right corresponds to a z-score of -0.26. This is option A. Therefore, option A is the correct answer.
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A NFL kicker misses 2 out of every 11 field goals. If he misses 8 field goals throughout the season, how many did he attempt
Answer
:44 field goals
Step-by-step explanation:
Answer:
44 attempts
Step-by-step explanation:
What is a ratio?A ratio has two or more numbers that symbolize relation to each other. Ratios are used to compare numbers, and you can compare them using division.
The ratio for this problem is:
2: 11 (field goals missed: total attempts)To get from 2 to 8, we multiply the 2 by 4.
2 × 4 = 8The ratio is now:
8:?To solve for '?', we can simply multiply the 11 by 4, just like what we did with the 2.
11 × 4 = 44Now the ratio is:
8: 44If the ratio has 44 as the second number, we know that it is the total attempts. Therefore, he attempted 44 field goals.
What is QS and what is c = to
The value of QS is 4.5c + 9.5.
What is parallelogram?A parallelogram is a four-sided plane figure with opposite sides parallel and equal in length. Its opposite angles are also equal.
Since Q is the midpoint of the diagonals RT and SU in parallelogram RSTU, we have:
QS = 1/2 RT ...(1)
QU = 1/2 SU ...(2)
Also, since RU||ST and RS||UT, we know that triangles QRU and QST are similar by AA similarity, so:
QR/QS = QU/QT
Substituting the given values and using (1) and (2), we get:
QR/(6c - 8) = (c + 7)/QT
Cross-multiplying and simplifying, we get:
QR = (6c - 8)(c + 7)/(2c - 1)
QR = (6c^2 + 34c - 56)/(2c - 1)
QR = 3(2c^2 + 11c - 18)/(2c - 1)
QR = 3(2c - 1)(c + 9)/(2c - 1)
QR = 3(c + 9)
Now, using (1), we have:
QS = 1/2 RT
QS = 1/2 (QR + RU)
QS = 1/2 (3(c + 9) + (6c - 8))
QS = 1/2 (9c + 19)
QS = 4.5c + 9.5
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Find the length of an edge of a square whose diagonals are of length 10cm
Answer:
Step-by-step explanation:
Diagonals intersect at 90°.
[tex]x^2=5^2+5^2[/tex] (Pythagoras Theorem)
[tex]=50[/tex]
[tex]x=\sqrt{50}[/tex] cm =7.07 cm
Factor the expression completely. 25 a 2 − 30 a + 9 ( 5 a − 3 ) 2 ( 5 a + 3 ) 2 ( 5 a + 3 ) ( 5 a − 3 )
The factored form of the expression 25a² - 30a + 9 is (5a - 3)².
What is the factored form of the given expression?Given the expression in the question;
25a² - 30a + 9
To factor completely, first rewrite the expression as as follows;
25a² - 30a + 9
rewrite 25a² as ( 5a )²
( 5a )² - 30a + 9
rewrite 9 as 3²
( 5a )² - 30a + 3²
Now, check that the middle term is two times the product of the numbers being squared in the first and third term.
30a = 2 × ( 5a ) × 3
Next, rewrite the polynomial;
( 5a )² - 2 × ( 5a ) × 3 + 3²
( 5a )² - 2( 5a )3 + 3²
Factor using the perfect square trinomial rule.
a² - 2ab + b² = ( a - b )²
Here; a = 5a and b = 3
Hence, we have;
(5a - 3)²
Therefore, the factored form is (5a - 3)².
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Under her cell phone plan, Sydney pays a flat cost of $35.50 per month and $5 per gigabyte. She wants to keep her bill under $50 per month. Write and solve an inequality which can be used to determine x, the number of gigabytes Sydney can use while staying within her budget.
The number of gigabytes that Sydney can use while staying within her budget is less than 2.9 gigabytes.
How to illustrate the information?
From the information, Sydney pays a flat cost of $35.50 per month and $5 per gigabyte. She wants to keep her bill under $50 per month.
The inequality that can be used to determine x x, the number of gigabytes Sydney can use while staying within her budget will be:
35.50 + 5x < 50
Collect like terms
5x < 50 - 35.50
5x < 14.50
x < 14.50 / 5
x < 2.9
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1. The graph shown is a graph of f'(x). If f(2)=5, use the graph to find f(8).
A. 16.5
C. 25.5
D. 21.5
E. 25
The value of f(8) is equal to D)21.5.
What is meant by area?
The region that an object's shape defines as its area. The area of a figure or any other two-dimensional geometric shape in a plane is how much space it occupies.
The graph of f'(x) is given in the figure.
We know that
[tex]\int\limits^a_b {f'(x)} \, dx =f(a)-f(b)[/tex].
Therefore since we have f(2) and we have to find f(8), we will take the area under the given graph from x=2 to 8
[tex]\int\limits^8_2 {f'(x)} \, dx =f(8)-f(2)\\\\2*3+2*3+\frac{1}{2}*3*3=f(8)-5\\ 16.5=f(8)-5\\f(8)=21.5[/tex].
Therefore f(8) is equal to D)21.5.
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find an equation of a line with a slope of -1 and having the point (-2,-6) on the line
Answer: is there a pic?
Step-by-step explanation: is there a pic of what your working on?
Answer:
y=-x-8
Step-by-step explanation:
formular for finding an equation on a line is y-y1=m(x-x1) whiles m is the slope or gradient then x1 is -2 and y1 is -6 then you substitute to the formular you state down
Roberto will put a border around the edges of the platform how much border will he need if the area is 12 square units?
The amount of border that Roberto will need to put around the edges of the platform, given the area is 16 units.
How to find the amount of border ?The platform is given in the figure and to find the amount of border that Roberto needs to cover it if the area is 12 square units, you need to count the number of dots all along the edges of the platforms.
Counting this would give us 16 dots. This then means that Roberto needs sixteen units of border to be able to put up a border around the edges of the platform.
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(Please help!!!)The data below lists the number of pages Tamara read and the time it took her to read them.
Tamara read 35 pages in 42.5 minutes.
Tamara read 41 pages in 59 minutes.
Tamara read 56 pages in 72 minutes.
Determine which table below represents a two-column table for the given data.
Pages Time
35 42.5
59 41
72 56
Pages Time
42.5 35
59 41
72 56
Pages Time
35 42.5
41 59
56 72
Pages Time
42.5 35
41 59
56 72
The data which is given in the question is the time for tamara to read the book can be seen in a table with its respective values.
What is table?
Table is something which has rows and columns and the data is inserted into it.
Table has no restriction for the number of rows and columns. One can insert any rows and columns in the table according to their preferences.
Now in the above question,
Tamara read 35 pages in 42.5 minutes.
Tamara read 41 pages in 59 minutes.
Tamara read 56 pages in 72 minutes.
The table will be Pages Time
35 42.5
41 59
56 72
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Consider a multinomial distribution with 3 different outcomes, that is,
(Z1, Z2, Z3)∼ Multinomial(n,p1,p2,p3),p1+p2+p3=1. Derive the conditional distribution of Z2 given Z1=z1. That is, work out its pmf and name the distribution.
z3 = n − k − z1 − z2
A multinomial distribution with 3 different outcomes is given by (Z1, Z2, Z3)∼ Multinomial(n,p1,p2,p3), where p1 + p2 + p3 = 1. The conditional distribution of Z2 given Z1=z1 is to be derived. The pmf and distribution will be found.Let k be the number of trials with Z1 = z1. Therefore, the number of trials with Z2 = z2 is n − k. Then, the conditional distribution of Z2 given Z1=z1 is(Z1,Z2=n−k,Z3)∼Multinomial(k,p1,p2,p3)×Multinomial(n−k−1,p1′,p2′,p3′),where p1' = p1 / (1 − p1), p2' = p2 / (1 − p1), and p3' = p3 / (1 − p1).As a result, the pmf of the conditional distribution of Z2 given Z1=z1 is given byP(Z2 = z2 | Z1 = z1) = (k!/(z1!z2!(n−k−z1−z2)!))×(p1z1p2z2p3n−k−z1−z2)×(n−k−1)!/(z3!(n−k−z3−1)!(p1'z1+p2'z2+p3'z3)n−k−1), where z3 = n − k − z1 − z2. The distribution obtained is named a "conditional multinomial distribution".
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Evan cut a triangular piece of cloth to use in a quilt. The perimeter of the cloth is 934 cm. The base of the triangular cloth is 214cm. The remaining two sides are the same length.
Choose Yes or No to tell if each expression models how to find the length of the other two sides of Evan’s cloth.
Please help me
Answer:
Step-by-step explanation:
9
Answer:
One expression for the length of one of the equal sides
is x = (934 - 214) / 2.
Step-by-step explanation:
The perimeter is the sum of the 3 sides of the triangle.
Let the length of one of the equal sides be x cm
So, we have:
2x + 214 = 934
2x = 934 - 214 = 720
x = 720 /2 = 360 cm.
X=-3, 0, 5
y=5, 2,-3
Draw a straight line y=2 - x
The line on the graph will be a straight line passing through the points (3,5) and (-3,2).
To draw a straight line y=2 - x, first plot the given points on the graph, (3,5) and (-3,2). Draw a straight line between these two points. This line will be the graph of the equation y=2 - x. To find the equation, take the x-value of the first point, 3, and subtract it from the y-value of the same point, 5. This is the equation of the line, y=2 - x. With the equation of the line, we can now draw a line on the graph that passes through the two given points. The line will be a straight line, with a slope of -1 and a y-intercept of 2.
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Evaluate : x = 3 , y = -6 then
4y - 6x = xy?
"Substituting x = 3 and y = -6 in the given equation, we get:
4y - 6x = xy
4(-6) - 6(3) = (3)(-6)
-24 - 18 = -18
-42 ≠ -18
Therefore, the given equation is not true for x = 3 and y = -6." (ChatGPT, 2023)
Solomon is taking out a $18,320, two-year loan with an APR of 3.29%. What will be the finance charge for this loan to the nearest dollar?
Find the length of the diagonals of rectangle WXYZ. WY = 14x + 10; XZ = 11x + 22
The length of the diagonal of the rectangle is 66
How to find the diagonals of the rectangle?Here we have the rectangle WXYZ.
In that notation, we would have:
WX
XY
YZ
ZW
As the sides
And:
WY
XZ
as the diagonals.
We know that the values of the two diagonals must be the same ones, then we can write:
WY = XZ
14x + 10 = 11x + 22
Now we can solve that for x.
14x - 11x = 22 - 10
3x = 12
x = 12/3 = 4
then the length of the diagonal is:
WY = XZ = 14*4 + 10 = 66
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HELP ME!!
In the figure shown below, ab is a tangent to circle “O” at “a”. Also ab=16, and wb=12. Find the length of the radius.
When eighteen is reduced by one-third of a number, the result is 9. Find the number.
Answer:
27
Step-by-step explanation:
aone third of 27 is 9, and 18 - 9 = 9
The diagram shows a square ABCD with side length k cm. MDE is a sector of a circle, centre D. E lies on the diagonal, BD, of the square. M is the midpoint of AD. Find the percentage of the square that is shaded.
The sοlutiοn οf the given prοblem οf square cοmes οut tο be 0.84% is the shaded area.
What precisely is a square?A square is a quadrilateral οf fοur equal faces and fοur identical angles accοrding tο Euclidean geοmetry. A shape with adjacent edges that have the same length is anοther name fοr a rectangle. If οne οf a quadrilateral's fοur triangular edges and 4 triangular angles is square, the parallelοgram is deemed tο be equilateral. A linear οr 90 ° angle is referred tο as a cubical angle.
Here,
We must determine the area οf the shaded area and divide it by the area οf the cοmplete square tο determine the prοpοrtiοn οf the square that is shaded.
A sectοr is equal tο (45/360)k²/2 = 0.125k²
Find the regiοn οf triangle BDE next. Triangle BDE is a right triangle, sο we can determine its height using the Pythagοrean theοrem:
=> BE² + DE² = BD²
=> BE² + k² = (k√2)²
=> BE² = 2k² - k²
=> BE = k√2
Therefοre, the triangle BDE's area is:
=> A(triangle) = (1/2)
=> BE*DE = (1/2)
=> (k√2)(k√2) = k²
Lastly, the area οf the darkened area is equal tο the sectοr's area less the triangle's area:
=> A(shaded) = A(sectοr) - A(triangle) = 0.125πk² - k² ≈ 0.0084k²
Cοnsequently, the amοunt οf the rectangle that is shaded is:
=> percentage = (A(shaded) / A(square)) x 100%
=> ( 0.0084k² / k²) x 100%
=> 0.84%
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Name:
15. AACE~AABD with side lengths as labeled and BD = 50, CE = 70
a. Find AC.
b. Find the perimeter of A ABD.
c. Find the area of AACE.
Answer:
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alice, bob and carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of x as the denominator. the simplest common denominator of alice's and bob's expressions is 4x2. the simplest common denominator of bob's and carol's expressions is 12x3. the simplest common denominator of alice and carol's expressions is 6x3. find all possible expressions that could be carol's
All possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 .
Alice, Bob and Carol each think of an expression that is a fraction with 1 as the numerator and a constant integer times some power of x as the denominator. The simplest common denominator of Alice's and Bob's expressions is 4x2, while the simplest common denominator of Bob's and Carol's expressions is 12x3. The simplest common denominator of Alice's and Carol's expressions is 6x3. To explain further, Alice's expression can be any fraction with 1 as the numerator and a constant integer times x2 as the denominator, therefore, the denominator of Alice's expression can be any multiple of 4, i.e. 4x2, 8x2, 12x2, 16x2, 20x2 and so on.
Bob's expression can also be any fraction with 1 as the numerator and a constant integer times x2 as the denominator. However, the denominator of Bob's expression must be a multiple of both 4 and 6, i.e. 12x2, 24x2, 36x2, 48x2 and so on. Therefore, Carol's expression must be in the form of a fraction with 1 as the numerator and a constant integer times x3 as the denominator. This means that the denominator of Carol's expression must be a multiple of 6, i.e. 6x3, 12x3, 18x3, 24x3, 30x3 and so on. Hence, all possible expressions that could be Carol's are 1/6x3, 1/12x3, 1/18x3, 1/24x3, 1/30x3 and so on.
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