a) Work out the prime factor decomposition of 396. Give your
answer in index form.
b) The prime factor decompositions of five numbers are shown
below. Use your answer to part a) to work out which two of
these numbers are factors of 396.
88=2³×11
121=11²
9=3²
14=2×7
22=2×11

This final result as a fractiοn and a decimal 115075 using the given fractiοns 6/2 and 4/8

A fractiοn is a mathematical representatiοn οf a part οf a whοle οr a ratiο between twο quantities. It is typically represented as a tοp number (numeratοr) divided by a bοttοm number (denοminatοr), separated by a hοrizοntal line.

Fοr example, the fractiοn 3/4 represents three parts οut οf fοur equal parts οf a whοle. The numeratοr (3) represents the number οf parts, while the denοminatοr (4) represents the tοtal number οf parts that make up the whοle.

Accοrding tο the questiοn:

Let's first find hοw many gummys Jaly has:

Jaly has 241 times greater number οf gummys than Deens, whο has 463. Sο Jaly has:241 x 463 = 111583 gummys Jaly then gave 50 away and received 3542 gummys.

Therefοre, the tοtal number οf gummy's Jaly nοw has is:111583 - 50 + 3542 = 115075We can express this final result as a fractiοn and a decimal using the given fractiοns 6/2 and 4/8.

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The value of sec2(pi/5) - tan2(pi/5) is equals to  1.

What is trigonometric identity ?

A trigonometric identity is a mathematical equation that relates the values of trigonometric functions of an angle. These identities are true for all values of the angle for which both sides of the equation are defined.

We can use the trigonometric identities to simplify this expression. Recall that:

sec(x) = 1/cos(x)

tan(x) = sin(x)/cos(x)

sin2(x) + cos2(x) = 1

Using these identities, we can rewrite the expression as:

sec2(pi/5) - tan2(pi/5)

= (1/cos2(pi/5)) - (sin2(pi/5)/cos2(pi/5))

= (1 - sin2(pi/5))/cos2(pi/5)

= cos2(pi/5)/cos2(pi/5) (using the identity sin2(x) + cos2(x) = 1)

= 1

Therefore, sec2(pi/5) - tan2(pi/5) = 1.

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The value of x is 60.

We can use the fact that the opposite angles of a parallelogram are equal to find the value of x. Angle BCD is therefore equal to angle ABD, and angle CBD is equal to angle CAD.

Based on the given angles, we can construct two equations:

Angle ABD + angle CBD = 100 (because angle BCD = angle ABD) Angle ABD + angle ACD = 80 (because angle CAD = angle CBD)

When we substitute the given values, we get:

x + 40 = 100 x + 20 = 80

When we solve for x in both equations, we get:

x = 60 x = 60.

We can be confident that our answer is correct because both equations give us the same value for x. As a result, the value of x is 60.

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Answer: 0.9 is the answer hope I helped.

Step-by-step explanation:

The require value of side of rectangle(x) is equal to 0.9 in.

A quadrilateral with parallel sides that are equal to one another and four equal vertices is known as a rectangle. It is also known as an equiangular quadrilateral for this reason. Rectangles can also be referred to as parallelograms because their opposite sides are equal and parallel.

We have;

length of longer side = 6.4 in and loneger side of small rectangle = 1.6 in.

Then,

Scale factor = 6.4/1.6  = 4

It means Sides of big rectangle is 4 times of small rectangle.

We also have smaller side of big rectangle = 3.6 in

3.6 = 4x

x = 3.6/4

x = 0.9

Thus, required value of x is 0.9 in.

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The Confidence interval starts at 177.28 and the Confidence interval ends at 187.72. Thus option c is correct.

The given data is as follows:

Number of coffee, N = 28

Mean, µ = 182.5

Standard deviation, α = 14.1

Confidence Level, CL = 95%

z-value for 95% confidence interval = 1.96

The confidence interval is calculated by using the formula,

Confidence interval = μ ± Ζ (α/√n )

Confidence interval = 182.5 ± 1.96*(14.1/√28)

Confidence interval = 182.5 ± 5.22

Confidence interval = (177.28, 187.72)

Therefore we can conclude that the Confidence interval starts at 177.28 and the Confidence interval ends at 187.72.

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The creation of the bin.matrix using cbind() allows us to organize and manipulate our data in a way that is conducive to statistical analysis.

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, with a constant probability of success.

First, let's define what a binomial distribution is. For example, if we toss a fair coin 10 times, the number of heads that come up is described by a binomial distribution.

Now, let's talk about the cbind() function. cbind() is short for "column bind". It is used to combine vectors or matrices into a single matrix, where each input vector or matrix becomes a column in the final matrix.

In this exercise, we have created 7 vectors, each with a different success probability for a binomial distribution, ranging from 0.2 to 0.8. We will use cbind() to combine these vectors into a matrix called bin.matrix.

The dimension of the bin.matrix is

=> 300 x 7.

This means that the matrix has 300 rows and 7 columns. Each row represents a trial, and each column represents a different success probability. For example, the first column might represent a trial with a success probability of 0.2, while the second column might represent a trial with a success probability of 0.3, and so on.

By creating this matrix, we can easily perform statistical analyses on the data contained within it. We can calculate summary statistics, such as the mean and variance, for each column, or we can compare the distributions of the different columns to see if there are any significant differences between them.

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9. a)The required value of h is 4

9.b) The required value of h is 4

10.a)The required value of h is -6

10.b) They are linearly dependent for all values of h.

Here the given vectors are:

v1 = [tex]\left[\begin{array}{ccc}1\\-3\\2&\end{array}\right][/tex]

v2 = [tex]\left[\begin{array}{ccc}-3\\10\\-6&\end{array}\right][/tex]

v3 = [tex]\left[\begin{array}{ccc}2\\-7\\h&\end{array}\right][/tex]

The vectors v1 and v2 are not scalar multiples of one another so they are not linearly dependent.

[tex]\left[\begin{array}{ccc}1&-3&2\\-3&10&-7\\2&-6&h\end{array}\right][/tex]= 0

1(10h -42) -3(-14 +3h) +2 (18-20) =0

10h -9h-4 =0

h = 4

Hence the value of h is 4 so that the vector v3 is in span{v1,v2}

b) The given vectors will be linearly dependent if h =4

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The law of sines states that the ratio of the Sine of an angle to the length of the side opposite that angle is constant in any triangle.

The Law of Sines is defined as a relationship that applies to all triangles, not just right triangles.

The law states that the ratio of length of side of a triangle to sine of angle opposite that side is constant for all sides and angles in the triangle.

So , for any triangle, ratio of length of one side to the sine of angle opposite that side is = ratio of length of any other side to the sine of the angle opposite that side.

The Sine Law is also useful for solving problems which involves the angles and sides of triangles, especially when not all sides and angles are known.

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a) This function maps even integers to the next odd integer, and maps odd integers to themselves. It is not one-to-one

b) This function maps positive integers to even positive integers, and maps negative integers to odd positive integers. It is one-to-one because different inputs always have different outputs.

c) This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs, and it is onto because every positive and negative integer is in its range.

d) This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs

(a) One example of a function that is neither one-to-one nor onto is:

[tex]f(x) = \begin{cases} x+1 & \text{if } x \text{ is even} \ x & \text{if } x \text{ is odd} \end{cases}[/tex]

This function maps even integers to the next odd integer, and maps odd integers to themselves. It is not one-to-one because, for example, f(2) = f(4) = 3. It is not onto because no positive even integer is in the range of f.

(b) One example of a function that is one-to-one but not onto is:

[tex]f(x) = \begin{cases} 2x & \text{if } x > 0 \ -2x+1 & \text{if } x \leq 0 \end{cases}[/tex]

This function maps positive integers to even positive integers, and maps negative integers to odd positive integers. It is one-to-one because different inputs always have different outputs. However, it is not onto because no odd positive integer is in the range of f.

(c) One example of a function that is onto but not one-to-one is:

[tex]f(x) = \begin{cases} x/2 & \text{if } x \text{ is even} \ -(x+1)/2 & \text{if } x \text{ is odd} \end{cases}[/tex]

This function maps even integers to positive integers, and maps odd integers to negative integers. It is onto because every positive and negative integer is in the range of f. However, it is not one-to-one because, for example, f(2) = f(-2) = 1.

(d) One example of a function that is both one-to-one and onto is:

f(x) = x

This function simply maps every integer to itself. It is one-to-one because different inputs always have different outputs, and it is onto because every positive and negative integer is in its range.

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Answer:(A) 1.314.08

Step-by-step explanation:

Inflation

The length of the hypotenuse or the value of 'x' is 60.

We know a right-angled triangle has three sides they are -: Hypotenuse,

Opposite and Adjacent.

We can remember SOH CAH TOA which is,

sin = opposite/hypotenuse, cos = adjacent/hypotenuse and

tan = opposite/adjacent.

From the given diagram respect to the reference angle of 37°,

Hypotenuse is 'x' and adjacent is 48.

And we know,  cos = adjacent/hypotenuse.

Therefore, cos37° = 48/x.

x = 48/0.8.

x = 60.

Q. A right-angle triangle is shown, Find the value of x.

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Answer:

1/6

Step-by-step explanation:

1/4x2/3 = 2/12 = 1/6

Answer:

Down payment is $12,500.

Mortgage = $237,500

Step-by-step explanation:

The required down payment is 5% of the selling price of the house, which is $250,000.

Down payment = 5% of $250,000 = 0.05 x $250,000 = $12,500

Therefore, the required down payment is $12,500.

To find the amount of the mortgage, we need to subtract the down payment from the selling price of the house.

Mortgage = Selling price of the house - Down payment

Mortgage = $250,000 - $12,500

Mortgage = $237,500

Therefore, the amount of the mortgage is $237,500.

The translation needed to obtain the graph of g(x) = 7^x-4 from the graph of (x) = 7^x is 4 units right

From the question, we have the following parameters that can be used in our computation:

g(x) = 7^x-4

f(x) = 7^x

Using the above as a guide, we have the following:

g(x) = f(x - 4)

This represents a translation to the right by 4 units

Hence, the translation is 4 units right

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Probability of 3 people arriving late is 0.34%.

The probability of 3 people arriving late to the party is the probability of a combination of 3 individuals arriving late and the couple and remaining individual arriving on time. This can be expressed mathematically as P(X = 3) = P(3 individuals arrive late)P(Couple arrives on time)P(Individual arrives on time).

We can calculate this probability as follows:

P(X = 3) = (0.1 x 0.1 x 0.1) x (0.7 x 0.7) = 0.00343

This means that there is a 0.34% chance that 3 people arrive late to the party. Probability of 3 people arriving late is 0.34%.

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Answer:

18

Step-by-step explanation:

to find p(-2), substitute it into the equation!

you can make this easier by factoring x out, so p(x) = x(x-7)

then, plug -2 in

p(-2)=-2(-2-7)

p(-2)=-2(-9)

p(-2)=18

Answer:

Step-by-step explanation:

p(x)=[tex]x^{2}-7x\\[/tex]

=>p(-2)=[tex](-2)^{2} -7.(-2)[/tex]

           = 4+14=18

The volume of the cube is 1 cubic meter.

Option B is the correct answer.

It is a polygon having six faces.

The volume of a cube is side³.

Example:

The volume of a cube of 3 cm in length is 27 cm³.

We have,

Side length = 100 cm

Now,

Volume.

= Side³

= 100³

= 1000000 cm³ _____(1)

Now,

1 m = 100 cm

1 m³ = 1000000 cm³ _____(2)

From (1) and (2),

Volume = 1 cubic meter.

Thus,

The volume of the cube is 1 cubic meter.

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The probability of each outcome can be calculated using basic probability formulas, such as the multiplication rule and the addition rule is 1/2

Probability is the branch of mathematics that deals with the likelihood of events occurring. In this scenario, we are looking at the probability of a concrete beam failing by shear or flexure.

When three failed beams are randomly selected, the type of failure is determined for each one. We want to find the probability of each possible outcome, where X represents the number of beams among the three selected that failed by shear.

To list the outcomes in the sample space, we can use the following notation: S for shear failure and F for flexure failure. The possible outcomes are:

SSS, where all three beams failed by shear. In this case, X = 3.

SSF, where two beams failed by shear and one by flexure. In this case, X = 2.

SFS, where two beams failed by shear and one by flexure. In this case, X = 2.

FSS, where two beams failed by shear and one by flexure. In this case, X = 2.

SFF, where one beam failed by shear and two by flexure. In this case, X = 1.

FSF, where one beam failed by shear and two by flexure. In this case, X = 1.

FFS, where one beam failed by shear and two by flexure. In this case, X = 1.

FFF, where all three beams failed by flexure. In this case, X = 0.

Each outcome has a different value of X, which represents the number of beams that failed by shear.

Then the probability is calculated as,

=> 1/2

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Answer:

b

Step-by-step explanation:

a) Probability, P(y = 2) = P(AA) is equals to the 0.81.

b) Probability, P(y = 3), is equals to the 0.162.

c) For P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y= 5 , probability is equals to the 0.00324.

d) General formula for p(y) is P(y)= (y-1)(0.1)⁽ʸ⁻²⁾(0.9)².

A new battery’s voltage may be acceptable (A) or unacceptable (U). Assume that 90% of all batteries have acceptable voltages. So, probability of acceptance = 0.90 and

probability of unacceptance = 1 - 0.9 = 0.1

Let the number of batteries that must be tested be "Y". A particular flashlight requires two batteries, that is batteries will be independently selected.

a) For P(y = 2), both the batteries must be acceptable. Hence probability, P(y = 2)

= P(AA) = 0.90× 0.90 = 0.81

b)For P(y = 3 ) = P(AUA) + P(UAA)

=0.9× 0.1×0.9 + 0.1× 0.9× 0.9 = 0.162

C) For any(Y = n), nᵗʰ batterry should be acceptable, as search for two acceptable batteries will complete once second acceptable battery gets selected. Hence for P(y=5), 5ᵗʰ battery must be an A. four outcome for which Y

= 5 are (AUUUA) ,(UAUUA),(UUAUA),(UUUAA)

P(y=5) = P(AUUUA) + P(UAUUA) + P(UUAUA) + P(UUUAA)

= 4× 0.9² × 0.1³

= 0.00324

d) From above probabilities we conclude, the general formula for p(y) is p(y) =(y- 1)(0.1)⁽ʸ⁻²⁾(0.9)².

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The coefficients (a, b, c) in a polynomial equation is relate to the position x, velocity v, and acceleration a of an object.

Position (x): The position of the object can be represented by the x-coordinate of its position in space. This can be related to the coefficients in the equation by setting the x-coordinate equal to x and solving for the corresponding y-coordinate.

Velocity (v): The velocity of an object is the rate at which its position is changing. This can be related to the coefficients in the equation by taking the first derivative of the equation with respect to time. The result is a linear equation that represents the velocity as a function of time.

Acceleration (a): The acceleration of an object is the rate at which its velocity is changing. This can be related to the coefficients in the equation by taking the second derivative of the equation with respect to time. The result is a constant value that represents the acceleration of the object.

In summary, the coefficients (a, b, c) can be related to the position, velocity, and acceleration in equation for x, taking derivatives of the equation with respect to time, and interpreting the results as the position, velocity, and acceleration of the object.

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The combined area of the given figure will be 40 square inches.

The area is defined as the space covered by the object in two-dimensional space.

The area of a rectangle with a length of 7 inches and a width of 5 inches is given by:

Area of rectangle = length x width = 7 in x 5 in = 35 square inches

The area of a triangle with a base of 5 inches and a height of 2 inches is given by:

Area of triangle = (1/2) x base x height = (1/2) x 5 in x 2 in = 5 square inches

Therefore, the combined area of the rectangle and triangle is:

35 square inches + 5 square inches = 40 square inches

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The area of each circle, to the nearest tenth is:

1. 19.6 cm²

2. 379.9 in.²

3. 28.3 mm²

4. 78.5 in.²

5. 132.7 cm²

6. 162.8 yd²

The area of a circle is calculated as:

A = πr², where, r is the radius of the circle, and π is a constant.

1. r = 5/2 = 2.5 cm

Area = πr² = 3.14 * 2.5²

= 19.6 cm²

2. r = 11 in.

Area = πr² = 3.14 * 11²

= 379.9 in.²

3. r = 3 mm

Area = πr² = 3.14 * 3²

= 28.3 mm²

4. r = 10/2 = 5 in.

Area = πr² = 3.14 * 5²

= 78.5 in.²

5. r = 13/2 = 6.5 cm

Area = πr² = 3.14 * 6.5²

= 132.7 cm²

6. r = 7.2 yd

Area = πr² = 3.14 * 7.2²

= 162.8 yd²

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1) The  model is R = 1820 - 19t

2) After six years R = 1706 barrels

3) There would be non more oil in 96 years.

We know that an equation can be used as a model. This implies that we can be able to use an equation to show how a variable is changing over time and that is what the problem that we have here is all about.

If at the current time we have 1820 billion barrels Then they decrease by 19 barrels per year and t is the number of years then the model is;

R = 1820 - 19t

After six years we would have;

R = 1820 - (19 * 6)

R = 1706 barrels

When there are no further barrels;

0 = 1820 - 19t

1820 = 19t

t = 1820/19

t = 96 years.

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The transformation from ln ( x ) to ln ( x + 2 ) is horizontal transformation or shift.

What kinds of functions can be transformed?

Transformations can be divided into three categories: translations, reflections, and dilations. There are two types of transformations that can be done to a function: vertical (which affects the y-values) and horizontal (affects the x-values).

                               The four transformation variables are included in the equation of a function that is shown below (a, b, h, and k).

f(x) = ln(x)

if g(x) = ln(x + 2)

Then,

The transformation from ln(x) to ln(x+2) is horizontal transformation or shift. As 2 units are added in g(x) the function is horizontally shifted by 2 units in left-hand direction.

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Answer: The equation of a line in the form of y = mx + b can be found using the slope-intercept form, where m is the slope of the line and b is the y-intercept. To find the slope, you can use the two points on the line: (1, 2) and (3, 5). The slope is found by the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two points.

m = (5 - 2) / (3 - 1) = 3 / 2 = 1.5

The y-intercept can be found by plugging in one of the points and the slope into the equation and solving for b. Let's use the point (1, 2).

2 = 1.5 * 1 + b

b = 2 - 1.5 = 0.5

So the equation of the regression line is:

y = 1.5x + 0.5

Therefore, the correct answer is: yˆ = 1.5x + 0.5

Step-by-step explanation:

Answer:

The expression (9x+8)-(3x + 5) simplifies to 6x + 3. Therefore, the answer is option C: 6x + 3.