prove that in any set of n integers, either one of these integers is a multiple of n or the sum of several of them is a multiple of n.

The numbers of cups that macaroni will he use will he use if he wishes makes 27 bowls is 9 cups.

If Mike makes 27 bowls of macaroni salad, we can find out how many cups of macaroni he will need by multiplying the amount of macaroni needed for one bowl by the number of bowls:

= 1/3 cup of macaroni per bowl x 27 bowls

= 9 cups of macaroni

Therefore, Mike will need 9 cups of macaroni to make 27 bowls of macaroni salad.

Full question "Mike is making macaroni salad. For each bowl of macaroni salad, he needs 1/3 cup of macaroni. How many cups of macaroni will he use will he use if he makes 27 bowls of?"

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The height of the triangle is 3/5.

A triangle is said to be right-angled if one of its inner angles is 90 degrees, or if any one of its angles is a right angle.

Given:

A right-angled triangle.

Let the values of the hypotenuse in parts be m and n.

So,

(1)² = (5/4)m

m = 4/5

And n = 9/16 x 4/5

n = 9/20

So, the height of the triangle is,

h² = 9/20 x 4/5

h² = 9/25

h = 3/5

Therefore, h = 3/5.

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Answer:

x = 2.4

Step-by-step explanation:

[tex]2x + 4 \times \frac{1}{5} = 9 \\ = 2x + \frac{21}{5} = 9 \\ = 5 \times 2x + 5 \times \frac{21}{5} = 9 \times 5 \\ = 10x + 21 = 45 \\ = 10x = 45 - 21 \\ = 10x = 24 \\ \frac{10x}{10} = \frac{24}{10} \\ \times = 2.4[/tex]

The measure of variation include range, variance, iqr etc

In statistics, a measure of variation is a numerical value that describes how spread out or dispersed a set of data is. The most common measures of variation are:

Range: The range is the difference between the maximum and minimum values in a data set. It gives an idea of the spread of the data, but is sensitive to outliers.

Interquartile range (IQR): The IQR is the range of the middle 50% of the data. It is less sensitive to outliers than the range.

Variance: The variance is the average of the squared differences of each data point from the mean. It measures how much the data is spread out from the mean.

Standard deviation: The standard deviation is the square root of the variance. It is a common measure of variation and indicates the typical amount that each data point deviates from the mean.

Coefficient of variation: The coefficient of variation is the ratio of the standard deviation to the mean, expressed as a percentage. It is used to compare the variation of data sets with different means.

An overview was given due to incomplete information.

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We must establish both of the following statements in order to demonstrate that t(n) is unusual if and only if n is a perfect square.

First instruction: If t(n) is odd, n is a perfect cube.

Assume that t(n) is strange. All the positive divisors of n should be d 1, d 2, ldots, and d k. So, we understand that k=t(n) is unusual. The divisors can be combined into frack2 pairs, with a sum of n for each pair:

(d 1, d k), (d 2, d k-1)

If k is odd, only one divisor remains, which, if n is a perfect cube, is the square root of n. The conclusion is that n must be a perfect square if t(n) is unusual.

Second instruction: If t(n) is odd, then n is a perfect square and n is.

Let's assume that n is a perfect square, such as n=m2. Then, the positive divisors of n appear in pairs, denoted by (d, fracnd), where d spans all the divisors of m. We only need to tally the divisor d when d=fracnd because the product d cdot fracnd = n is not a perfect square if d is not equal to fracnd. Since m has an odd number of divisors, t(n) is only odd if and only if m.

We can look at m's prime factorization to understand why it has an odd amount of divisors. Write m=p 1,p 2,a 1,a 2,ldots,p k where p 1,p 2,a 1,a 2,ldots,p k are distinct prime numbers and a 1,a 2,ldots,a k are positive integers.

As a result, we have demonstrated that t(n) is odd only when n is a perfect square.

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The two-column method table to prove that the triangles ΔLMN and ΔLJK are similar, ΔLMN ~ ΔLJK, where LM/LJ = LN/LK can be completed as follows

Statements             [tex]{}[/tex]                         Reasons

LM/LJ = LN/LK     [tex]{}[/tex]                            Given

∠L ≅ ∠L [tex]{}[/tex]                                          Reflexive property of congruency

ΔLMN ~ ΔLJK [tex]{}[/tex]                                 SAS similarity theorem

Similar triangles are triangles that have the same shape, and interior angles, but which may have different side lengths such that the proportion of the corresponding sides are equivalent.

The details of the reasons used to prove the similarity of triangles ΔLMN and ΔLJK can be presented as follows;

Reflexive property of congruency

The reflexive property of congruency states that a figure, such as an angle or a length is congruent to itself

SAS similarity theorem

The SAS, Side-Angle-Side similarity theorem states that if two sides in one triangle are proportional to two sides in another triangle, and the included angle between the two sides in both triangles are congruent, then the two triangles are similar.

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The equation of the plane passing through P1(12,3), P2(3,2,1), and perpendicular to the plane 4−y+2z=7 is -3x + 51y - 8z = -39.

To find a plane that passes through the points P1(12,3) and P2(3,2,1) and is perpendicular to the plane 4−y+2z=7, we can follow these steps:

Find the normal vector of the given plane by taking the coefficients of x, y, and z in the equation 4−y+2z=7. The normal vector is (1, -1, 2), because the coefficients of x, y, and z are 1, -1, and 2 respectively.

Find the direction vector of the line passing through P1 and P2 by subtracting the coordinates of P1 from P2. The direction vector is (-9, -1, -2), because P2 - P1 = (3-12, 2-3, 1-0) = (-9, -1, -2).

Find the cross product of the normal vector of the given plane and the direction vector of the line passing through P1 and P2 to get the normal vector of the plane we are looking for. The cross product is:

(1, -1, 2) x (-9, -1, -2) = (-3, -17, -8)

Plug one of the given points, say P1(12,3), and the normal vector we found in step 3 into the point-normal form of the equation of a plane:

-3(x - 12) - 17(y - 3) - 8(z - 0) = 0

Simplifying, we get:

-3x + 51y - 8z = -39

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The probability that X is between 90 and 110 = 0.2 if a normally distributed random variable x has a mean of 100 and P(x < 90) = 0.40.

The majority of the observations are centered around the middle peak of the normal distribution, which is a continuous probability distribution that is symmetrical around its mean.

The probabilities for values that are farther from the mean taper off equally in both directions. Extreme values in the distribution's two tails are likewise rare. Not all symmetrical distributions are normal, even though the normal distribution is symmetrical.

A normally distributed random variable x has a mean of 100

P(x < 90) = 0.4

The probability that X is between 90 and 110

Mean is 100

Hence P (< 100) = 0.5  as Mean is centered

Hence

P  (90 < X < 100) = 0.5 - 0.4 = 0.1

The difference between 100 and 90 is 10

The difference between 100 and 110 is 10

and 100 is Mean

Hence P  (100 < X < 110) = P  (90 < X < 100) = 0.1

Adding Both

P ( 90 < X < 110) = 0.1 + 0.1 = 0.2

The probability that X is between 90 and 110 = 0.2

Using z score table.

Z  = ( Value - Mean) /SD

Z score for 0.4 = -0.254

-0.253 = (90 - 100)/SD

SD ≈ 39.53

Z = ( 110 - 100)/39.53

Z = 0.253

For z score  < 0.253, the probability is 0.6

Hence Probability in Between is 0.6 - 0.4 = 0.2

The probability that X is between 90 and 110 = 0.2

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The standard deviation expressed as a percent of the mean is the D: coefficient of variation.

The coefficient of variation (CV) is a statistical measure that expresses the standard deviation of a data set as a percentage of the mean. It is particularly useful for comparing the variability of different data sets with different units or scales, and for identifying the degree of variation relative to the mean. The formula for the coefficient of variation is:

CV = (standard deviation / mean) x 100%

So, the coefficient of variation is a dimensionless quantity that measures the relative dispersion of the data.

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Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).

It seems like some of the numbers in the table are cut off. Nevertheless, we can determine if Jeremiah was correct in saying that g(x) increases more gradually than f(x).

From the given information, we know that f(x) = IlxIl (the greatest integer function). This function increases by 1 whenever x crosses an integer value.

Jeremiah's function is g(x) = [x] (the floor function), which is the greatest integer less than or equal to x. This function increases by 1 only when x is an integer.

Since g(x) increases less frequently than f(x), Jeremiah is correct in saying that g(x) increases more gradually than f(x).

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The following sets of functions are subspaces of v, if v be the (real) vector space of all functions f from r into r is: all f such that f(x²) = f(x²), all f which are continuous.

V is a Vector- Space of all real- valued functions over field of real numbers R and W consists of all real- valued even functions which are bounded also as a subset of V.

Let f , g belong to W then f , g both are even & bounded. Hence ;

(1) (f + g ) is even & bounded because ,

(f + g )(-x ) = f(- x )+ g(-x) = f( x) + g( x )

=( f+g)( x) and for all x€ R and

= | f (x) + g(x) | </= |f (x) | + | g (x) |

</= (c +d) = C(constant)

and similarly for any scalar k€ R , (kf ) will be an even function and it will be bounded also.

A set whose elements, frequently termed vectors, can be added to and multiplied ("scaled") by figures known as scalars is referred to as a vector space (also known as a linear space). Real numbers make up scalars most of the time, but they can also be complex numbers or, more broadly, components of any field. Certain conditions, referred to as vector axioms, must be met by the operations of vector addition and scalar multiplication.

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The correlation between X and Y is -0.648, rounded to two decimal places.

First, we need to calculate the means of X and Y:

mean(X) = (1 + 1.5 + 1.5 + 2.5 + 3) / 5 = 2

mean(Y) = (1/4 + 1/8 + 1/4 + 1/4 + 1/8) / 5 = 0.15

Next, we can calculate the covariance using the formula:

cov(X, Y) = E[(X - mean(X)) * (Y - mean(Y))] = Σ[(X - mean(X)) * (Y - mean(Y))] / (n - 1)

where n is the number of data points. Substituting the values, we get:

cov(X, Y) = [(1 - 2) * (1/4 - 0.15) + (1.5 - 2) * (1/8 - 0.15) + (1.5 - 2) * (1/4 - 0.15) + (2.5 - 2) * (1/4 - 0.15) + (3 - 2) * (1/8 - 0.15)] / 4

= -0.035

Therefore, the covariance between X and Y is -0.035.

Finally, we can calculate the correlation coefficient using the formula:

corr(X, Y) = cov(X, Y) / (stddev(X) * stddev(Y))

where stddev is the standard deviation. We can calculate the standard deviations as follows:

stddev(X) = sqrt([Σ(X - mean(X))^2] / (n - 1)) = 0.8292

stddev(Y) = sqrt([Σ(Y - mean(Y))^2] / (n - 1)) = 0.0632

Substituting the values, we get:

Corr(X, Y) = -0.035 / (0.8292 * 0.0632) = -0.648

Therefore, the correlation between X and Y is -0.648, rounded to two decimal places.

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P (X=4)  = 0.0028

P(X=3) or P(X=4) = 0.0326

P (X<=4) = 0.9999

P(X>=4) 0.0029

The standard deviation is 0.8198

Standard deviation is a statistical measure of how to spread out a set of data is from its mean or average value. It measures the degree of variation or dispersion of a dataset, which helps in understanding how much the individual data points deviate from the average.

To solve for:

1. P(X=4) = [tex](\frac{5}{4}) (0.16)^4 (1-0.16)^5^-^4[/tex]

= 0.0028.

2. P(X=3) or P(X=4) = [tex](\frac{5}{3}) (0.16)^3 ((1-0.16)^2\\[/tex]

= 0.0326

3. P(X<=4) = [tex](\frac{5}{x})(0.16)^x(1-0.16)5^-^x[/tex]

= 0.9999

4. P(X>=4) = P(X=4) + P(X=5)

= [tex](\frac{5}{4}) (0.16)^4(0.84)^1 + (\frac{5}{5})(0.16)^5[/tex]

= 0.0029.

5. u = np = 0.8

[tex]\sqrt{np(1-p)}[/tex]

= 0.8198

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3/12 =1/4  hopefully this answer works for you

the graph of the function is attached below.

A relationship between a group of inputs and one output is referred to as a function. In plain English, a function is an association between inputs in which each input is connected to precisely one output. A domain, codomain, or range exists for every function. Typically, f(x), where x is the input, is used to represent a function.

Given a function,  f(x) = 3√x -2

To graph the function f(x) = 3√x -2, we can follow these steps:

f(0) = 3√0 - 2 = -2

f(1) = 3√1 - 2 = 1

f(8) = 3√8 - 2 ≈ 4.9

f(27) = 3√27 - 2 ≈ 7.2

Here, the graph of f(x) = 3√x -2 is attached below.

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Answer:

10

Step-by-step explanation:

20% = 0.2

What is 20% of 50?

We take

50 x 0.2 = 10

So, 20% of 50 is 10

The probability that the last person to board sits in the assigned seat is always 1/2, regardless of the number of people n in the line.

The issue can be tackled utilizing numerical enlistment.

Base Case:

At the point when n = 2, there are two travelers and two seats. The principal traveler takes an irregular seat, and the subsequent traveler will sit in his doled out seat with likelihood 1/2 or take the other seat with likelihood 1/2. Consequently, p2 = 1/2.

Inductive Speculation:

Accept that for some certain number k, pk = 1/2.

Inductive Step:

Think about n = k+1 travelers. The principal traveler takes an irregular seat. There are two cases to consider:

Case 1: The primary traveler sits down relegated to the k+1-th traveler. For this situation, the last traveler will be ensured to sit in his doled out seat, and the leftover k travelers can be considered as a subproblem with similar circumstances. By the inductive speculation, the likelihood that the last traveler in the subproblem sits in his alloted seat is 1/2. Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is 1/(k+1) + (k/(k+1)) * 1/2 = (k+2)/(2(k+1)).

Case 2: The primary traveler sits down other than the one doled out to the k+1-th traveler. For this situation, the issue lessens to the case with k travelers, and the likelihood that the last traveler in the decreased issue sits in his doled out seat is pk = 1/2 (by the inductive speculation). Thusly, the likelihood that the last traveler in the first issue sits in his doled out seat for this situation is (k/(k+1)) * 1/2 = k/(2(k+1)).

Joining the two cases, we have:

pk+1 = (k+2)/(2(k+1)) + k/(2(k+1)) = (k+1)/(2(k+1)) = 1/2.

Consequently, by numerical acceptance, the likelihood that the last individual to board sits in the appointed seat is dependably 1/2, no matter what the quantity of individuals n in the line.

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The function that models the data is [tex]y = 42.(\frac{1}{2})^x[/tex].

The formula for exponential growth is [tex]y = y_0e^{(kt)}.[/tex]

The formula for exponential decay is [tex]y = y_0e^{(-kt)}.[/tex]

Let, The given exponential be [tex]y = a(b)^x[/tex] .

Now, At (0, 42).

42 = ab⁰.

a = 42.

At (1, 21).

21 = 42.b¹.

b = 1/2.

Therefore, [tex]y = 42.(\frac{1}{2})^x[/tex]is the required function.

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(a) - A runs the race at a constant speed, never speeding up or slowing down. B accelerates throughout the race, starting out slower than A and, by the end, running faster than A.

(b) - Based on the graph, it appears that they are furthest apart after 8 seconds, when they are approximately 30 meters apart.

(c)- The two graphs appear to have the same slope (i.e., velocity) 9 or 10 seconds into the race.

graph attached below,

constant speed

When an object travels the same distance in the same period of time, it is said to be traveling at a constant speed. At constant speed, an object travels a uniform distance in an equal interval of time. The equation of the speed can be given as: S = d t.

Slope

Slope is a measure of the steepness of a line.

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The four points on the graph of this function f(x) = √(x + 1) + 4 has been plotted on the graph shown in the image attached below.

In Mathematics, a graph can be defined as a type of chart that is typically used for the graphical representation of data points or ordered pairs on both the horizontal and vertical lines of a cartesian coordinate, which are the x-axis and y-axis respectively.

Next, we would use an online graphing calculator to plot the given absolute value function f(x) = √(x + 1) + 4 as shown in the graph attached below.

By observing critically the graph (see attachment), the four ordered pairs (points) that fit on its axes include the following:

Ordered pair = (-1, 4), which is the leftmost.

Ordered pair = (0, 5).

Ordered pair = (3, 6).

Ordered pair = (8, 7).

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Answer:3x^3−4x^2+2x+6

Step-by-step explanation:

1).subtract the numbers

2).combine like terms

The pairs of linear equations that are parallel are:

2 and 4.

Two linear equations:

y = a*x + b

y = c*x + d

Are parallel if the two slopes are equal and the y-intercepts are different, then:

a = c

b ≠ d

Let's write all the lines in the slope-intercept form:

1) 4x + 3y = 15

  3y = 15 - 4x

   y = (-4/3)*x + 15/3

   y = (-4/3)*x + 5

2) 3x - 4y = -8

  -4y = -8 - 3x

     y = 2 + (3/4)x

So we can see that lines 2 and 4 are parallel.

If we rewrite line 3 we will get:

y + 1 = (4/3)*(x - 6)

y = (4/3)*x - 8  -1

y = (4/3)*x - 9

Then we conclude that only lines 2 and 4 are parallel.

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Answer:

400.059

How do you write this problem as a decimal?

So the first part of this question is four hundred:

400.

But then.. It says fifty nine hundredths. So lets write it out.

The closer we get to the decimal it gets " smaller " example the hundredths is all the way in back and the tenths is all the way to the front! So with that knowledge lets figure it out:

( to make it nice and simple for you ):

0.059

The " 9 " is in the hundredths zone and the " 5 " is in the tenths zone. So just like on top it looks like that.

0.059+400=

400.059

Thus, your answer is 400.059

If r(t) is the position vector for a smooth curve C, and Î (t), Ñ(t), and B(t) are unit tangent vector, principal unit normal vector, and binormal unit vector, respectively, then (1) B(t) · Î(t) = 0, (2) Þ(t) · B(t) = 0, (3) ÎN(t) · (B(t) – 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0 and (4) Þ(t) x Î (t) = B(t).

(1) Since B(t) is the cross product of Î(t) and Ñ(t), it is perpendicular to both Î(t) and Ñ(t). Therefore, B(t) · Î(t) = 0.

(2) Þ(t) is the derivative of r(t), and B(t) is defined as the cross product of Î(t) and Ñ(t). Therefore, Þ(t) and B(t) are both orthogonal to Î(t). Hence, Þ(t) · B(t) = 0.

(3) ÎN(t) is the cross product of Î(t) and Ñ(t), and B(t) is also the cross product of Î(t) and Ñ(t). Therefore, B(t) - 5ÊN(t) is parallel to ÎN(t). Hence, ÎN(t) · (B(t) - 5ÊN(t)) = |ÎN(t)| |B(t) - 5ÊN(t)| cos(π/2) = 0.

(4) The cross product of two vectors is orthogonal to both of the vectors. Therefore, Þ(t) x Î(t) is orthogonal to both Þ(t) and Î(t), and hence it is parallel to Ñ(t). Therefore, Þ(t) x Î(t) is equal to B(t). So the answer is B.

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The number of cells in the Petri dish after 10 hours is 2¹⁰ = 1024 cells in the dish.

The process in mathematics to operate and interpret the function to make the function or expression simple or more understandable is called simplifying and the process is called simplification.

Here,
The number of cells in the Petri dish after each hour can be represented by the expression 2ⁿ, where n is the number of hours that have passed.

So, after 1 hour, there are 2¹ = 2 cells in the dish.

After 2 hours, there are 2² = 4 cells in the dish.

After 3 hours, there are 2³ = 8 cells in the dish.

And so on.

Therefore, after 10 hours, there are 2¹⁰ = 1024 cells in the dish.

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If the length of a rectangle is four times its width. if the area of the rectangle is 256 cm2, the perimeter of the rectangle is 80 cm.

Let's first use the given information to write two equations relating the length and width of the rectangle:

The length L is four times the width W: L = 4W

The area A is 256 cm²: A = LW = 256

Substituting equation 1 into equation 2, we get

4W^2 = 256

Solving for W, we get:

W^2 = 64

W = 8 (since we're looking for a positive value)

Substituting this value back into equation 1, we get:

L = 4W = 32

Therefore, the length and width of the rectangle are 32 cm and 8 cm, respectively.

The perimeter of the rectangle is twice the sum of its length and width:

P = 2(L + W) = 2(32 + 8) = 80 cm

So the perimeter of the rectangle is 80 cm.

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The type and degree of association is As the time a basketball player practices increases, the number of points scored in a game increases with a strong nonlinear association, the correct option is B

The correlation coefficient is the degree of association between two quantities in term of linear relation.

The range of correlation coefficient is -1 to 1

when the correlation is -1, then that means  as the one quantity increases, the other quantity decreases (linearly)

when the correlation is 0, then there is no linear relationship between two variables.

when the correlation is 1, then that means  as the one quantity increases, the other quantity increases(linearly) and vice versa for decrement.

Given the graph

Now, a function is an expression, that describes the relationship between one variable (the independent variable) and another variable (the dependent variable) .

Linear function, the graph is a straight line

Therefore, by correlation coefficient answer will be B

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The system of equations y+4x=7 and −2y−4=8x has no solutions.

Two or more expressions with an Equal sign is called as Equation.

The given system of equations are

y+4x=7 ...(1)

−2y−4=8x...(2)

2y+8x=-4

From equation 1

y=7-4x

Simplifying and solving for x, we get:

-14 + 8x - 4 = 8x

-18 = 0

This is a contradiction, since -18 is not equal to 0. Therefore, there is no solution that satisfies both equations.

Hence, the system of equations has no solutions.

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on solving the provided question we can say that the function at  (h, k) = (1, - 3), will be [tex]g(x) = (x - 1)^2 - 3- > B[/tex]

Mathematics deals with numbers and their variants, equations and associated structures, forms and their positions, and locations where they might be found. The term "function" describes the connection between a group of inputs, each of which has a corresponding output. A function is an association between inputs and outputs where each input results in a single, unique outcome. A domain and a codomain, or scope, are assigned to each function. Functions are often denoted by the letter f. (x). The input is an x. On functions, one-to-one functions, many-to-one functions, within functions, and on functions are the four primary categories of functions that are available.

The graph of g(x) has its vertex at (1, - 3)

The equation of a parabola in vertex firm is

[tex]y = a(x - h)^2 + k[/tex]

(h, k) =coordinates of the vertex

and

a is a multiplier

(h, k) = (1, - 3),

[tex]g(x) = (x - 1)^2 - 3- > B[/tex]

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Answer:

1. 40

2. 351

3. 49

Step-by-step explanation:

just multiply the 2 numbers!

5 times 8 is 40

13 times 27 is 351

7 times 7 is 49

(square is the same on every side so every side is 7)