Given f(x) = x³ + kx + 9, and the remainder when f(x) is divided by x − 2 is 7,
then what is the value of k?

The bases for the row space and null space of A, we put A into reduced row echelon form and solve for the null space. The dot product of basis vectors shows they are orthogonal.

To find the bases for the row space and null space of A, we perform row operations on A until it is in reduced row echelon form:

[ 1 -1  3 |  5 ]    [ 1 -1  3 |  5 ]

[ 2  1 -5 | -9 ] -> [ 0  3 -11 | -19]

[-1 -1  2 |  2 ]    [ 0  0  0  |  0 ]

[ 1  1 -1 | -1 ]    [ 0  0  0  |  0 ]

The reduced row echelon form of A tells us that there are two pivot columns, corresponding to the first and second columns of A. The third and fourth columns are free variables. Therefore, a basis for the row space of A is given by the first two rows of the reduced row echelon form of A:

[ 1 -1  3 |  5 ]

[ 0  3 -11 | -19]

To find a basis for the null space of A, we solve the system Ax = 0. Since the third and fourth columns of A are free variables, we can express the solution in terms of those variables. Setting s = column 3 and t = column 4, we have:

x1 - x2 + 3x3 + 5x4 = 0

2x1 + x2 - 5x3 - 9x4 = 0

-x1 - x2 + 2x3 + 2x4 = 0

x1 + x2 - x3 - x4 = 0

Solving for x1, x2, x3, and x4 in terms of s and t, we get:

x1 = -3s - 5t

x2 = s + 2t

x3 = s

x4 = t

Therefore, a basis for the null space of A is given by the vectors:

[-3  1  1  0]

[ 5  2  0  1]

To verify that every vector in the row space of A is orthogonal to every vector in the null space of A, we compute the dot product of each basis vector for the row space with each basis vector for the null space:

[ 1 -1  3 |  5 ] dot [-3  1  1  0] = 0

[ 1 -1  3 |  5 ] dot [ 5  2  0  1] = 0

[ 0  3 -11 | -19] dot [-3  1  1  0] = 0

[ 0  3 -11 | -19] dot [ 5  2  0  1] = 0

Since all dot products are equal to zero, we have verified that every vector in the row space of A is orthogonal to every vector in the null space of A.

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_____The given question is incomplete, the complete question is given below:

in excercises 7 and 8 find bases for the row space and null space of a. verify that every vector in the row(a) is orthogonal to every vector in null(a). a = [ 1 -1 3   5 2 1   0 1 -2   -1 -1 1]

Answer:    

The radius of the circle is 3.

The center of the circle is (-2, 1).

Step-by-step explanation:

   Rewrite the equation in standard form:

   We need to rewrite the given equation in standard form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is the radius. To do this, we complete the square for both the x and y terms.

x² + 4x + y²-2y = 20

(x² + 4x + 4) + (y² - 2y + 1) = 20 + 4 + 1

(x + 2)² + (y - 1)² = 25

   Identify the center and radius:

   Now that we have the equation in standard form, we can identify the center and radius of the circle.

The center is the point (-2, 1), which we can read directly from the equation.

The radius is the square root of the number on the right side of the equation, which is 5. Therefore, the radius is sqrt(5) or approximately 2.236.

Alternatively, we can also use the Desmos graphing calculator to plot the equation and visually determine the center and radius. When we plot the equation, we see that it forms a circle with center (-2, 1) and radius 3.

The slope of tangent line is, m= [tex]-\frac{1}{14}[/tex]

Equation of tangent line, for m= [tex]-\frac{1}{14}[/tex] and b= [tex]\frac{53}{7}[/tex] is, 14y = -x + 106

The slope of a line is a measure of how steeply it rises or falls as it moves horizontally. It is calculated by dividing the change in the vertical coordinate by the change in the horizontal coordinate between two points on the line.

Slope of tangent, m = [tex]\frac{dy}{dx}[/tex]

f(x) = y = [tex]\sqrt{57-x}[/tex]

y = [tex](57-x)^{\frac{1}{2}}[/tex]  

Differentiate the above equation y with respect to x.

[tex]\frac{dy}{dx} = \frac{1}{2} * (57-x)^{1-\frac{1}{2} }* (-1)[/tex]

[tex]\frac{dy}{dx} = -\frac{1}{2} * (57-x)^{-\frac{1}{2} }[/tex]

[tex]\frac{dy}{dx} = -\frac{1}{2\sqrt{57-x} }[/tex]

Therefore, the slope (m) of tangent line f(x) at point (8, 7) is,

[tex]\frac{dy}{dx} | _{(8, 7)} = -\frac{1}{2\sqrt{57-8} } = - \frac{1}{14}[/tex]

Equation of tangent line f(x) at point (8, 7) is,

y = mx + b

7 = [tex]-\frac{1}{14}[/tex] × 8 + b

b = [tex]\frac{53}{7}[/tex]

So, Equation is, y = [tex]-\frac{1}{14}x + \frac{53}{7}[/tex]

Therefore, 14y = -x + 106

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The mean bonus paid by large companies to their executives is $5 million and the modal class is 3.5-6.5.

To find the mean, we need to find the midpoint of each class and multiply it by the frequency, then add up all of these values and divide by the total frequency:

Class boundaries Midpoint Frequency Midpoint x Frequency

0.5-3.5 2                          11                                         22

3.5-6.5 5                          12                                        60

6.5-9.5 8                           4                                        32

9.5-12.5 11                           2                                       22

12.5-15.5 14                           1                                        14

Total 150  

Mean = (Midpoint x Frequency) / Total Frequency

Mean = 150 / 30

Mean = 5

Therefore, the mean bonus paid by large companies to their executives is $5 million.

To find the modal class, we need to look for the class with the highest frequency. In this case, the class with the highest frequency is 3.5-6.5, with a frequency of 12. Therefore, the modal class is 3.5-6.5.

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The area of the composite shape using the area formula for the different shapes is 460.48ft².

The area of composite shapes refers to the space occupied by any composite shape. A composite shape is a shape that is made by connecting a few polygons to form the required shape.

These figures or shapes can be built from a wide range of shapes, such as triangles, squares, quadrilaterals, etc. Divide a composite item into basic forms such a square, triangle, rectangle, or hexagon to get its area.

Now in the question,

First let us find the area of the semi-circle.

Area of semi-circle = πr²/2

= [3.14 × (16/2) ²]/2

= (3.14 × 8²)/2

= 200.96/2

= 100.48ft²

Now coming to the rectangle,

area of the rectangle = l × b

= 20 × 15

= 300ft²

Now for calculating the area of the triangle,

area = 1/2 × b × h

= 1/2 × 12 × 10

= 60ft²

Therefore, the area of the total figure = 100.48 + 300 + 60 = 460.48ft².

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Answer:

No

Step-by-step explanation:

Since a polygon has straight sides, with 3 or more, it cannot be a polygon since one side is curved.

If Jackson feels confident that he can score higher than 69 on the final exam, then he should take it. Otherwise, he would be better off not taking the final exam.

What is probability?

Probability is a branch of mathematics that deals with the study of random events or phenomena. It is the measure of the likelihood or chance of an event or set of events occurring.

If Jackson does not take the final exam, the average of his three highest scores would be:

(72 + 73 + 70)/3 = 71.67.

If Jackson takes the final exam, there are two possibilities:

If Jackson scores lower than any of his previous exam scores, then his lowest score will be dropped, and his grade will be calculated based on his four highest scores, which would be:

(73 + 72 + 70 + X)/4.

where X is his score on the final exam. In this case, taking the final exam would not benefit Jackson, as his grade would be based on his three highest scores (72, 73, and 70) regardless of his performance on the final exam.

If Jackson scores higher than any of his previous exam scores, then his lowest score will be the lowest of his first four exams, and his grade will be calculated based on his four highest scores, which would be:

(73 + 72 + X1 + X2)/4.

Therefore, If Jackson feels confident that he can score higher than 69 on the final exam, then he should take it. Otherwise, he would be better off not taking the final exam.

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Answer:

To find the equation of the ellipse, we need to use the standard form of the equation for an ellipse centered at the origin:

((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1

where (h, k) is the center of the ellipse, a is the distance from the center to the end of the major axis, and b is the distance from the center to the end of the minor axis.

Step 1: Find the center of the ellipse

The center of the ellipse is halfway between the two foci:

center = ((2+2)/2, (4-8)/2) = (2,-2)

Step 2: Find the length of the major axis

The distance between the two foci is 12 units (the absolute value of the difference in the y-coordinates):

c = 12

The length of the minor axis is the distance between the two co-vertices, which is 16 units:

2b = 16

b = 8

To find the length of the major axis, we use the relationship between a, b, and c in an ellipse:

c^2 = a^2 - b^2

a^2 = b^2 + c^2

a^2 = 8^2 + 12^2

a^2 = 256

a = 16

Step 3: Plug in the values to the standard form of the equation

((x-2)^2)/16^2 + ((y+2)^2)/8^2 = 1

Therefore, the equation of the ellipse is:

((x-2)^2)/256 + ((y+2)^2)/64 = 1

Answer: The correct answer is C, the X.

Answer:

B) False

Step-by-step explanation:

Perpendicular lines form right angles. It does not form acute angles.

Answer:

Step-by-step explanation:

Since means cannot be smaller than 0, the sampling distribution of the mean is always right skewed.

No matter the sample size, the form of the sampling distribution of means is always the same as the population distribution.

We require two assumptions in order to apply the sampling distribution model to sample proportions: The selected values must be independent of one another, according to the independence assumption. The Sample Size Assumption demands that the sample size, n, be sufficiently large.

While doing a t-test, it is typical to make the following assumptions: the measuring scale, random sampling, normality of the data distribution, sufficiency of the sample size, and equality of variance in standard deviation.

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the actual question is :

Which of the following is true about the sampling distribution of the mean?

a. It is an observed distribution of scores

b. It is a hypothetical distribution

c. It will tend to be normally distributed with a

standard deviation equal to the population

standard deviation

d. The mean will be estimated by the standard

error

e. Both (a) and (b)

"The rate at which the bottom of the ladder moving along the ground at the point in time when the bottom of the ladder is 4 feet from the wall is calculated to be 3.464 ft/s."

At a pace of 2 feet per second, the lower end of the ladder is being pulled away from the wall.

At a specific moment, when the lower end of the ladder is 4 feet from the wall, we should determine the rate at which the bottom of the ladder is lowering.

From the point t, the bottom of the ladder is x m, the top of the ladder is y m from the wall.

x² + y² = 64

Differentiating the given relationship with regard to t,

2x dx/dt + 2y dy/dt = 0

x dx/dt + y dy/dt = 0

We need to find out dx/dt at x = 4.

dy/dt = -2

At x = 4, we have,

x² + y² = 64

16 + y² = 64

y² = 48

y = 4√3

Put in the known values to find out dx/dt,

x dx/dt + y dy/dt = 0

4 dx/dt + 4√3 (-2) = 0

4 dx/dt = 8√3

dx/dt = 2√3 = 3.464

Thus, the bottom of the ladder is calculated to be moving at the rate 3.464 ft/s.

The figure can be drawn as shown in the attachment.

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Hence the output variable (total weekly earnings) is a function of the input variable (number of hours worked as a payroll clerk) and may be expressed by the equation y = 16x + 120.

A variable is anything that may be altered in the context of a mathematical notion or experiment. Variables are frequently denoted by a single symbol. .

a. Input variables: hours worked as a cashier, payroll clerk, and assistant.

The total amount earned each week is the output variable.

b. To determine the weekly total, multiply the number of hours worked at each job by the hourly rate and put them together. As a result, the equation would be:

Total weekly wage = (hours worked as a cashier x cashier hourly rate) + (hours worked as a payroll clerk x payroll clerk hourly rate) + (hours worked as an aide x rate per hour as an aide)

c. The amount of hours worked as an assistant is always 10 hours fewer than that of a payroll clerk. Therefore, if x denotes the number of hours worked as a payroll clerk, then the number of hours worked as an assistant may be denoted as (x - 10).

d. The equation describing the total money earned each week is as follows:

(20 x 9.5) + (x x 9) + ((x - 10) x 7) = y

This expression is simplified as follows:

y = 190 + 9x - 70 + 7x

y = 16x + 120

Hence the output variable (total weekly earnings) is a function of the input variable (number of hours worked as a payroll clerk) and may be expressed by the equation y = 16x + 120.

f. The realistic replacement values for x would be determined by the maximum number of hours you may work at each job as well as the amount of time available to work. Working 8 hours as a payroll clerk may be a reasonable substitute value if you have restricted availability, but 50 hours is unlikely.

g. If you do not work as an aide, you may calculate your total weekly income by setting the number of hours worked as an aide to zero in the calculation for the total amount earned each week. This results in:

y = 16x + 120 + 0

y = 16x + 120

As a result, the total weekly wage would be determined only by the number of hours performed as a cashier and payroll clerk.

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The probability of obtaining a reading between -0.08°C and 1.68°C is approximately 0.4854 or 48.54%.

Probability is the branch of mathematics concerned with the occurrence of a random event, and there are four types of probability: classical, empirical, subjective, and axiomatic.

The readings at freezing on a set of thermometers are normally distributed, with a mean () of 0°C and a standard deviation () of 1.00°C. We want to know how likely it is that we will get a reading between -0.08°C and 1.68°C.

To solve this problem, we must use the z-score formula to standardise the values:

z = (x - μ) / σ

where x is the value for which we want to calculate the probability, is the mean, and is the standard deviation.

The lower bound is -0.08°C:

z1 = (-0.08 - 0) / 1.00 = -0.08

1.68°C is the upper bound:

z2 = (1.68 - 0) / 1.00 = 1.68

We can now use a standard normal distribution table or calculator to calculate the probabilities for each z-score.

The probability of obtaining a z-score of -0.08 or less is 0.4681, and the probability of obtaining a z-score of 1.68 or less is 0.9535, according to the table. We subtract the probability associated with the lower bound from the probability associated with the upper bound to find the probability of obtaining a reading between -0.08°C and 1.68°C:

P(-0.08°C x 1.68°C) = P(z1 z z2) = P(z 1.68) minus P(z -0.08) = 0.9535 - 0.4681 = 0.4854

As a result, the chance of getting a reading between -0.08°C and 1.68°C is approximately 0.4854 or 48.54%.

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Step-by-step explanation:

the volume of container B is Travers from A to B.

so, the volume of the empty space in A is exactly the volume of container B.

the volume of a cylinder is

base area × height = pi×r² × height.

the reside is as always half of the diameter.

r = 8/2 = 4 ft

the volume of the empty space in A = the volume of container B =

= pi×4² × 20 = pi×16 × 20 = 320pi = 1,005.309649... ≈

≈ 1,005.3 ft³

Answer:

B) 33 times.

Step-by-step explanation:

The total amount of pebbles is 50. There is 22 yellow pebbles.

Note that 3/2 * 50 is 75. 3/2 * 22 = 33.

He should expect to choose a yellow pebble B) 33 times.

The percent of the number 2160 which is 270 is 12.5%.

Given a number 2160.

It is required to find the percent of this number which is 270.

Let x be the percent of 2160 which is 270.

Then, this can be written as:

2160 × (x/100) = 270

2160 × x = 270 × 100

2160 × x = 27000

x = 27000 / 2160

  = 12.5

Hence the required percentage is 12.5%.

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A) Relative frequency is number of times an event happened over total number of events:

Answer is 21/50 = 0.42

B) On a 6 sides die, there are 3 even numbers and 3 odd numbers, so the theoretical probability of landing on odd would be 3/6 = 0.50

C) Because the die has an equal amount of chance landing on even or odd, both are 3/6, then the dice is not biased.

Answer:

Debra has $10.50, Alan has $3.50, and Shen has $108.

Step-by-step explanation:

Set up a system of equations to represent the situation.

Let a represent Alan's money, d represent Debra's money, and s represent Shen's money.

1. a + d + s = 122

2. a = d - 7

3. d = 3a

Use equation's 1 and 2 alone to calculate a and d. Substitute Eq 3 into Eq 2 such that

a = 3a - 7

-2a = -7

a = $3.50

Use Alan's money to calculate Debra's money.

d = 3(3.5)

d = $10.50

Now use values of a and d to calulate Shen's money, given the total.

s + 3.5 + 10.5 = 122

s + 14 = 122

s = $108

So, Debra has $10.50, Alan has $3.50, and Shen has $108.

Answer:

angle 4 and angle 1 are vertical angles

Step-by-step explanation:

vertical angles are opposite each other

By looking at the vertex of the graph, we can see that the fourth graph is the correct option.

Here we want to see which one of the given graphs represents the given absolute value function.

Remember that for the absolute value function:

f(x) = |x - a| + b

Has a vertex at the point (a, b) and opens up.

Then in this particular case, with the function f(x)=∣x+1∣−3, the vertex will be at the point (-1, -3), so we just need to identify which one of the given graphs has that vertex, we can see that the correct option is the fourth option.

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This is the linear approximation of f(x) = 5th root of x when x = 32. To find the linear approximation of f(x) = 5th root of x when x = 32, we need to first find the equation of the tangent line at x = 32.

The derivative of f(x) = 5th root of x is: f'(x) = 1/(5x⁴/⁵)

So, at x = 32, we have f(32) = 2 and f'(32) = 1/80.

Using the point-slope form of a line, we can write the equation of the tangent line as:

y - 2 = (1/80)(x - 32)

Simplifying this equation, we get:

y = (1/80)x + (79/40)

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Answer:

Step-by-step explanation:

The range rule of thumb states that the range of a dataset is approximately four times the standard deviation. Using this rule, we can estimate the range of brain volumes in this dataset:

Range ≈ 4 × standard deviation = 4 × 120.4 cm3 = 481.6 cm3

To identify the limits separating values that are significantly low or significantly high, we can add and subtract half the range to and from the mean:

Lower limit = mean - (range/2) = 1190.7 cm3 - (481.6 cm3 / 2) = 949.9 cm3

Upper limit = mean + (range/2) = 1190.7 cm3 + (481.6 cm3 / 2) = 1431.5 cm3

Therefore, any brain volume below 949.9 cm3 or above 1431.5 cm3 would be considered significantly low or significantly high, respectively.

A brain volume of 1411.5 cm3 is within the range of values that are not considered significantly high or low. Its z-score can be calculated as follows:

z-score = (1411.5 - 1190.7) / 120.4 = 1.84

Using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 1.84 or higher is approximately 0.0336. This means that a brain volume of 1411.5 cm3 would be considered uncommon, but not extremely rare.

The volume of the frustum of right circular cone =  7730 cube

The volume of the frustum of the right circular cone:

The formula for the Volume of Frustum of Cone (V)=[tex]\frac{1}{3} \pi H (R^2+r^2+Rr )[/tex]

where,

         H = Height of frustum

         R = Radius of lower base

         r = Radius of top base

According to given question,

      Height of frustum(H) = 30 units

       Radius of lower base(R) = 21 units

        Radius of top base (r) = 11 units

Substituting all the given values in the formula of volume of the frustum of the cone we will get,

    The volume of a frustum of a right circular cone(V)   =[tex]\frac{1}{3} \pi H (R^2+r^2+Rr )[/tex]

                                                                            [tex]=\frac{1}{3}*30(21^2+11^2+21*11)\\\\=10*(441+121+211)\\=10*(773)\\=7730 unit^3[/tex]  

The volume of the frustum of the right circular cone =  271.22

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Answer:

I am not good at this but i gess itis 11 √121 equals 11

Answer:

Step-by-step explanation:

I don’t know how to do that it