PLEASEE HELP ME THIS IS DUE AN IN HOUR!!

The polynomial for the number of miles between the two cars is d(t) = 5t + 150

Given that

Car A: 7t²+50t+150

Car B: 7t²+45t

The distance function is represented as

d(t) = 7t²+50t+150 - 7t² - 45t

So, we have

d(t) = 5t + 150

This means that t =3

So, we have

d(3) = 5 * 3 + 150

d(3) = 165

This means that d(t) = 200

So, we have

5t + 150 = 200

Evaluate the difference

5t = 50

So, we have

t = 10

Hence, the time is 10 hours

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The degree of the function (1)/(2)=(x+2)(x-1)^(2)(5x-2) is 4, as there are 4 total x terms in the equation.

The end behavior of the function is determined by the leading term, which is (1/2)(5x^4). As the degree is even and the leading coefficient is positive, the end behavior is that the function will rise to the right and rise to the left.

The x-intercepts of the function are the values of x that make the function equal to zero. These can be found by setting each factor equal to zero and solving for x:

x+2=0 -> x=-2

x-1=0 -> x=1

5x-2=0 -> x=2/5

The x-intercepts are -2, 1, and 2/5.

The y-intercept is the value of the function when x=0. Plugging in 0 for x gives:

(1/2)(0+2)(0-1)^2(5(0)-2) = (1/2)(2)(-1)^2(-2) = -2

The y-intercept is -2.

The zeroes of multiplicity are the values of x that make the function equal to zero and the number of times they appear as a factor in the equation. In this case, the zeroes of multiplicity are:

-2 with a multiplicity of 1

1 with a multiplicity of 2

2/5 with a multiplicity of 1

A few midinterval points can be found by plugging in values of x between the x-intercepts and solving for the function value. For example, plugging in x=0.5 gives:

(1/2)(0.5+2)(0.5-1)^2(5(0.5)-2) = (1/2)(2.5)(-0.5)^2(-0.5) = -0.3125

So one midinterval point is (0.5, -0.3125).

Another midinterval point can be found by plugging in x=-1:

(1/2)(-1+2)(-1-1)^2(5(-1)-2) = (1/2)(1)(-2)^2(-7) = -7

So another midinterval point is (-1, -7).

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Answer:

x^2 + 10x + 20

Step-by-step explanation: When you multiply binomials, you are practically taking each term and multiplying it with others until you are out of combinations. However, when the product is represented in standard form, it requires the terms to be set within a specific order that prompts the term at the highest order/power placed on the furthest left side of the expression. The first term x^2 can be referred to as a term of the second power, 10x is seen as one to the first power (x^1 = 1st), and 20 as 0, since it is a constant.

Given the slope of the line and the intercept it forms with the y-axis, one of the mathematical forms used to derive the equation of a straight line is called the slope intercept form. Y = mx + b, where m is the slope of the straight line and b is the y-intercept, is the slope intercept form.

The equation of the line, as determined by the slope-intercept formula, is: y = mx + b, where m is the line's slope and b is its y-intercept. As x and y represent each point on the line, they must be maintained as variables when using the formula above.

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The equation of exponential function=y = 0.054 × [tex]1.028^{x}[/tex]

As the name implies, exponents are utilised in exponential functions. But remember that a variable serves as the exponent instead of a constant as the basis of an exponential function.

A function is a power function rather than an exponential function if its base is a variable and its exponent is a constant.

If the graph of exponential function passes through the points (1960,19.005) and (2010,19.548), then the coordinates of these points satisfy the equation, y = [tex]ab^{x}[/tex]

Now, 19.005 = a × [tex]b^{1960}[/tex]

⇒  [tex]b^{1960}[/tex] = 19.005/a

Similarly, 19.548 = a × [tex]b^{2010}[/tex]

⇒ [tex]b^{2010}[/tex] = 19.548/a

⇒  [tex]b^{1960}[/tex] × [tex]b^{50}[/tex] = 19.548/a

⇒ [tex]b^{50}[/tex] = 19.548/a × a/19.005

         = 1.028

Putting this value to find a,

we get a = 0.054

So, the equation of exponential function=

y = 0.054 × [tex]1.028^{x}[/tex]

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To find the number of coins the statement made by Elena is correct.

The exponent of a number indicates how many times a number has been multiplied by itself. For instance, 34 indicates that we have multiplied 3 four times. Its full form is 3 3 3 3. Exponent is another name for a number's power. It might be an integer, a fraction, a negative integer, or a decimal.

Elena is on point. The formula 228 = 28 x 2 = 56 doesn't make sense in this situation since it suggests that they only counted for 28 days and received 228 coins, when the problem states that they counted for 28 days and received 228 coins. The fact that they counted for 28 days and came up with a total of 28 times 28 coins, or 784, makes the equation 228 = 28 x 28 make sense. Elena is accurate as a result.

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Answer:

37

Step-by-step explanation:

4.61 x 10¹⁷ atoms of radon means that is the total number of atoms in the  element and is dependent on the number of mole.

This is referred to as the proportionality factor that relates the number of constituent particles in a sample with the amount of substance in that sample.

One mole of a substance is equal to 6.022 × 10²³ units of that substance (such as atoms, molecules, or ions). The number 6.022 × 10²³ is known as Avogadro's number or Avogadro's constant. The concept of the mole can be used to convert between mass and number of particles.

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By answering the supplied question, we may infer that the response is slope As a result, the equation's y-intercept is equal to -8.

A line's slope determines how steep it is. The gradient is mathematically expressed as gradient overflow. By dividing the vertical change (run) between two spots by the height change (rise) between the same two locations, the slope is determined. The equation for a straight line, y = mx + b, is written as a curve form of an expression. When the slope is m, b is b, and the line's y-intercept is located (0, b). For instance, the y-intercept and slope of the equation y = 3x - 7 (0, 7). The location of the y-intercept is where the slope of the path is m, b is b, and (0, b).

The slope of the line is represented by the coefficient of x in the equation Y = -6x - 8.

As a result, the equation's slope is -6.

-8, the line's y-intercept, serves as the equation's constant term.

As a result, the equation's y-intercept is equal to -8.

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The solutions in radian units are:
Ω = 0.928, 2.213, 4.070, 5.355
The solutions should be entered as a comma-separated list:
Ω = 0.928, 2.213, 4.070, 5.355

The equation cos(Ω) = -cos(2Ω) can be solved by using the double angle formula for cosine. The double angle formula for cosine is cos(2Ω) = 1 - 2sin^2(Ω).

Substituting the double angle formula into the original equation gives:
cos(Ω) = - (1 - 2sin^2(Ω))

Rearranging the equation gives:
2sin^2(Ω) = 1 + cos(Ω)

Squaring both sides of the equation gives:
4sin^4(Ω) - 4sin^2(Ω) - cos^2(Ω) = 0

Using the identity cos^2(Ω) = 1 - sin^2(Ω) gives:
4sin^4(Ω) - 4sin^2(Ω) - (1 - sin^2(Ω)) = 0

Simplifying the equation gives:
4sin^4(Ω) - 3sin^2(Ω) - 1 = 0

Using the quadratic formula gives:
sin^2(Ω) = (3 ± √(9 + 16))/8

Solving for sin(Ω) gives:
sin(Ω) = ±√(7/8)

Taking the inverse sine of both sides gives:
Ω = sin^-1(±√(7/8))

The solutions in radian units are:
Ω = 0.928, 2.213, 4.070, 5.355

The solutions should be entered as a comma-separated list:
Ω = 0.928, 2.213, 4.070, 5.355

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a) y = 1/2 √ −3(x + 1) + 4
- The function is multiplied by 1/2, indicating a vertical compression by a factor of 1/2.
- The function is multiplied by -3 inside the square root, indicating a horizontal compression by a factor of 1/3 and a reflection across the y-axis.
- The function is shifted 1 unit to the left, indicated by the (x + 1) inside the square root.
- The function is shifted 4 units up, indicated by the + 4 outside the square root.

a) y = (2(x − 1))^3 − 5
- The function is multiplied by 2 inside the cube, indicating a horizontal compression by a factor of 1/2.
- The function is shifted 1 unit to the right, indicated by the (x - 1) inside the cube.
- The function is shifted 5 units down, indicated by the - 5 outside the cube.

The transformations to f(x) = √x are as follows:
a) y = 1/2 √ −3(x + 1) + 4
- The function is multiplied by 1/2, indicating a vertical compression by a factor of 1/2.
- The function is multiplied by -3 inside the square root, indicating a horizontal compression by a factor of 1/3 and a reflection across the y-axis.
- The function is shifted 1 unit to the left, indicated by the (x + 1) inside the square root.
- The function is shifted 4 units up, indicated by the + 4 outside the square root.

The transformations to f(x) = x^3 are as follows:
a) y = (2(x − 1))^3 − 5
- The function is multiplied by 2 inside the cube, indicating a horizontal compression by a factor of 1/2.
- The function is shifted 1 unit to the right, indicated by the (x - 1) inside the cube.
- The function is shifted 5 units down, indicated by the - 5 outside the cube.

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Carter will have to work about 33 hours in order to save $300.

To find out how many hours Carter will have to work in order to save $300, we can use the following steps:

1. First, we need to determine how much Carter saves per hour. Since he saves (3)/(5) of his earnings, we can multiply his hourly wage by (3)/(5) to find out how much he saves per hour:

$15.25 * (3)/(5) = $9.15

2. Next, we need to determine how many hours Carter will have to work in order to save $300. To do this, we can divide $300 by the amount he saves per hour:

$300 / $9.15 = 32.79 hours

3. Since Carter can't work a fraction of an hour, we'll round up to the nearest whole hour:

33 hours

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The volume of the left-hand cylinder is less than the volume of the right-hand cylinder.

What is a cylinder?

One of the most fundamental curvilinear geometric shapes, a cylinder has historically been a three-dimensional solid. It is regarded as a prism with a circle as its base in basic geometry. One of the fundamental three-dimensional shapes in geometry is the cylinder, which has two distant, parallel circular bases. At a predetermined distance from the centre, a curved surface connects the two circular bases. The axis of the cylinder is the line segment connecting the centres of two circular bases. The height of the cylinder is defined as the distance between the two circular bases.

The volume of the cylinder is given by:

V = π r² h

where r = radius

         h = height

Consider the left-hand side cylinder.

radius = h

height = g

Then the volume is V1 =  π h²g

Now consider the right-hand side cylinder

radius = g

height = h

Then the volume is V2 =  π g²h

It is given that g > h

Taking V1/V2 = π h²g/π g²h = h/g = k

where k is a constant.

Now V1 = k V2

This means that V2 will be k times V1.

So right-hand side cylinder has the largest volume among the two.

Therefore the volume of the left-hand cylinder is less than the volume of the right-hand cylinder.

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We have shown that ∫_((a,b])▒〖G(x)dF(x)=F(b)G(b)-F(a)G(a)-∫_((a,b])▒〖F(x)dG(x)〗〗 as required.

The given statement is that F and G are two cumulative distribution functions on the real line, and they have no common points of discontinuity in the interval (a, b). We need to show that ∫_((a,b])▒〖G(x)dF(x)=F(b)G(b)-F(a)G(a)-∫_((a,b])▒〖F(x)dG(x)〗〗

First, we can use the fact that F and G are cumulative distribution functions to write the integral of G(x)dF(x) as the difference of the product of F and G at the endpoints of the interval:

∫_((a,b])▒〖G(x)dF(x)=F(b)G(b)-F(a)G(a)〗

Similarly, we can write the integral of F(x)dG(x) as the difference of the product of F and G at the endpoints of the interval:

∫_((a,b])▒〖F(x)dG(x)=F(b)G(b)-F(a)G(a)〗

Subtracting the second equation from the first gives us:

∫_((a,b])▒〖G(x)dF(x)-∫_((a,b])▒〖F(x)dG(x)=F(b)G(b)-F(a)G(a)-F(b)G(b)+F(a)G(a)〗

Simplifying the right-hand side of the equation gives us:

∫_((a,b])▒〖G(x)dF(x)-∫_((a,b])▒〖F(x)dG(x)=0〗

Therefore, we have shown that ∫_((a,b])▒〖G(x)dF(x)=F(b)G(b)-F(a)G(a)-∫_((a,b])▒〖F(x)dG(x)〗〗 as required.

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A. My sister is a graduate of Polytechnic University and a border services officer who lives in North Delta.
B. Yesterday, Jasmine left to visit her friend in Alberta.
A. They listened to relaxing music while they did their homework, which helped them concentrate.
B. To help them focus on their homework, they listened to relaxing music.

Combining sentences is a useful writing technique that helps make sentences concise and easy to read. Combining two or more sentences into one sentence can also help to create smoother transitions between ideas. When combining sentences, it is important to make sure the sentence remains grammatically correct, and to retain important information from the original sentences.

Another way to combine sentences is to delete certain words or phrases. For example, in the sentence “Yesterday, Jasmine left to visit her friend in Alberta”, the phrase “yesterday” can be removed without changing the meaning of the sentence.

You can also combine sentences by changing the order of the words. For example, in the sentence “They listened to relaxing music while they did their homework, which helped them concentrate”, the order of the words can be changed to “They did their homework while they listened to relaxing music, which helped them concentrate”.

Finally, it is important to make sure that the sentence remains grammatically correct. When combining sentences, it is important to make sure that the sentence still has a subject and verb, and that the verb is still in the correct tense.

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Therefore, 13.509 is roughly the standard deviation of the number of attendees who would favour chicken.

The degree of variation or dispersion in a set of data is measured by standard deviation. It calculates how far the data points, on average, deviate from the mean (average) of the data collection. Finding the square root of the data's variation yields the standard deviation. In statistics, the standard deviation is frequently used to characterise the distribution of data and is significant in areas like science, finance, and economics. It can be used to assess the validity of statistical data and to contrast different groups of data.

given

The following formula must be used to determine the standard deviation of attendees who would favour chicken as the main course:

σ = √(np(1-p))

When we change the numbers, we obtain:

σ = √(780 x 0.38 x 0.62) (780 x 0.38 x 0.62)

σ = √(182.616) (182.616)

σ = 13.509

Therefore, 13.509 is roughly the standard deviation of the number of attendees who would favour chicken.

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the area of the shaded region:

[tex]= 2(the \ area \ of \ a \ quarter \ circle) - (the \ area \ of \ the \ square)\\[/tex]

[tex]= 2\Big(\dfrac{\pi 12^{2} }{4}\Big) - 12^{2} = \dfrac{144 \pi}{2} - 144 = 72 \pi - 144\\[/tex]

[tex]= 72 (\pi - 2) \ m^{2}[/tex]

Answer:

  72π-144 m²

Step-by-step explanation:

You want the area of a shaded region consisting of the overlap of two quarter circles in a 12 m square.

If we draw a diagonal from upper left to lower right through the figure, the shaded area is divided into two 90° segments of a circle of radius 12 m.

The formula for the area of a segment is ...

  A = 1/2r²(θ -sin(θ))

where θ is the measure of the central angle.

For θ = π/2 radians, this is ...

  A = 1/2r²(π/2 -1) . . . . . half the shaded area

Then the whole shaded area is ...

  2 × 1/2r²(π/2 -1) = (12 m)²(π/2 -1) = 72π -144 m²

The area of the shaded region is 72π -144 m².

__

Additional comment

If we expand the shaded area formula, we get ...

  A =1/2πr² -r²

This is recognizable as twice the area of a quarter circle, less the area of the square.

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The inverse of the following matrix X is X-1 = ​3/416/4-1/4​​-3/44/30/4​-2/41/41/4​​.

The inverse of a matrix is a matrix that, when multiplied by the original matrix, produces the identity matrix. To find the inverse of the following matrix X, we need to use the formula X-1 = 1/|X| adj(X), where |X| is the determinant of X and adj(X) is the adjoint of X.

First, let's find the determinant of X:
|X| = (1)(-12)(-3) + (2)(-1)(-2) + (2)(3)(-1) - (-2)(-1)(2) - (1)(3)(-2) - (4)(-1)(-3)
|X| = -36 + 4 + -6 + 4 - -6 - -12
|X| = -16

Since the determinant of X is not equal to 0, the inverse of X exists.

Now, let's find the adjoint of X:
adj(X) = ​-12-2-1​​34-30​24−11−3​​

Finally, let's use the formula X-1 = 1/|X| adj(X) to find the inverse of X:
X-1 = 1/(-16) ​-12-2-1​​34-30​24−11−3​​

X-1 = ​3/416/4-1/4​​-3/44/30/4​-2/41/41/4​​

Therefore, the inverse of the following matrix X is X-1 = ​3/416/4-1/4​​-3/44/30/4​-2/41/41/4​​.

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Answer: A, D, and F

Step-by-step explanation:

SOH CAH TOA

Sin = Opposite/Hypotenuse

Cos = Adjacent/Hypotenuse

Tan = Opposite/Adjacent

Answer:

tanθ=6/8, sinθ=6/10, cosθ= 8/10

Step-by-step explanation:

the ratio of tan is opposite over adjacent from the angle's perspective, and 6 is opposite whereas 8 is adjacent so tan(x)=6/8

the ratio of sin is opposite over hypotenuse from the angle's perspective, where 6 is opposite from it whereas the hypotenuse will always be the side opposite to 90 so it's 10 so sin(x)=6/10

the ratio of cos is adjacent over the hypotenuse from the angle's perspective, where 8 is adjacent to it whereas 10 is the hypotenuse so the ratio will be cos(x)=8/10

The angle LZW, given that angles Izw and Wzo are supplementary, would be an Obtuse angle.

If the angles Izw and Wzo are supplementary, then their sum is equal to 180 degrees.

Since Wzo is an acute angle (meaning it measures less than 90 degrees), we know that Izw must be an obtuse angle, so that their sum is 180 degrees. An obtuse angle is an angle that measures greater than 90 degrees and less than 180 degrees. It is "obtuse" because it is more than a right angle (90 degrees), which is considered "sharp" or "acute".

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The value of x in the sequence notation is 27/4

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

[tex]\sum\limits^4_{j=-2} {2x(-\frac{3}{2})^j = \frac{1389}{32}[/tex]

When the above expression is expanded, we have the following equation

2x[(-3/2)⁻² + (-3/2)⁻¹ + (-3/2)⁰ + (-3/2)¹ + (-3/2)² + (-3/2)³ + (-3/2)⁴] = 1389/32

Evaluate the exponents the expressions

So, we have

2x[4/9 - 2/3 + 1 - 3/2 + 9/4 - 27/8 + 81/16] = 1389/32

Evaluate the fractions

2x * 463/144 = 1389/32

Evaluate the products

x * 463/72 = 1389/32

Solve for x

x = 72/463 * 1389/32

Evaluate

x = 27/4

Hence, the value of x is 27/4

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The solution(s) of the equation are x = 0 and the solutions of the cubic equation x^(3)+3x^(2)+25x+75 = 0.

The solution(s) of the equation can be found by rearranging the terms and then factoring. Here are the steps:

Step 1: Rearrange the terms to have all the x terms on one side of the equation:
-4x^(4)+25x^(2)+75x+5x^(4)+3x^(3) = 0

Step 2: Combine like terms:
x^(4)+3x^(3)+25x^(2)+75x = 0

Step 3: Factor out the common factor of x:
x(x^(3)+3x^(2)+25x+75) = 0

Step 4: Use the zero product property to find the solutions:
x = 0 or x^(3)+3x^(2)+25x+75 = 0

The first solution is x = 0. The other solutions can be found by solving the cubic equation x^(3)+3x^(2)+25x+75 = 0. This equation does not have any rational solutions, so the solutions will be irrational or complex.

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To solve this problem, we can use a system of equations. Let's call the number of main-floor seats x and the number of balcony seats y.
The first equation can be written as:
10x + 4y = 5800
The second equation can be written as:
0.3x + 0.5y = 1900

Now we can use the elimination method to solve for one of the variables. Let's multiply the second equation by -10 to eliminate the x variable:
-3x - 5y = -19000
Adding the two equations together gives us:
7x - y = 3900
Now we can use substitution to solve for one of the variables. Let's solve for y in the first equation:
y = (5800 - 10x)/4
And substitute this value into the second equation:
7x - (5800 - 10x)/4 = 3900
Multiplying both sides by 4 gives us:
28x - 5800 + 10x = 15600
Solving for x gives us:
38x = 21400
x = 563.16
And plugging this value back into the first equation to solve for y gives us:
y = (5800 - 10(563.16))/4
y = 609.21
So there are approximately 563 main-floor seats and 609 balcony seats.

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The line's y-intercept is equal to -3.

An inequality to express the possible distances Jim and Lee can travel in the taxi is 1.75x + 3.00 < 45.

In Mathematics, the y-intercept is sometimes referred to as an initial value or vertical intercept and the y-intercept of any graph such as a linear function, generally occur at the point where the value of "x" is equal to zero (x = 0).

Based on the information provided about the line on the graph, we have the following:

y-intercept of function P = (0, -3).

Next, we would determine the possible distances Jim and Lee can travel in the taxi:

1.75x + 3.00 < 45.

1.75x < 45 - 3

1.75x < 42

x < 42/1.75

x < 24.

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The required equation of the line perpendicular to y=4 that passes through (-5,-2) is: x = -5

The equation of the line y=4 is a horizontal line that passes through the point (0,4).

A line perpendicular to this line would be a vertical line passing through the point (-5,-2).

The equation of a vertical line passing through the point (-5,-2) can be written in the form x = -5,

which means that the x-coordinate of any point on this line must be -5.

Therefore, the equation of the line perpendicular to y=4 that passes through (-5,-2) is: x = -5

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rank(A) = 1

Given that A = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$, where $a \neq 0$.Let's find the rank of A:Rank(A) = number of leading 1's in the row echelon form of A.= $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$=> $\left[\begin{array}{cc}a & b \\ 0 & 0\end{array}\right]$There are two leading 1's in row 1. Hence, rank(A) = 1.Now, let's find the basis of the null space of A:Null space of A is the set of all solutions to the equation $Ax=0$.So, $Ax$ = $\left[\begin{array}{cc}a & b \\ a & b\end{array}\right]$ $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}ax+by \\ ax+by\end{array}\right]$= $\left[\begin{array}{c}(a+b)x \\ (a+b)y\end{array}\right]$= $\left[\begin{array}{c}0 \\ 0\end{array}\right]$=> (a + b) x = 0 and (a + b) y = 0=> $x = \frac{-b}{a}y$=> $\left[\begin{array}{c}x \\ y\end{array}\right]$= $\left[\begin{array}{c}\frac{-b}{a}y \\ y\end{array}\right]$= $y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right]$=> Null space of A = $\{y \left[\begin{array}{c}\frac{-b}{a} \\ 1\end{array}\right] | y \in R \}$Let's take a = 1, b = 2. So, the null space of A = $\{y \left[\begin{array}{c}-2 \\ 1\end{array}\right] | y \in R \}$Therefore, the basis of the null space of A = $\left[\begin{array}{c}-2 \\ 1\end{array}\right]$Now, let's find the basis of the row space of A:Row space of A is the span of the rows of A.=> Row space of A = span $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$Since rank(A) = 1, there is only 1 non-zero row. Therefore, a basis of the row space of A is $\left\{\left[\begin{array}{cc}a & b\end{array}\right]\right\}$.

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The statement about cosecant, secant, and cotangent values that is correct is Kayla's: The cosecant and secant of an acute angle are always greater than 1, but the cotangent can be greater than 1, less than 1 or equal to 1.

In Trigonometry, the sine of an acute angle is a ratio of the length of an opposite side of a right-angled triangle to the length of its hypotenuse. Also, the sine of an angle generally lies between -1 and 1.

In Trigonometry, the cosecant (cosec) of an acute angle is a ratio of the length of its hypotenuse to the length of an opposite side of a right-angled triangle. Thus, the cosecant of an angle is the reciprocal of the sine function and as such lies outside -1 and 1.

The cotangent of an acute angle is a ratio of the length of its adjacent side to the length of an opposite side of a right-angled triangle.

In conclusion, Kayla's statement about the values of cosecant, secant, and cotangent is correct.

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Answer:

3m-4

Step-by-step explanation:

marcie's age be x

her father age is four less than three times

the equation tht describes her father age is 3m-4=40

The value of x would be equal to 20 degrees.

Linear pairs of angles are produced when two lines intersect each other at a point. The sum of angles of the linear pair is always 180 degrees.

We are given the parameters as;

7x and (x + 20)

The sum of angle is equal to 180.

7x + (x + 20) = 180

8x + 20 = 180

8x = 180 - 20

8x = 160

x = 160/8

x = 20

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The points are not on the same side. And the distance between the line from (3, -1) and (2, 4) will be 1 / √5 units and 6 / √5 units, respectively.

Let (x₁, y₁) be the point and ax + by + c = 0 be the line. Then the distance between a point and a line will be given as,

[tex]\rm d = \dfrac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}[/tex]

The diagram is given below.

The points are not on the same side of the line y = 2x - 6. Then the distance from a line to (3, -1) is given as,

d = |2 × 3 - (-1) - 6| / [√(2² + (-1)²)]

d = |7 - 6| / √5

d = 1 / √5

And the distance from a line to (2, 4) is given as,

d = |2 × 2 - 4 - 6| / [√(2² + (-1)²)]

d = |-6| / √5

d = 6 / √5

The points are not on the same side. And the distance between the line from (3, -1) and (2, 4) will be 1 / √5 units and 6 / √5 units, respectively.

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