1: Here is an inequality: -2x>10.

List some values for x that would make this inequality true.



2: How are the solutions to the inequality -2x10 different from the solutions to -2x>10? Explain your reasoning.

To calculate the chi-square test statistic, we need to create a contingency table with the observed frequencies for each combination of seed brand and germination status, as well as the expected frequencies if there were no association between the two variables.

We can use the following formula to calculate the chi-square test statistic:

χ² = ∑ (Oᵢ - Eᵢ)² / Eᵢ

where Oᵢ is the observed frequency for category i, Eᵢ is the expected frequency for category i, and the sum is taken over all categories.

Using the given data, we can construct the following contingency table:

                   Germinated         Failed to germinate            Total

Brand A              77                             23                               100

Brand B              43                             17                                  60

Total                  120                            40                                160

To calculate the expected frequencies, we can assume that there is no association between seed brand and germination status, and thus the expected frequency for each cell is the product of the row and column totals divided by the grand total.

For example, the expected frequency for the cell in the first row and first column is:

E₁₁ = (100/160) x (120/160) x 160 = 75

We can calculate the expected frequencies for all cells in the contingency table:

                     Germinated          Failed to germinate        Total

Brand A                75                              25                          100

Brand B                45                               15                          60

Total                     120                              40                        160

Using these values, we can calculate the chi-square test statistic:

χ² = [(77-75)²/75 + (23-25)²/25 + (43-45)²/45 + (17-15)²/15]

   = 1.013

To determine whether this value is statistically significant at the given level of significance, we need to compare it to the critical value from the chi-square distribution with (r-1) x (c-1) degrees of freedom, where r and c are the numbers of rows and columns in the contingency table.

For a level of significance of 0.05 and one degree of freedom (since there are two levels of the categorical variable), the critical value is 3.84.

Since our calculated value of 1.013 is less than the critical value, we would fail to reject the null hypothesis and conclude that there is no significant association between seed brand and germination status at the given level of significance.

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To evaluate the limit of the function f(x) = 4x^3 + 3x^2 + 2x - 2 as x approaches 1, we can substitute x = 1 into the function and calculate the resulting value.

When we substitute x = 1 into the function, we get f(1) = 4(1)^3 + 3(1)^2 + 2(1) - 2 = 4 + 3 + 2 - 2 = 7. This means that as x approaches 1, the function f(x) approaches the value of 7. In other words, the limit of f(x) as x approaches 1 is equal to 7.

Geometrically, we can interpret this result as follows. The function f(x) is a polynomial function of degree 3, which means that it is a cubic function. When we plot the graph of f(x), we get a curve that has a specific shape.

As x approaches 1 from both sides, we can see that the curve gets closer and closer to the horizontal line y = 7. This indicates that the limit of f(x) as x approaches 1 is indeed equal to 7.

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(a) is: Let t denote the number of text messages and c the charge, in dollars, (b) The formula is c(t) = 49.99 + 0.10(t - 100).

(a) In this case, we will let t denote the number of text messages and c represent the charge, in dollars.

(b) To write a formula for the cell phone charges as a function of the number of text messages (assuming t is at least 100), we will consider the flat monthly rate of $49.99 and the extra charges of $0.10 per text message after the first 100 messages. The formula will be:

c(t) = 49.99 + 0.10(t - 100)

This formula calculates the total charge, c(t), based on the number of text messages, t. The first 100 text messages are included in the flat monthly rate of $49.99, so we subtract 100 from the total number of text messages, t, and multiply the result by the extra charge per text message, $0.10. Then we add the flat monthly rate of $49.99 to get the total charge, c(t).

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Answer:

Coefficient of variation is not a measure of relative position. It is a measure of relative variability, which is the ratio of the standard deviation to the mean. It is commonly used to compare the variability of two or more datasets with different units or scales.

On the other hand, quartile, percentile, and median are all measures of relative position. Quartiles divide a dataset into four equal parts, while percentiles divide a dataset into 100 equal parts.

Median is the value that divides the dataset into two equal parts. These measures help to understand the distribution of the dataset and the position of a particular observation within that distribution.

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The expression (6 - √8) / (√2 - 1) can be written in the form a + b√2, where a = 4 and b = 6.

We have,

To write (6 - √8) / (√2 - 1) in the form a + b√2, we need to rationalize the denominator.

Multiplying both the numerator and denominator by the conjugate of the denominator.

(6 - √8) / (√2 - 1) x (√2 + 1)/(√2 + 1)

Simplifying the numerator and denominator:

= [(6 x (√2 + 1)) - (√8 x (√2 + 1))] / [(√2)² - 1²]

= [(6√2 + 6) - (2√2)] / (2 - 1)

= (6√2 + 4) / 1

= 6√2 + 4

Therefore,

(6 - √8) / (√2 - 1) can be written in the form a + b√2, where a = 4 and b = 6.

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A proportional relationship is represented by an equation in the form of y = kx, where k is a constant. This means that y is directly proportional to x, and as x increases or decreases, y will change by a constant ratio.

Out of the given options, the equation that represents a proportional relationship is:

y equals negative one third times x

This equation follows the form y = kx, where k = -1/3.

The function a(x) = 1/(x-1) has an inverse function a^-1(x) that can be found by switching the x and y variables and solving for y. The resulting equation is a^-1(x) = 1/(x-1) + 1.

To find the inverse function of a(x) = 1/(x-1), we switch the x and y variables and solve for y. This means we start with the equation:

x = 1/(y-1)

And solve for y:

xy - x = 1

xy = x + 1

y = (x+1)/x

So the inverse function of a(x) is:

a^-1(x) = (x+1)/x

However, we can simplify this expression by multiplying the numerator and denominator of the fraction by x-1:

a^-1(x) = [(x+1)/(x-1)] * [(x-1)/x]

a^-1(x) = (x+1)/(x-1) + 1/x

So the final equation for the inverse function is:

a^-1(x) = 1/(x-1) + 1

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Answer:

boring

Step-by-step explanation:

To prove the given identity, we'll start with the left side and simplify it step by step until we obtain the right side of the equation:

We have:

((sec y - tan y)(sec y + tan y)) / sec y

Using the difference of squares, we can rewrite the numerator:

((sec y)^2 - (tan y)^2) / sec y

Recall the trigonometric identity: (sec x)^2 - (tan x)^2 = 1

Substituting this identity into our expression:

(1) / sec y

Using the reciprocal identity: sec x = 1 / cos x

We can rewrite the expression as:

1 / (1 / cos y)

Simplifying further by multiplying by the reciprocal:

cos y / 1

Which simplifies to:

cos y

Therefore, we have shown that the left side of the identity ((sec y - tan y)(sec y + tan y)) / sec y simplifies to cos y, which is the same as the right side of the equation.

The power is 430.2 watts. To calculate the power, we need to first find the instantaneous value of the product of voltage and current, which is the power at any given time.

Then we can take the time-average of this product to find the average power over a period.

Given, v = 150cos(ωt - 50°) and i = 20cos(ωt - 15°), the product of voltage and current is:

p(t) = v(t) i(t) = 150cos(ωt - 50°) × 20cos(ωt - 15°)

p(t) = 3000cos(ωt - 50°)cos(ωt - 15°)

Using the trigonometric identity cos(a)cos(b) = (1/2)[cos(a+b) + cos(a-b)], we can simplify this expression as:

p(t) = (1500/2)[cos(35°) + cos(2ωt - 65°)]

The average power over a period T is given by:

p = (1/T) ∫[p(t)]dt from t=0 to t=T

Since the time-average of the second term (cos(2ωt - 65°)) over a period is zero, we can ignore it and focus only on the first term. The integral becomes:

p = (1/T) ∫[750cos(35°)]dt from t=0 to t=T

p = (750/T) ∫[cos(35°)]dt from t=0 to t=T

Since cos(35°) is a constant, we can take it outside the integral and evaluate the integral:

p = (750/T) [sin(35°)T - sin(35°)0]

Since sin(35°) = 0.5736, we get:

p = (750/T) [0.5736T] = 430.2 watts

Therefore, the power is 430.2 watts.

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To prove that a matrix A with n linearly independent eigenvectors also has n linearly independent eigenvectors, we need to use the fact that eigenvectors corresponding to distinct eigenvalues are linearly independent.

Let λ1, λ2, ..., λn be the n distinct eigenvalues of A with corresponding eigenvectors v1, v2, ..., vn, respectively. Since these eigenvectors are linearly independent, we know that any linear combination of them that equals the zero vector must have all coefficients equal to zero.

Suppose now that there exists a linear combination of these eigenvectors that equals the zero vector, say c1v1 + c2v2 + ... + cnvn = 0, where not all ci are zero. Without loss of generality, assume that c1 is nonzero. Multiplying both sides of this equation by A, we obtain

c1λ1v1 + c2λ2v2 + ... + cnλnvn = 0.

Since λ1, λ2, ..., λn are distinct, we know that λ1 cannot be equal to λ2, λ3, ..., λn. Therefore, we can divide both sides of the above equation by λ1 and obtain

c1v1 + (c2λ2/λ1)v2 + ... + (cnλn/λ1)vn = 0.

This equation shows that we have a nontrivial linear combination of the eigenvectors that equals the zero vector, contradicting the assumption that they are linearly independent. Therefore, our original assumption must be false, and we conclude that A has n linearly independent eigenvectors.

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The figures that can be precisely defined by using only undefined terms are: angle, line segment, and parallel lines. The correct answer is options (a), (d) and (e).

An angle is formed by two rays sharing a common endpoint, called the vertex. Rays are simply parts of lines, and lines are undefined terms in geometry. Thus, an angle can be defined using only undefined terms.
A line segment is a part of a line that has two distinct endpoints. Lines are undefined terms in geometry, so a line segment can be defined using only undefined terms.

Parallel lines are lines that are always equidistant and never intersect. The concept of lines is an undefined term in geometry, so parallel lines can be defined using only undefined terms.
In contrast, an arc and a circle rely on defined terms such as radius and circumference, so they cannot be precisely defined by using only undefined terms.

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a. The population of interest in this case is the general public of the United States who have knowledge about the criminal trial and its verdict. b. It is likely that the proportion of people unhappy with the verdict in the sample is similar to the proportion of individuals unhappy with the verdict in the population.

a. The population of interest in this case is the general public of the United States who have knowledge about the criminal trial and its verdict.

b. It is likely that the proportion of people unhappy with the verdict in the sample is similar to the proportion of individuals unhappy with the verdict in the population. This is because the television news program invited viewers to go to their website and indicate their opinion, which means that only a certain segment of the population who watched that particular news program and had access to the internet were able to participate in the poll. Additionally, people who feel strongly about the verdict may be more likely to participate in the poll, leading to potential bias in the sample. Therefore, it is important to acknowledge that the poll results may not be entirely representative of the entire population and should be interpreted with caution. In order to obtain a more accurate estimate of the proportion of people unhappy with the verdict in the population, a larger, more diverse and random sample should be used.

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The probability of a point in the red triangle is 0.64 or approximately 64%.

To find the likelihood that a haphazardly chosen point inside the circle falls losing money concealed triangle, we really want to decide the proportion of the region of the triangle to the region of the circle.

We should initially track down the region of the circle. The range of the circle is 4 units, so its region is π [tex](4)^2[/tex] = 16π square units.

Then, we should track down the region of the triangle. The foundation of the triangle is 8 units, and the level is likewise 8 units (since the triangle is an isosceles right triangle). So the region of the triangle is (1/2)(8)(8) = 32 square units.

Consequently, the likelihood that a haphazardly chosen point inside the circle falls bleeding cash concealed triangle is:

32/16π ≈ 0.64

Adjusting to the closest 100th, the response is 0.64. So the likelihood is around 64%.

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The image of B after applying the translation rule is (7, -4). Therefore, the correct answer is option A: (7, -4).

To find the image of point B(2,5) using the translation (x,y)-(x+5,y-9), we need to add 5 to the x-coordinate and subtract 9 from the y-coordinate.

Therefore, the image of point B(2,5) would be (2+5,5-9) which simplifies to (7,-4). This means that the point B(2,5) has been translated 5 units to the right and 9 units downwards to the point (7,-4).

Hence, the correct answer is A. (7,-4).

The point B(2, 5) and the translation rule (x, y) -> (x + 5, y - 9). To find the image of B, we simply need to apply the translation rule to its coordinates.

Start by taking the x-coordinate of B, which is 2, and add 5 according to the translation rule. This gives us 2 + 5 = 7. Next, take the y-coordinate of B, which is 5, and subtract 9, as stated in the rule. This gives us 5 - 9 = -4.

So, the image of B after applying the translation rule is (7, -4). Therefore, the correct answer is option A: (7, -4).

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a) To draw a direction field, we need to calculate the slope at several points in the x-y plane. For the system of equations given by:

$\begin{pmatrix} x' \ y' \end{pmatrix} = \begin{pmatrix} 4 & -3 \ 8 & -6 \end{pmatrix} \begin{pmatrix} x \ y \end{pmatrix}$

We can calculate the slope at a point (x,y) as:

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{-3x-6y}{4x-8y}$

Using this formula, we can calculate the slope at several points and plot the corresponding line segment with that slope, resulting in a direction field.

b) To find the general solution of the system, we can begin by finding the eigenvalues and eigenvectors of the coefficient matrix:

[tex]$\begin{pmatrix} 4 & -3 \ 8 & -6 \end{pmatrix} \begin{pmatrix} v_1 \ v_2 \end{pmatrix} = \lambda \begin{pmatrix} v_1 \ v_2 \end{pmatrix}$[/tex]

This gives us the system of equations:

$4v_1 - 3v_2 = \lambda v_1$

$8v_1 - 6v_2 = \lambda v_2$

which we can rewrite as:

[tex]$(4-\lambda)v_1 - 3v_2 = 0$$8v_1 - (6+\lambda)v_2 = 0$[/tex]

This system has nontrivial solutions only if the determinant of the coefficient matrix is 0:

[tex]$\begin{vmatrix} 4-\lambda & -3 \ 8 & -6-\lambda \end{vmatrix} = (4-\lambda)(-6-\lambda) - (-3)(8) = \lambda^2 - 2\lambda - 24 = 0$[/tex]

Solving this quadratic equation, we find that the eigenvalues are $\lambda_1 = 6$ and $\lambda_2 = -4$. To find the corresponding eigenvectors, we can solve the systems of equations:

$(4-6)v_{11} - 3v_{21} = 0$

$8v_{11} - (6+6)v_{21} = 0$

and

$(4+4)v_{12} - 3v_{22} = 0$

$8v_{12} - (6-4)v_{22} = 0$

which give us the eigenvectors[tex]$\begin{pmatrix} 3 \ 4 \end{pmatrix}$ and $\begin{pmatrix} 3 \ -2 \end{pmatrix}$,[/tex] respectively.

Using these eigenvalues and eigenvectors, we can write the general solution of the system as:

[tex]$\begin{pmatrix} x(t) \ y(t) \end{pmatrix} = c_1 e^{6t} \begin{pmatrix} 3 \ 4 \end{pmatrix} + c_2 e^{-4t} \begin{pmatrix} 3 \ -2 \end{pmatrix}$[/tex]

where $c_1$ and $c_2$ are constants determined by the initial conditions.

c) To draw a few of the trajectories, we can pick some initial conditions and plot the corresponding solutions on the x-y plane. For example, if we choose $c_1 = c_2 = 1$ and $x(0) = 0$, $y(0) = 1

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The formula solved for the variable x is x = √2(y + 2)

Subject of formula can be described as the variable that is made to stand out on one end of the equality sign.

The variable is allowed to stand alone.

It is the variable that is being sort out in an equation.

From the information given, we have that;

y=1/3 x2-2

To make 'x' the subject, we have;

y + 2 = 1/2x²

Divide both sides by the coefficient of x squared, we have;

2(y + 2) = x²

Now, find the square root of both sides, we have;

x = [tex]\sqrt{2(y + 2)}[/tex]

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The equation of the circle graphed in this problem is given as follows:

(x - 4)² + (y - 3)² = 29.

The equation of a circle of center [tex](x_0, y_0)[/tex] and radius r is given by:

[tex](x - x_0)^2 + (y - y_0)^2 = r^2[/tex]

The radius of a circle represents the distance between the center of the circle and a point on the circumference of the circle.

The coordinates of the center are given as follows:

(4,3).

Hence:

(x - 4)² + (y - 3)² = r².

Considering the center of (4,3) and the point (-1, 1) on the circumference of the circle, the square of the radius is given as follows:

r² = (-1 - 4)² + (1 - 3)²

r² = 29.

Hence the equation is:

(x - 4)² + (y - 3)² = 29.

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The required length of the missing side is 9.28 units for the given triangle.

As shown triangle:

∠A = 104°

∠B = 47°

∠C = 29°

Side c = 7 units

Let's denote the length of the missing side as x.

Using the Law of Sines:

sin(A)/a = sin(B)/b = sin(C)/c

We know the values of angles A, B, and C, as well as the length of side c. Plugging in these values, we get:

sin(104°)/x = sin(47°)/7

To find x, we can rearrange the equation and solve for x:

x = (7 × sin(104°)) / sin(47°)

Using a calculator, to get the value of x:

x ≈ 9.28

Therefore, the length of side a (in front of angle A) is approximately 9.28 units.

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Proof of the similarity of triangle AMN and Triangle ABC is shown below.

We have to given that;

In a figure,

⇒ MN is parallel to BC.

Since, A triangle is a three sided polygon, which has three vertices and three angles which has the sum 180 degrees.

And, Two rays (half-lines) combined into an angle have a single terminal. The latter is known as the vertex of the angle, while the rays are known as its sides, occasionally as its legs, and occasionally as its arms.

Now, By definition of alternate angle of parallel lines, we get;

⇒ ∠ AMN and ∠ ABC are equal.

⇒ ∠ ANM and ∠ ACB are equal.

And, In triangles AMN and ABC;

⇒ ∠ A is common angle.

Hence, By AAA postulates we can say that;

⇒ Δ AMN ≈ Δ ABC

Thus, triangle AMN and Triangle ABC are similar by AAA postulates.

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Answer:

-2/3

Step-by-step explanation:

The vertex is the point on the curves with minimum distance from focus and the center is the midpoint between the vertices.

A set of points on a plane such that they have a positive and constant absolute difference from its 2 fixed points is called a hyperbola.

The two separate curves of a hyperbola are called its branches. The point on each branch with a minimum distance from the focus is called the vertices of the hyperbola.

It is the midpoint between the vertice that is called the center.

Here, we can see that the center is marked by (h,k) while the left vertex is (h - a , k)and (h + a , k)

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To find fx, fy, we differentiate the function : a) fx(-2, 2) = -22, b) fy(-2, 2) = 22.

a) To find fx, we differentiate the function with respect to x while treating y as a constant:

fx(x, y) = d/dx (x² - 9xy + y²)

= 2x - 9y

Now, evaluate fx(-2, 2):

fx(-2, 2) = 2(-2) - 9(2)

= -4 - 18

= -22

Therefore, fx(-2, 2) = -22.

Partial derivatives are a type of derivative that measures the rate at which a function changes with respect to one variable while holding all other variables constant. In other words, it calculates the slope or rate of change of a function with respect to a specific variable, assuming that all other variables remain fixed.

b) To find fy, we differentiate the function with respect to y while treating x as a constant:

fy(x, y) = d/dy (x² - 9xy + y²)

= -9x + 2y

Now, evaluate fy(-2, 2):

fy(-2, 2) = -9(-2) + 2(2)

= 18 + 4

= 22

Therefore, fy(-2, 2) = 22.

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We know that 1044 people contracted the flu virus 14 days after it first began.

Since the rate of increase is proportional to the number of people that contract the flu virus at any time, we can use the formula:

N = N0 * e^(kt)

Where N is the number of people that contract the virus at any given time, N0 is the initial number of people that contracted the virus, t is the time elapsed (in days), k is the proportionality constant, and e is the mathematical constant approximately equal to 2.718.

Using the information given, we can solve for k:

743 = 500 * e^(k*7)

e^(k*7) = 1.486

k*7 = ln(1.486)

k = 0.075

Now we can use the formula to find the number of people that contracted the virus after 14 days:

N = 500 * e^(0.075*14)

N ≈ 1044

Therefore, approximately 1044 people contracted the flu virus 14 days after it first began.

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The strength of the correlation is indicated by the sign and the numerical value. Thus, the correct option is:

c. both a and b.

The strength of the correlation is determined by both its sign (+ or -) and its numerical value.

The sign indicates the direction of the relationship (positive or negative). A positive correlation coefficient (r) indicates a direct or positive relationship between the variables. A negative correlation coefficient (r) indicates an inverse or negative relationship between the variables.

The numerical value represents the magnitude of the correlation, with values closer to 1 or -1 indicating stronger correlations. A correlation coefficient close to 0 indicates a weak correlation. When the correlation coefficient is close to 0, it suggests little to no relationship between the variables.

Therefore, to assess the strength of a correlation, we need to consider both the sign and the numerical value of the correlation coefficient.

Hence, the correct option is :

(c) both a and b.

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A major arc is an arc of a circle that measures more than 180 degrees, while a minor arc is an arc that measures less than 180 degrees.

A major arc spans more than half of the circle measuring greater than 180 degrees but minor arc spans less than half of the circle measuring less than 180 degrees.

The terms are relative to the size of the angle formed by the arc. In geometric terms, a major arc is an obtuse angle while a minor arc is an acute angle. The concept are used in geometry and trigonometry to describe size of angles and their corresponding arcs on a circle.

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The correct option is B  [tex]x^{2} - x-10=0[/tex]

Given quadratic equation [tex]\frac{2}{3x-6} - \frac{1}{3x+6} =\frac{1}{3}[/tex]

On solving, [tex]\frac{2(3x+6)-1(3x-6)}{(3x-6)(3x+6)} = \frac{1}{3}[/tex]

Further, [tex]\frac{6x+12-3x+6}{9x^{2} -6^{2} } = \frac{1}{3}[/tex]

[tex]\frac{3x+18}{9x^{2} -36} = \frac{1}{3}[/tex]

Taking 3 common from numerator and 9 common from denominator, we have

[tex]\frac{3(x+6)}{9(x^{2} -4)} = \frac{1}{3}[/tex]

[tex]\frac{(x+6)}{3(x^{2} -4)} = \frac{1}{3}[/tex]

[tex]\frac{x+6}{x^{2} -4} = 1[/tex]

[tex]x+6=x^{2} -4[/tex]

On rearranging in decreasing powers of x we get, [tex]x^{2} - x-10=0[/tex]

Hence on solving, We get [tex]x^{2} - x-10=0[/tex]

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The image of -2,5 is 1,1 . The image of 3,2 under the same transition is

(6,-2).

The image of the point (-2,5) is (1,1) under a certain translation. To find the image of the point (3,2) under the same translation, we need to calculate the difference between the original point and its image.

For (-2,5) to (1,1), the change in the x-coordinate is 1 - (-2) = 3, and the change in the y-coordinate is 1 - 5 = -4. Apply this same translation to the point (3,2): add 3 to the x-coordinate (3 + 3 = 6) and subtract 4 from the y-coordinate (2 - 4 = -2). The image of the point (3,2) under the same translation is (6,-2).

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Answer:

= 70.5 ^2 , 70.6^2

Step-by-step explanation:

[tex]a = \frac{given \: degree}{360} \times \pi {r}^{2} [/tex]

= 165°

= 7

= 3.14

= 165°/360° × 3.14 × (7)^2

= 70.5 ^2

The p-value of 0.03 indicates that there is evidence to reject the null hypothesis (h0) and support the alternative hypothesis (ha) that the difference between the means of the two populations (μ1 - μ2) is greater than zero. This suggests that there is a significant difference between the two groups being compared. However, it is important to note that the results of the t-test do not provide information on the magnitude or practical significance of this difference. Additional analysis may be necessary to determine the practical implications of the findings.

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The p-value of the two-sample t-test is 0.03.

In this case, the null hypothesis (H0) states that there is no significant difference between the means of the two populations (μ1 - μ2 = 0). The alternative hypothesis (Ha) suggests that the difference between the means of the two populations is greater than zero (μ1 - μ2 > 0). The p-value of 0.03 indicates the probability of obtaining a result as extreme as, or more extreme than, the observed difference in sample means if the null hypothesis were true.

Since the p-value (0.03) is typically compared to a predetermined significance level (usually 0.05), we can conclude that the p-value of 0.03 is less than the significance level of 0.05. This suggests that there is significant evidence to reject the null hypothesis (H0) in favor of the alternative hypothesis (Ha), indicating that the difference between the means of the two populations is likely greater than zero.

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Answer: 1

Step-by-step explanation: look at the number beside the 0 or wtv number it is 5 and above give it a shove 4 and below leave it alone

Answer:

Rounded to the nearest thousandth: 1.008

Rounded to the nearest hundredth: 1.01

Rounded to the nearest tenth: 1.0

Rounded to the nearest whole number: 1

Step-by-step explanation: