1. Determine if the function is an exponential function.
f(x) = ab^z

The  volume of the pyramid is determined as 266.67 cm³.

The volume of the pyramid is calculated by applying the following formula as shown below;

V = ¹/₃ Bh

where;

The  volume of the pyramid is calculated as;

V = ¹/₃ x (10 cm)² x 8 cm

V = 266.67 cm³

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Yes, it is true that if A and B are square and invertible matrices, then AB is also invertible, and its inverse is given by[tex](AB)^{(-1) }= B^{(-1)}A^{(-1).[/tex]

To see why this is true, consider the product[tex](AB)(B^{(-1)}A^{(-1)}).[/tex]

Using the associative property of matrix multiplication, we can rearrange this expression as

[tex](A(BB^{(-1)})A^{(-1)}) = (AIA^{(-1)}) = AA^{(-1)} = I,[/tex]

where I is the identity matrix.

Similarly, we can show that [tex](B^{(-1)}A^{(-1)})(AB) = I,[/tex] which means that

[tex](AB)^{(-1) }= B^{(-1)}A^{(-1).[/tex]

Therefore, we conclude that if A and B are square and invertible matrices, then AB is invertible, and its inverse is given by [tex](AB)^{(-1)} = B^{(-1)}A^{(-1).[/tex]

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The expression can be simplified to get:

[tex]3*(a^0*b^2) = 3b^2[/tex]

Remember that for any real:

x > 0

We have that the power when the exponent is zero, is equal to one.

[tex]x^0 = 1[/tex]

Here we know that:

a > 0, b > 0.

And we have the expression:

[tex]3*(a^0*b^2)[/tex]

The first power can be removed, because that is equal to 1, then we can simplify our expression to get.

[tex]3*(a^0*b^2) = 3b^2[/tex]

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(i) To find the number of different department codes possible, we need to determine the number of choices available for each character in the code.

The first two characters can be any of the 26 capital letters of the alphabet, so there are 26 choices for each of these characters. The third character must be a digit, so there are 10 choices for this character. Therefore, the total number of different department codes possible is: 26 x 26 x 10 = 6,760.

(ii) If no letter appears more than once, then we have to choose three distinct letters for the code. There are 26 choices for the first letter, 25 choices for the second letter (since we've already used one letter), and 24 choices for the third letter (since we've already used two letters).

The third character must be a digit, so there are 10 choices for this character. Therefore, the total number of different department codes possible is: 26 x 25 x 24 x 10 = 15,600.

(iii) If no letter or digit appears more than once, then we have to choose three distinct characters (either letters or digits) for the code. There are 26 choices for the first character (since it can be any letter), 25 choices for the second character (since it can be any letter except the one we chose for the first character or any digit),

and 10 choices for the third character (since it can be any digit except the one we chose for the second character). Therefore, the total number of different department codes possible is:
26 x 25 x 10 = 6,500

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The margin of error associated with the 95% confidence interval for the population mean volume is: D. 5.62.

To calculate the 95% confidence interval for the population mean volume, we can use the formula:

CI = x ± (t * (s/√n))

Where CI represents the confidence interval, x is the sample mean, t is the t-score associated with the desired confidence level (95%), s is the sample standard deviation, and n is the sample size.

In this case, x = 64 cc, s = 12 cc, and n = 20 orangutans. We need to find the t-score for a 95% confidence interval with 19 degrees of freedom (n-1). Using a t-table, we find that the t-score is approximately 2.093.

Now we can calculate the margin of error:

Margin of Error = t * (s/√n) = 2.093 * (12/√20) ≈ 5.62

Therefore, the margin of error associated with the 95% confidence interval for the population mean volume is: D. 5.62.

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The partial sum for the power series of arctan[tex](x^2)[/tex] consisting of the first four non-zero terms can be written as: arctan[tex](x^2) ≈ x^2 - (1/3)x^6 + (1/5)x^10 - (1/7)x^14[/tex]

The power series representation of the function f(x) = arctan[tex](x^2)[/tex] is given by:

[tex]f(x) = ∑n=0^∞ (-1)^n x^(2n+1) / (2n+1)[/tex]

To find the partial sum consisting of the first four non-zero terms, we can simply substitute n = 0, 1, 2, and 3 into the above expression, and add up the resulting terms:

[tex]f(x) ≈ x - x^3/3 + x^5/5 - x^7/7[/tex]

This is the desired partial sum for the power series representation of arctan[tex](x^2)[/tex], consisting of the first four non-zero terms. Note that as we add more terms to this sum, we get a better and better approximation to the function f(x) over a wider range of x values.

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The equivalent postfix (reverse Polish notation) expression for the infix expression "A-B+C*(DE-F)/(G+HK)" is option B: AB-CDEF-+GHK*+/.

The equivalent postfix (reverse Polish notation) expression for the given infix expression "A-B+C*(DE-F)/(G+HK)" is

AB-CDEF-+GHK*+/

Therefore, the correct answer is B), expression can be obtained by following the order of operations for postfix notation start from the left and push operands onto a stack until an operator is encountered. When an operator is encountered, pop the top two operands, perform the operation, and push the result back onto the stack.

Repeat until the end of the expression is reached, and the final result is the top value on the stack.

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--The given question is incomplete, the complete question is given

"  Consider the postfix expression: A-B+C*(D*E-F)/(G+H*K). The equivalent postfix (reverse Polish notation) expression is:

A.

AB-C+DE*F-GH+K**/

B.

AB-CDE*F-*+GHK*+/

C.

ABC+-E*F-*+GHK*+/

D.

None of these"--

To find the number that maximizes the product, we need to use calculus. Let's start by setting up the problem. Let x be the first number and y be the second number. We know that: x + y^3 = 864. We want to find the maximum value of xy.

Now, substitute this expression for x into our product function:
P(y) = (864 - y^3) * y

To find the maximum, we need to take the derivative of P(y) with respect to y and set it equal to 0:
dP/dy = -3y^4 + 864y - y^3(dy/dy)

dP/dy = -3y^4 + 864y - y^3(1)

Now, set dP/dy = 0 and solve for y:
0 = -3y^4 + 864y - y^3

It's a challenging polynomial to solve, but we can use numerical methods or technology to find the critical points of y.

Once we have the value(s) of y, we can plug it back into the equation x = 864 - y^3 to find the corresponding value(s) of x.

Finally, determine whether the critical points result in a maximum by checking the second derivative or analyzing the function's behavior around the critical points. The pair of numbers (x, y) will yield the maximum product.

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Answer:

x = -9

Step-by-step explanation:

3x - 9 = 5x + 9

3x - 5x = 9 + 9

-2x = 18

x = 18 / - 2x = -9

Answer:

x=-9

Step-by-step explanation:

(3 x -9) - 9 = -36

(5 x -9) + 9 = -36

Answer:

x < -4

Step-by-step explanation:

-17>6x + 7

Switch it to: 6x + 7 <-17

6x + 7 < -17

Now, Subtract the opposite of 7 by both sides.

6x = -24

Divide both sides by 6

Answer : x < -4

The curve f(x) is f(x) = -2x + 3.

Given that dy/dx = -2, we can integrate both sides with respect to x to obtain:

dy/dx = -2

dy = -2 dx

Integrating both sides, we get:

y = -2x + C

where C is the constant of integration. To find the value of C, we use the initial condition f(0) = 3:

y = -2x + C

f(0) = 3

-2(0) + C = 3

C = 3

Thus, the equation of the curve is:

y = -2x + 3

Therefore f(x) is -2x + 3

We can sketch the curve by plotting a few points. For example, when x = 0, y = 3 (as required by the initial condition). When x = 1, y = 1. When x = -1, y = 5. We can also note that the slope of the curve is always -2, meaning that the curve is a straight line with a negative slope.

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We can always find a monochromatic rectangle by reducing the problem to finding a monochromatic rectangle in a smaller area by moving the bottom of the rectangle up by one unit method.

What is rectangle?

A rectangle is a quadrilateral (a 2-dimensional shape with four sides) with four right angles. This means that opposite sides of a rectangle are parallel and congruent, and all four angles are equal to 90 degrees.

Let us consider a rectangle with sides parallel to the coordinate axes. Such a rectangle can be uniquely determined by two pairs of points that define its opposite corners. If all four of these points are the same color, then we have found a monochromatic rectangle. Otherwise, there are three cases to consider:

The two points at the top of the rectangle are the same color, and the two points at the bottom are the other color. In this case, we can reduce the problem to finding a monochromatic rectangle in a smaller area by moving the bottom of the rectangle up by one unit.

The two points on the left side of the rectangle are the same color, and the two points on the right side are the other color. In this case, we can reduce the problem to finding a monochromatic rectangle in a smaller area by moving the left side of the rectangle to the right by one unit.

Both pairs of points are of mixed color. In this case, we can reduce the problem to finding a monochromatic rectangle in two smaller areas by dividing the rectangle into four equal parts with a vertical or horizontal line.

By repeating this process on the smaller rectangles obtained in each case, we can eventually find a monochromatic rectangle. Since the rectangles we consider at each step have half the area of the previous ones, this process terminates after at most log_2(A) steps, where A is the area of the original rectangle.

Therefore, we can always find a monochromatic rectangle with this method.

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Complete question:

Each point in the x-y plane colored red or blue. show that there is a rectangle whose corners are all the same color.

Based on the information given, we can use a hypothesis test to determine if the proportion of households in the country that own a pet is actually less than 73%.

Let p be the true proportion of households that own a pet in the country.

The null hypothesis (H0) is that p = 0.73, meaning that the proportion of households that own a pet is 73%.

The alternative hypothesis (Ha) is that p < 0.73, meaning that the proportion of households that own a pet is less than 73%.

We can use a z-test for proportions to test this hypothesis.

The test statistic is:

z = (P - p) / sqrt(p(1-p)/n)

where P is the sample proportion, p is the hypothesized proportion (0.73), and n is the sample size (400).

Plugging in the values given, we get:

z = (0.7 - 0.73) / sqrt(0.73 * 0.27 / 400) = -1.96

Using a standard normal distribution table, we find that the p-value (the probability of getting a test statistic as extreme as -1.96 or more extreme, assuming the null hypothesis is true) is 0.025.

Since this p-value is less than the significance level of 0.05, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that less than 73% of households in the country own a pet.

In other words, the sample provides evidence that the proportion of households that own a pet is less than 73%. However, we cannot say with certainty that the true proportion is any particular value, only that it is less than 73%.

Based on your question, the Humane Society claims that less than 73% of households in a certain country own a pet. In a random sample of 400 households, 280 households own a pet. To determine if this claim is accurate, we can calculate the proportion of pet owners in the sample:

Proportion of pet owners = (Number of pet owners) / (Total households) = 280 / 400 = 0.7 or 70%

Since the sample proportion (70%) is less than the claimed proportion (73%), it appears that the Humane Society's claim is supported by the sample data. However, for a more rigorous test, you may want to consider conducting a hypothesis test or calculating a confidence interval to determine the statistical significance of this difference.

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If "square-field" area is 40000 ft square, and Mr. Blackburn is planning to put a fence diagonally in field, then the length of fence should be 282.84 ft.

The area of the Blackburn's square-field is = 40000 ft²,

First, we equate this with area formula,

On Equating ,

We get,

⇒ (side)² = 40000,

⇒ side = 200,

substituting the "side-length" of the field as 200 ft, in the diagonal of a square formula,

we get,

⇒ Length of field's diagonal is = (side)√2,

⇒ Length of field's diagonal is = (200)√2,

⇒ Length of field's diagonal is ≈ 282.84 ft.

Therefore, the length of field's fence is 282.84 ft.

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The locus of points in a plane that are 3 inches from point A is a circle centered at point A with radius 3 inches.

The locus of points in a plane that are 3 inches from point A forms a circle centered at point A with a radius of 3 inches. This is because a circle is defined as the set of all points in a plane that are equidistant from a fixed point called the center. In this case, the fixed point or the center is point A, and the distance from any point on the circle to point A is 3 inches.

To see this, consider any point P in the plane that is 3 inches away from point A. The distance from point P to point A is given by the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

where (x1, y1) and (x2, y2) are the coordinates of points A and P, respectively. Since P is 3 inches away from A, we have d = 3. Substituting the coordinates of A and P into the distance formula and simplifying yields:

(x2 - x1)² + (y2 - y1)² = 3²

This equation represents a circle with center (x1, y1) = A and radius 3.

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None of the statements about quantitative versus qualitative research are true.

This is because the comparison between quantitative and qualitative research is not a matter of one being objectively better or more subjective, richer, or more time-consuming than the other. They are different approaches to research, each with its strengths and limitations, and the choice between them depends on the research question, the nature of the data, and the research design.

Quantitative research is characterized by the collection of numerical data that can be analyzed using statistical methods to identify patterns, relationships, and trends. This approach is used when the research question requires an objective measurement of variables that can be quantified and compared across groups. For example, a quantitative study may examine the relationship between height and weight in a population, using statistical techniques to identify the strength of the relationship and its significance.

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The probability that his/her forecast is

(a) at least 3.3 percent 0.93

(b) at most 7 percent 0.0274.

(c) between 3.3 percent and 7 percent is 0.7337.

To calculate the probability of a single analyst's forecast being at least 3.3 percent, we need to find the z-score associated with that value. The z-score is a measure of how many standard deviations a data point is from the mean of a distribution. Using the formula:

z = (x - μ) / σ

where x is the value we want to find the z-score for, μ is the mean of the distribution, and σ is the standard deviation of the distribution, we can calculate the z-score for 3.3 percent:

z = (3.3 - 6) / 1.4 = -1.93

To find the probability of a single analyst's forecast being at least 3.3 percent, we need to find the area under the normal curve to the left of the z-score of -1.93. This can be done using a standard normal distribution table or a calculator. The probability is approximately 0.0274.

To calculate the probability of a single analyst's forecast being at most 7 percent, we need to find the z-score associated with that value. Using the same formula as before, we can calculate the z-score for 7 percent:

z = (7 - 6) / 1.4 = 0.71

To find the probability of a single analyst's forecast being at most 7 percent, we need to find the area under the normal curve to the left of the z-score of 0.71. This can also be done using a standard normal distribution table or a calculator. The probability is approximately 0.7611.

To calculate the probability of a single analyst's forecast being between 3.3 percent and 7 percent, we need to find the area under the normal curve between the z-scores of -1.93 and 0.71. This can also be done using a standard normal distribution table or a calculator. The probability is approximately 0.7337.

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The populations of the two towns will be equal in the year 2048.

Let's assume that the population of Appleville in the year "t" is "A(t)", and the population of Treehill in the year "t" is "T(t)". We know that the population of Treehill is increasing linearly by 100 per year. Therefore, we can express the population of Treehill in terms of "t" as follows:

T(t) = 41820 + 100t

The population of Appleville is also growing linearly, but we do not know the exact rate of growth. However, we can calculate this rate using the population data we have. The change in population from 2010 to 2020 is 35000 - 28000 = 7000. This change occurred over a period of 10 years. Therefore, the average annual growth rate is:

(35000 - 28000) / 10 = 700

We can use this rate to express the population of Appleville in terms of "t" as follows:

A(t) = 28000 + 700t

Now, we can set these two equations equal to each other to find the year when the populations of the two towns will be equal:

41820 + 100t = 28000 + 700t

Simplifying this equation, we get:

420t = 13820

t = 33

Therefore, the populations of the two towns will be equal in the year 2015 + 33 = 2048.

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Answer: c. 6

Step-by-step explanation:  

Option-C is correct that is 6 number from the set {4 , 5, 6, 7} makes the equation 4(8 - x) = 8 true.

Given that,

We have to find which value of the x from the set {4 , 5, 6, 7} makes the equation 4(8 - x) = 8 true.

We know that,

A group of well defined objects is referred to as a set. Only on the basis of simplicity are the objects of a set considered to be distinct.

A family or collection of sets are common names for a group of sets.

So,

The set has 4 numbers by substituting the numbers we get to know if the equation is true or false.

Take the equation,

4(8 - x) = 8

If x = 4,

4(8 - 4) = 8

4 × 4 = 8

16 ≠ 8

Now, If x = 5,

4(8 - 5) = 8

4 × 3 = 8

12 ≠ 8

Now, If x = 6,

4(8 - 6) = 8

4 × 2 = 8

8 = 8

And, If x = 7,

4(8 - 7) = 8

4 × 1 = 8

4 ≠ 8

Therefore, Option-C is correct that is number 6.

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The probability of rolling a higher number on the second roll is 15/36 or approximately 0.42.

When rolling a die, each roll has an equal chance of landing on any of the six possible outcomes. Therefore, the probability of rolling any specific number on the first roll is 1/6, and the probability of rolling a number higher than the first on the second roll is also 1/6.

To find the probability of both events happening, we can use the multiplication rule of probability, which states that the probability of two independent events occurring together is equal to the product of their individual probabilities.

So, the probability of rolling a number on the first roll and then rolling a higher number on the second roll is:

P = (1/6) x (1/6)

However, since the question doesn't specify a particular number for the first roll, we need to consider all the possible outcomes for the first roll. These outcomes are mutually exclusive and have an equal chance of happening, so we can add their probabilities to find the overall probability of rolling a higher number on the second roll:

P = (1/6) x (5/6) + (1/6) x (4/6) + (1/6) x (3/6) + (1/6) x (2/6) + (1/6) x (1/6) + (1/6) x 0

Simplifying this expression, we get:

P = 15/36

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the probability of the sample containing at least one green ball when selecting 7 balls randomly is approximately 0.9779 or 97.79%.

To find the probability that the sample contains at least one green ball, it's easier to calculate the probability of the complementary event (not getting any green balls) and then subtract that from 1. The complementary event would be selecting all 7 balls as white balls.

1. Total balls in the urn = 13 white balls + 7 green balls = 20 balls
2. Calculate the probability of selecting 7 white balls:
  - Number of ways to choose 7 white balls out of 13 = C(13,7) = 1716
  - Number of ways to choose 7 balls out of 20 = C(20,7) = 77520
  - Probability of selecting 7 white balls = 1716 / 77520 = 0.0221267
3. Calculate the probability of getting at least one green ball:
  - Probability of at least one green ball = 1 - Probability of 7 white balls
  - Probability of at least one green ball = 1 - 0.0221267 = 0.9778733

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If z varies with y and inversely with and z = 6 when x = 4 and y = 3, then the value of Proportionality Constant is given by 8.

Proportion is a relation between two mathematical variables. If two variables vary directly that states if one increases another will also decrease and same for decrease.

If two variables are in inverse relation that states that if one variable increases then another decreases and if one variable decreases then another increases.  

Given that, z varies with y and inversely with x. So,

z = k*(y/x), where k is the proportionality constant.

Given that, z = 6 when x = 4 and y = 3. So,

6 = k*(3/4)

k = (6*4)/3

k = 2*4

k = 8

Hence the value of Proportionality Constant is 8.

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The question is incomplete. The complete question will be -

"z varies with y and inversely with x when z=6, x=4, and y=3. Find the value of Proportionality Constant."

The area of the given shape above would be = 228

To calculate the area of the shape above, the shape is divided into three separate shapes and then added up together. This would be calculated using area of rectangle = length×width.

First shape; length = 15; width = 9

Area = 15×9 = 135

Second shape; Length = 17; width = 3

Area = 17×3 = 51

Third shape ; length = 14 width = 3

Area = 14×3 = 42

Therefore, area of the shape = 135+51+42 = 228

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No, according to the triangle inequality theorem he will not be able to create the triangular component. That is option C

The triangular inequality theorem states that the sum of any two sides of a triangle is greater than the length of the third side.

That is;

a+b > c

a+ c > b

b+c > a

The lengths given include the following;

a = 4 in

b = 5 in

c = 1 in

Using the theorem;

a+b = 4+5 = 9 > 1

a+c = 4+1 = 5 = 5

b+c = 5+1 = 6> 4

Therefore he won't be able to create a triangular component using the three lengths of the tooth pick because is goes against the triangle inequality theorem rule.

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Answer:

Step-by-step explanation:

the answer is A

i took the text

The answer  is that the probability that the maximum speed differs from the mean value by at most 2.5 standard deviations can be calculated using the normal distribution.

We first need to calculate the mean value and standard deviation of the speed data. Then, we can use the formula for the normal distribution to find the probability that the maximum speed falls within a certain range from the mean value.

Specifically, we can use the formula:

P(mean - 2.5SD < max speed < mean + 2.5SD) = P(z < 2.5) - P(z < -2.5)

where z is the z-score of the maximum speed, calculated as (max speed - mean)/SD.

We can find the probabilities P(z < 2.5) and P(z < -2.5) using a standard normal distribution table or calculator.

In general, if the maximum speed is normally distributed and the mean and standard deviation are known, we can use this formula to calculate the probability that the maximum speed falls within a certain range from the mean value.

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Answer:

-----------------

To find the distance you can use the formula:

Given:

First, we need to convert the time from minutes to hours:

Now, set up the equation using the given values and units:

Answer:

869

Step-by-step explanation:

There are 9,000,000 different possible license plates.

Since the license plate has one letter followed by five digits, we need to consider two cases separately:

Case 1: The letter can be any of the 24 letters of the alphabet, excluding 'i' and 'o'.

Case 2: Each digit can be any of the 10 digits from 0 to 9.

For the first position, there are 24 choices for the letter. For the second position, there are 10 choices for the digit. Similarly, there are 10 choices for the third, fourth, fifth, and sixth positions.

Therefore, the total number of possible license plates is:

24 x 10 x 10 x 10 x 10 x 10 = 24 x 10^5 = 9,600,000

However, we need to exclude the license plates that contain 'i' or 'o'. There are 2 choices for the first position, and 10 choices for the remaining positions.

Therefore, the total number of license plates that contain 'i' or 'o' is:

2 x 10 x 10 x 10 x 10 x 10 = 2 x 10^5 = 200,000

So the final answer is:

24 x 10^5 - 2 x 10^5 = 9,600,000 - 200,000 = 9,000,000

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Answer:

Step-by-step explanation:

Precision is defined as the ratio of true positives (TP) to the total predicted positives (TP + FP). In this case, the confusion matrix shows 46 TP and 4 FP, so the precision can be calculated as:

precision = TP / (TP + FP) = 46 / (46 + 4) = 0.92

Rounding to 2 decimal places, the precision ratio is 0.92.

The block will cover the distance 12m before it stops.

According to Newton's second law of motion, the acceleration a

of a body of mass m is determined by the net force [tex]F_n_e_t[/tex] acting on it:

[tex]F_n_e_t[/tex]  = ma.

The mass of a block, m = 8 kg

The initial speed of the block on rough horizontal surface , u = 6 m/s

The force of friction, F = 12N

We know that, the Newton's 2nd law: [tex]F_n_e_t[/tex]  = ma.

Therefore, its final velocity is zero, v = 0

Since the only force that is acting on the body along the horizontal direction is the kinetic frictional force

[tex]F_n_e_t[/tex]  = ma. ([tex]F_n_e_t[/tex]  is acting in opposite direction to that of object motion)

a = [tex]\frac{f_n_e_t}{m}[/tex] = [tex]\frac{-12}{8} = -1.5m/s^2[/tex]

Negative sign shows that it is in the opposite direction to that of initial velocity.

By using the third kinematic equation, we get:

[tex]v^2=u^2+2as\\\\= > s = \frac{0-6^2}{2(-1.5)}\\ \\s = \frac{36}{3} = 12m[/tex]

Therefore, the block will cover the distance 12m before it stops.

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