Let F=6xyi+6y2j be a vector field in the plane, and C the path y=3 joining (0, 0) to (1, 3)in the plane.A. Evaluate ∫CF⋅dr.B. Does the intergrant in part (A) depend on the path joining (0, 0)to (1, 3)?Line Integral:The following formula used for calculating the line integral with the given path C:∫CF⋅dr=∫CMdx+NdyWhere,F=⟨M,N⟩The line integral is dependent on the given path C joining from (a, b) to (c, d), if the following condition is satisfied:Nx≠My

The rearranged form of the given integral is ∫[0,4]∫[0,20]f(x,y)dydx + ∫[0,20]∫[-y,4]f(x,y)dydx where a = 0, b = 4, g₁(x) = 0, g₂(x) = 20, c = 0, d = 20, g₃(x) = 4, g₄(x) = -y

What is integral?

An integral is a mathematical concept that represents the area under a curve or the accumulation of a quantity. It is a fundamental operation in calculus and is denoted by the symbol "∫" (integral symbol).

The given integral is ∫[20]∫[4-y]√[0]f(x,y)dxdy, where the limits of integration for x are from 0 to 20, and for y, they are from 0 to 4-y.

To change the order of integration, we need to determine the new limits of integration. In the given expression, x varies from 0 to 20, and y varies from 0 to 4-y.

Therefore, the new limits of integration become:

a = 0, b = 4: These are the new limits for y integration.

g₁(x) = 0, g₂(x) = 20: These are the new limits for x integration within the range of y.

c = 0, d = 20: These are the limits for y integration in the second integral.

g₃(x) = 4: This is the lower limit of y within the range of x.

g₄(x) = -y: This is the upper limit of y within the range of x.

By rearranging the order of integration, we obtain the sum of two integrals: ∫[a,b]∫[g₂(x),g₁(x)]f(x,y)dydx + ∫[c,d]∫[g₄(x),g₃(x)]f(x,y)dydx. Plugging in the new limits of integration, we arrive at the final form.

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The period of the given function can be given by 12.

Finding the horizontal distance between two consecutive troughs or crests is necessary to calculate the period of a function given its trough and crest.

The time (T) is equal to twice the wavelength, and this distance is known as the wavelength (λ).

Given that the trough is at -3 and the crest is at 3, we can calculate the wavelength as follows:

Wavelength (λ) = Crest - Trough

= 3 - (-3)

= 6

The period (T) is then twice the wavelength:

Period (T) = 2 x Wavelength

= 2 x 6

= 12

Therefore, the period of the function is 12.

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The first four terms of the recursive sequence in this problem are given as follows:

12, 13, 15, 19.

The recursive sequence in the context of this problem is defined as follows:

[tex]a_n = 2a_{n - 1} - 11[/tex]

The first term is given as follows:

[tex]a_1 = 12[/tex]

The second term is then given as follows:

2(12) - 11 = 13.

The third term is then given as follows:

2(13) - 11 = 15.

The fourth term is then given as follows:

2(15) - 11 = 19.

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The Factored form of 3x^2 - 11x + 6 is (x - 3)(3x - 2).

The quadratic expression 3x^2 - 11x + 6, we can use the method of factoring by grouping. Here's the step-by-step process:

Step 1: Multiply the coefficient of x^2 (3) by the constant term (6) in the expression.

  3 * 6 = 18.

Step 2: Find two numbers that multiply to give 18 and add up to the coefficient of x (-11).

  The numbers -2 and -9 fit this criteria because -2 * -9 = 18 and -2 + (-9) = -11.

Step 3: Split the middle term (-11x) into two terms using the numbers found in step 2.

  3x^2 - 2x - 9x + 6.

Step 4: Group the terms and factor out the greatest common factor (GCF) from each group.

  (3x^2 - 2x) - (9x - 6).

  x(3x - 2) - 3(3x - 2).

Step 5: Notice that the terms (3x - 2) are common in both groups. Factor it out.

  (x - 3)(3x - 2).

So, the factored form of 3x^2 - 11x + 6 is (x - 3)(3x - 2).

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The value 7.347 in the form of p/ q where p and q are integers and q≠ 0 is 7347/1000.

Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers. The inverse additives of the equivalent positive numbers are the negative numbers.

We will need to remove the decimal point by multiplying by 1000 as

7.347 × 1000

= 7347

Then to express in term of  p/ q, we we write it as 7347/1000 which implies that 7347/1000 =7.347

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correct question;

Express 7.347 in the form of p /q where p and q are integers and q≠ 0.

The number of Blake's scores that are within one standard deviation from the mean are; Five scores

The standard deviation is obtained from the square root of the squared average of the squared differences between each data point in the dataset and the mean.

The mean of the scores is; (100 + 85 + 55 + 95 + 75 + 100)/6 = 85

The square of the difference are;

(100 - 85)² = 225, (85 - 85)² = 0, (55 - 85)² = 900, (95 - 85)² = 100, (75 - 85)² = 100, and (100 - 85)² = 225

The sum of the squares of the difference from the mean is therefore;

225 + 0 + 900 + 100 + 100 + 225 = 1550

The variance = The average of the squared deviation is therefore;

Variance = 1550/6 = 258.[tex]\overline{3}[/tex]

The standard deviation is therefore; σ ≈ √(258.[tex]\overline{3}[/tex]) ≈ 16.07

The scores that are one standard deviation from the mean are;

85 - 16.07 ≤ Score ≤ 85 + 16.07

68.93 ≤ Score ≤ 101.07

Therefore, the scores that are within one standard deviation from the mean are; 75,  85, 95, 100, 100, therefore, five scores are within one standard deviation from the mean.

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We know that :

[tex] \sf2x + 15 + x + 10 + 3x + 5 = 180[/tex]

[tex] \sf6x + 30 = 180[/tex]

[tex] \sf6x = 180 - 30[/tex]

[tex] \sf6x = 150[/tex]

[tex] \sf \: x = 25[/tex]

Answer:

Question 1:   <BAC = 35 degrees

Question 2:   <ABC =  80 degrees

Question 3:  <ACB = 65 degrees

Step-by-step explanation:

Angle Sum Theorem:

<ABC + <ACB + <BAC = 180 degrees

(3x + 5) ^degrees + (2x + 15)^degrees +  (x + 10)^degrees = 180 degrees

<ABC = (3x + 5)^degrees

<ACB = (2x + 15)^degrees into <ABC + <ACB + <BAC = 180 degrees

x = 25

<BAC = (25 + 10)^degrees

<BAC = 35 degrees

Answer = <BAC = 35 degrees

<ABC + <ACB + <BAC = 180 degrees

(3x + 5) ^degrees + (2x + 15)^degrees +  (x + 10)^degrees = 180 degrees

<ABC = (3x + 5)^degrees

<ACB = (2x + 15)^degrees into <ABC + <ACB + <BAC = 180 degrees

x = 25

<ABC = (3 * 25 + 5)^degrees

<ABC = 80 degrees

Answer = <ABC =  80 degrees

Question 3:  Angle Sum Theorem

<ABC + <ACB + <BAC = 180 degrees

(3x + 5) ^degrees + (2x + 15)^degrees +  (x + 10)^degrees = 180 degrees

<ABC = (3x + 5)^degrees

<ACB = (2x + 15)^degrees into <ABC + <ACB + <BAC = 180 degrees

x = 25

<ACB = (2 * 25 + 15) ^degrees

<ACB = 65 degrees

Answer = <ACB = 65 degrees

Hope this helps!

Among the given options, the most reasonable conclusion that can be made when a facility employs mostly healthcare workers certified by the Association of Safe Patient Handling Professionals (ASPHP) is:

a. The facility will save money from reduced employee injuries.

Certification by ASPHP indicates that the healthcare workers have received specialized training in safe patient handling techniques. This training aims to prevent injuries to both the healthcare workers and the patients. By employing certified workers, the facility can expect a reduction in employee injuries, leading to potential cost savings associated with workers' compensation claims, medical expenses, and lost productivity.

It is important to note that while this conclusion is likely, it does not guarantee that all injuries will be eliminated or that the facility will be completely injury-free. However, employing certified healthcare workers is a positive step towards promoting a safer working environment.

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Answer:The correct answer is d. Neither the normal nor the t distribution applies.

Answer: 20pi,

Step-by-step explanation: 20pi, 62.8

and the diameter is 4, divide it by two, and you will get a radius of 2.

D/2=R

The next step is using the surface area formula of 4 times pi times radius squared. (4*pi*R^2) Treat Pi as a variable and do r squared which is 4, and then do 4*4 and you will get 16, then you multiply with pi and get 16pi

16pi divided by two will be a half circle. so we then get 8pi, but we're not done yet, we still have to find the bottom surface area, which is pi*R^2.

You square the radius (2) -> (4)

and will get 4pi. Add the total, 20pi.

And since it's asking for surface area without pi, simply do 20*3.14 and you will get 62.8.

Answer:

A. y = (x + 3)^ 2 - 1

Step-by-step explanation:

To represent the transformation of a horizontal shift left 3 and a vertical shift down 1 applied to the function y = x^2, you can use the following equation:

y = (x + 3)^2 - 1

In this equation, the term (x + 3) represents the horizontal shift left 3, and the term -1 represents the vertical shift down 1. The squared term remains the same as in the original function, y = x^2.

Answer:

(x – 4)^2 + (y +10)^2 = 64

Step-by-step explanation:

Recall that the formula for a circle is:  (x – h)^2 + (y – k)^2 = r^2

1. Find the Radius (r): Luckily, we can count the distance since the coordinates have the same y-value. 12 - 4 = 8. So, r^2 = 8^2 = 64

2.  Find h and k. These are the x and y coordinates of the center of the circle. So, h = 4, k = -10

3. Substitute the values in the equation:

(x – 4)^2 + (y – (-10))^2 = 8^2

(x – 4)^2 + (y +10)^2 = 64

Answer:

  (x -4)² +(y +10)² = 64

Step-by-step explanation:

Given a circle through point (12, -10) with center (4, -10), you want its equation.

The equation of a circle with center (h, k) and radius r is ...

  (x -h)² +(y -k)² = r²

You are given the center (h, k) = (4, -10). You only need to know the radius to finish the equation. That will be the value of r that makes the equation true at the given point:

  (x -4)² + (y +10)² = r²

  (12 -4) + (-10 +10)² = r² . . . . . . with (x, y) = (12, -10), the point on the circle

  8² + 0 = r²

The equation of the circle is (x -4)² +(y +10)² = 64.

__

Additional comment

The equation of a circle is essentially a statement of the distance formula. It is telling you that the circle consists of all points that are distance r from the center.

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In a graph represented by an adjacency matrix, you can find all the neighbors of a given vertex in O(V) operations, where V is the number of vertices in the graph.

To find the neighbors of a vertex using an adjacency matrix, you need to examine the corresponding row or column in the matrix. Each entry in the row or column represents the presence or absence of an edge between the given vertex and the other vertices in the graph.

By scanning the row or column of the given vertex in the adjacency matrix, you can identify all the vertices that share an edge with the given vertex, which are its neighbors. This process requires examining each entry in the row or column, which takes O(V) operations since there are V vertices in the graph.

The time complexity to find all the neighbors of a given vertex in a graph represented by an adjacency matrix is O(V).

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The percentage of the data values that are greater than or equal to 40 based on the five number summary of the data in the box-and-whisker plot is 50 percent

The box and whiskers plot indicates that the five number summary are;

The minimum value = 10

The first quartile, (The 25th percentile) Q₁ = 20

The median, (The 50th percentile), Q₂ = 40

The Third quartile, (The 75th percentile), Q₃ = 80
The maximum value = 90

Based on the five number summary, the median, which is the second quartile, (the 50th percentile), of the dataset is 40, therefore, the percentage of the values that are greater than 40 are 50%, which is half of the data in the dataset

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True. The statistics of each vehicle's random process will be represented by filter coefficients and a noise variance.

In this project, the statistics of each vehicle's random process will be represented by filter coefficients and a noise variance. This means that the filter coefficients will be used to model the vehicle's behavior over time, while the noise variance will represent the variability of the data. These values will be used to generate predictions about the vehicle's future behavior.

The project in question is likely related to modeling and predicting the behavior of a system that involves multiple vehicles. To do this, it is necessary to first collect data about each vehicle's behavior over time. This data can then be used to develop a statistical model that can be used to predict future behavior. One common approach to modeling vehicle behavior is to use a random process model. This involves modeling the vehicle's behavior as a stochastic process, which means that it is subject to random fluctuations over time. The goal is to estimate the statistical properties of this process, such as its mean and variance, in order to make predictions about future behavior.

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Answer:

Part A: 4 in.²

Part B: 35 in.²

Step-by-step explanation:

Part A:

The base of the triangle is 2 inches.

The height of the triangle is 4 inches.

A = bh/2

A = (2 in.)(4 in.)/2

A = 4 in.²

Answer: 4 in.²

Part B:

The non-shaded region of the figure can be broken down into several simple shapes:

Top right - a square 2 inches by 2 inches.

Below the square - a triangle congruent to the shaded triangle.

Bottom left - a rectangle 6 inches wide and 4 inches tall

Above the large rectangle - a smaller rectangle 3 inches by 1 inch.

The total non-shaded area is the sum of the areas of the simple shapes above.

All linear dimensions are inches, and the area is in square inches.

A = 2 × 2 + 4 + 6 × 4 + 3 × 1

A = 4 + 4 + 24 + 3

A = 35

Answer: 35 in.²

The answers to the question related to function and graphs of questions 45 and 46 are as follows:

45. Based on the given information, the function has zeros at x = 2 and x = 3, meaning that the graph intersects the x-axis at these points. It has a vertical asymptote at x = 5, indicating that the function approaches infinity or negative infinity as x approaches 5.

Therefore, the function has a horizontal asymptote at y = -3, meaning that as x approaches positive or negative infinity, the function approaches -3.

46. For the function y = g(x), the graph has two vertical asymptotes: one at x = -2 and one at x = 3. This means that as x approaches -2 or 3, the function approaches infinity or negative infinity. The function also has a horizontal asymptote at y = 0, which implies that as x approaches positive or negative infinity, the function approaches 0.

Therefore, the graph of g(x) crosses the x-axis once, specifically at x = 5, indicating that there is a single point where the function equals zero.

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(150^3) Divided by 600 equals 5,625.

The (150^3) divided by 600, the value of 150 raised to the power of 3.

150^3 means multiplying 150 by itself three times.

150^3 = 150 * 150 * 150

Calculating the multiplication:

150 * 150 = 22,500

22,500 * 150 = 3,375,000

So, 150 raised to the power of 3 is equal to 3,375,000.

Now, we can divide 3,375,000 by 600 to find the final result.

3,375,000 / 600 = 5,625

Therefore, (150^3) divided by 600 equals 5,625.

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The function f(x) = [tex]x^3[/tex] - 5[tex]x^{2}[/tex]- 2x + 24 has three real roots: approximately -3, 2, and 4.

To find the roots of the function f(x) = [tex]x^3[/tex] - 5[tex]x^{2}[/tex] - 2x + 24, we can use various methods such as factoring, the rational root theorem, or numerical methods like Newton's method. By trying different values for x, we can determine which values make the equation equal to zero.

By evaluating the function for different values of x, we find that f(-2) = 0, f(-3) = 0, f(2) = 0, and f(3) > 0, f(4) = 0. Therefore, the roots of the function are x = -2, x = -3, x = 2, and x = 4.

The first paragraph provides a concise summary of the answer, stating that the function f(x) = [tex]x^3[/tex] - 5[tex]x^{2}[/tex] - 2x + 24 has three real roots: approximately -3, 2, and 4.

The second paragraph explains the process of finding the roots of the function. It mentions different methods that can be used and then concludes by evaluating the function for different values of x to determine the roots. The paragraph also lists the specific values of x for which the function equals zero, confirming the roots as -2, -3, 2, and 4.

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The existence of two relative maximum points on the smooth curve signifies changes in the curve's slope and indicates a non-monotonic behavior in the corresponding interval.

Consider a smooth curve with no undefined points. If it has two relative maximum points, it implies that there are two distinct points on the curve where the slope changes from positive to negative.

A relative maximum point occurs when the curve reaches a local maximum value in a specific interval. At these points, the slope of the curve changes from positive to negative, indicating that the curve is increasing before the point and decreasing after the point.

The presence of two relative maximum points suggests that the curve undergoes an increase in slope, reaches a maximum value, then decreases in slope, reaches a lower value, and then increases in slope again, reaching a second maximum value.

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Based on the concept of real limits, the range of scores in the set is 8, which means that the highest score in the set is 16 and the lowest score is 8.

The real limits of the set would be 7.5 and 16.5, since these values represent the boundaries of each score interval. Therefore, any score between 7.5 and 8.5 would be rounded down to 8, and any score between 16.5 and 15.5 would be rounded up to 16. The range of 8 is the difference between the upper and lower real limits of the set.

Using the concept of real limits, the range of a set of scores is calculated as the difference between the highest and lowest scores. In this case, the high score is x = 16 and the low score is x = 8. The range can be found by subtracting the low score from the high score:

Range = High score - Low score
Range = 16 - 8
Range = 8

So, with the real limits concept, the range for this set of scores is indeed 8.

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When given the sets a = {1, 2, 3, 4, 5} and b = {0, 3, 6}, the operations yield the following results: a) a∪b = {0, 1, 2, 3, 4, 5, 6}, b) a∩b = {3}, c) a−b = {1, 2, 4, 5}, and d) b−a = {0, 6}.

To explain further, the union of sets a and b (a∪b) includes all the elements from both sets without repetition. In this case, a∪b = {0, 1, 2, 3, 4, 5, 6} because it contains all the elements from sets a and b.

The intersection of sets a and b (a∩b) refers to the common elements present in both sets. In this case, a∩b = {3} since 3 is the only element that appears in both sets a and b.

The set difference a−b consists of elements that are in set a but not in set b. Here, a−b = {1, 2, 4, 5} because these elements are present in set a but not in set b.

Finally, the set difference b−a represents elements that are in set b but not in set a. In this case, b−a = {0, 6} because these elements are present in set b but not in set a.

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To determine the optimum production cycle, we need to calculate the production run sizes (Q) for each item. Here's how you can fill in the table:

1. Unit Demand (Di): This is the annual demand divided by the number of working days: RF1: 2,500 / 250 = 10

RF2: 10,000 / 250 = 40

RF3: 5,000 / 250 = 20

RF4: 4,000 / 250 = 16

RF5: 15,000 / 250 = 60

2. Production Run Size (Qi): The production run size is calculated using the economic production quantity formula: Qi = sqrt((2 * Di * Ci) / Hi)

RF1: sqrt((2 * 10 * $200) / $1.50) ≈ 73

RF2: sqrt((2 * 40 * $250) / $0.60) ≈ 223

RF3: sqrt((2 * 20 * $250) / $1.00) ≈ 89

RF4: sqrt((2 * 16 * $100) / $2.50) ≈ 25

RF5: sqrt((2 * 60 * $400) / $1.00) ≈ 110

3. Production Cycle Length: The production cycle length is the sum of all production run sizes:

73 + 223 + 89 + 25 + 110 = 520 units

4. Slack Time per Run: The slack time per run is the total number of working days divided by the number of production runs:

250 / 5 = 50 days

To determine the optimum production cycle, we calculate the production run sizes for each item using the economic production quantity formula. This formula considers the annual demand, holding cost, and production run cost to find the optimal production quantity. Once we have the production run sizes for each item, we sum them up to determine the length of the complete production cycle. In this case, it amounts to 520 units. The slack time per run is then calculated by dividing the total number of working days by the number of production runs, which is 50 days in this scenario. This information helps in planning the production schedule efficiently.

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Rounded to four decimal places, the estimated area using left endpoints is approximately 8.2925.

(a) To estimate the area under the graph of f(x) = 2√x from x = 0 to x = 4 using four approximating rectangles and right endpoints, we can use the right Riemann sum.

The width of each rectangle is Δx = (4 - 0) / 4 = 1, since we are dividing the interval [0, 4] into four equal parts.

The right endpoints for the four rectangles are x = 1, 2, 3, and 4.

To find the height of each rectangle, we evaluate f(x) = 2√x at the right endpoints:

f(1) = 2√1 = 2

f(2) = 2√2 ≈ 2.8284

f(3) = 2√3 ≈ 3.4641

f(4) = 2√4 = 4

The area of each rectangle is the product of the width and height.

Area of first rectangle = 1 * 2 = 2

Area of second rectangle = 1 * 2.8284 ≈ 2.8284

Area of third rectangle = 1 * 3.4641 ≈ 3.4641

Area of fourth rectangle = 1 * 4 = 4

The estimate of the area under the graph is the sum of the areas of the four rectangles:

R4 = 2 + 2.8284 + 3.4641 + 4 ≈ 12.2925

Rounded to four decimal places, the estimated area is approximately 12.2925.

(b) To estimate the area under the graph using left endpoints, we use the left Riemann sum.

The left endpoints for the four rectangles are x = 0, 1, 2, and 3.

Using the same calculations as in part (a), we find the heights of the rectangles:

f(0) = 2√0 = 0

f(1) = 2√1 = 2

f(2) = 2√2 ≈ 2.8284

f(3) = 2√3 ≈ 3.4641

The area of each rectangle is the product of the width and height:

Area of first rectangle = 1 * 0 = 0

Area of second rectangle = 1 * 2 = 2

Area of third rectangle = 1 * 2.8284 ≈ 2.8284

Area of fourth rectangle = 1 * 3.4641 ≈ 3.4641

The estimate of the area under the graph using left endpoints is the sum of the areas of the four rectangles:

L4 = 0 + 2 + 2.8284 + 3.4641 ≈ 8.2925

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Answer:

8 kiwis

Step-by-step explanation:

We Know

James bought 32 kiwi fruit for $16.

16 / 32 = $0.50 per kiwi

How many kiwis can Lisa buy if she has $4?

We Take

4 / 0.50 = 8 kiwis

So, Lisa can buy 8 kiwis if she has $4.

Answer: 360

Step-by-step explanation:

done

Answer:

Slope = -1

Step-by-step explanation:

[tex]m= \frac{y2-y1}{x2-x1} \\\\m= \frac{-1-(-7)}{-1-(5)} \\\\\\m= \frac{6}{-1-(5)} \\\\\\m= \frac{6}{-6} \\\\\\m= -1[/tex]

Answer:

a.

y = (x+6) (x-2)

=x^2-2x+6x-12

=x^2+4x-12

=(x+6) (x-2)

b. x = (-6) (or) x=2

Answer:

  C.  1.875 J

Step-by-step explanation:

You want the work done stretching a spring from 25 cm to 30 cm, given that the force required to stretch it to 25 cm from its natural length of 15 cm is 30 N.

In newtons per meter, the spring constant is ...

  k = (30 N)/(0.25 m -0.15 m) = 300 N/m

Work is the product of force and distance. When the force is a function of distance, the work can be found by integrating:

  dW = k(x -0.15)dx

  [tex]\displaystyle W=\int_{0.25}^{0.30}300(x-0.15)dx = \dfrac{300}{2}((0.30-0.15)^2-(0.25-0.15)^2)\\\\=150(0.0125)=1.875\quad\text{joules}[/tex]

The work required to stretch the spring to 30 cm is 1.875 J.

__

Additional comment

The units of work (energy) used here are newton-meters (joules), so the distance dimensions need to be in meters. 15 cm = 0.15 m, for example.

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(i) To find the man's salary for the tenth month, we need to calculate the salary after 9 increases.

The salary after each increase can be calculated using the formula:

Salary = Initial Salary + (Number of Increases * Increase Amount)

Given:

Initial Salary = Rs. 5200

Increase Amount = Rs. 320

Number of Increases = 9

Using the formula, we can calculate the salary for the tenth month:

Salary = Rs. 5200 + (9 * Rs. 320)

Salary = Rs. 5200 + Rs. 2880

Salary = Rs. 8080

Therefore, the man's salary for the tenth month is Rs. 8080.

(ii) To calculate his total earnings during the first year, we need to sum up his monthly salaries for 12 months.

The first month's salary is Rs. 5200, and each subsequent month's salary increases by Rs. 320.

To calculate the total earnings for the first year, we can use the formula for the sum of an arithmetic series:

Total Earnings = (Number of Months / 2) * (2 * First Salary + (Number of Months - 1) * Increase Amount)

Given:

Number of Months = 12

First Salary = Rs. 5200

Increase Amount = Rs. 320

Using the formula, we can calculate the total earnings for the first year:

Total Earnings = (12 / 2) * (2 * Rs. 5200 + (12 - 1) * Rs. 320)

Total Earnings = 6 * (2 * Rs. 5200 + 11 * Rs. 320)

Total Earnings = 6 * (Rs. 10400 + Rs. 3520)

Total Earnings = 6 * Rs. 13920

Total Earnings = Rs. 83520

Therefore, the man's total earnings during the first year are Rs. 83,520.

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