Calculate the effective rate (APY) of interest for 1 year. The principal is $15,500 which an interest rate of 12%. The interest is compounded quarterly.

Based on the information provided, the inference that a significant number of students in the homeroom class plan to become soldiers is valid.

The teacher surveyed his homeroom class about their post-graduation plans. He found out that most of his students plan to become soldiers.

An inference is a conclusion that can be drawn from a given set of data. In this case, the inference that can be made is that a significant number of students in the homeroom class plan to become soldiers.Therefore, the inference is valid based on the data provided.

However, it is essential to keep in mind that the sample size in the homeroom class is relatively small and may not be representative of the entire population. If the teacher had surveyed a more extensive and diverse population, the results might have been different.

In conclusion, based on the information provided, the inference that a significant number of students in the homeroom class plan to become soldiers is valid. However, this may not apply to the entire population, and it is crucial to consider the sample size when drawing inferences.

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2 x 10^-8 is 5 x 10^-13 times smaller than 4 x 10^-4.In other words, 2 x 10^-8 is 500,000,000,000 (500 trillion) times Smaller than 4 x 10^-4.

To determine how many times smaller 2 x 10^-8 is than 4 x 10^-4, we can divide the two numbers and express the result in scientific notation.

First, let's divide 2 x 10^-8 by 4 x 10^-4:

(2 x 10^-8) / (4 x 10^-4)

To simplify this division, we can divide the numerical values and subtract the exponents:

2 / 4 = 0.5

10^-8 / 10^-4 = 10^(-8 - (-4)) = 10^-8 + 4 = 10^-12

Therefore, (2 x 10^-8) / (4 x 10^-4) simplifies to 0.5 x 10^-12.

Now, let's convert this result back to scientific notation:

0.5 x 10^-12 = 5 x 10^-13

Thus, 2 x 10^-8 is 5 x 10^-13 times smaller than 4 x 10^-4.

In other words, 2 x 10^-8 is 500,000,000,000 (500 trillion) times smaller than 4 x 10^-4.

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Answer:

None of the answer choices

Step-by-step explanation:

Answer:

x ≥ 4

Step-by-step explanation:

The given statement is "The difference of 10 and a number is at most six".

Let "x" be the unknown number.

"The difference of 10 and a number" can be represented as either "10 - x" or "x - 10", depending on whether x < 10 or x > 10.

"Is at most six" means it is less than or equal to six.

Therefore, we have two potential cases:

[tex]\begin{minipage}{3 cm}\underline{\sf Case\;1}:\;\textsf{If $x < 10$}\\\\\boxed{10 - x \leq 6}\end{minipage} \qquad \qquad\begin{minipage}{3 cm}\underline{\sf Case\;2}:\;\textsf{If $x > 10$}\\\\\boxed{x-10 \leq 6}\end{minipage}[/tex]

Solving the inequalities:

[tex]\boxed{\begin{aligned}\underline{\sf Case\;1}\\\\10 - x &\leq 6\\10 - x-10 &\leq 6-10\\-x &\leq -4\\x &\geq 4\end{aligned}}[/tex]      [tex]\boxed{\begin{aligned}\underline{\sf Case\;2}\\\\x-10&\leq 6\\x-10+10 &\leq 6+10\\x &\leq 16\\\\\end{aligned}}[/tex]

[Remember that when dividing both sides of an inequality by a negative number, we have to reverse the inequality sign.]

Therefore, the two possible solutions are:

If we take the statement "the difference between 10 and a number" as "10 - x" then the solution is x ≥ 4.

The ratio 2 to 7 written as a fraction in its lowest term is 2/7

To write the ratio 2 to 7 as a fraction in lowest terms, we can simply express it as a fraction thus :

Required Fraction = Numerator/ denominator = 2/7

Since there are no common factors other than 1 between 2 and 7, the fraction 2/7 is already in its lowest terms.

Hence, the fraction is 2/7

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The cotangent of the angle θ  is 2√3/3

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

sec θ = √7/2

To calculate the cotangent of the angle, we use the following identity

1 + tan² θ = sec² θ

So, we have

1 + tan² θ = (√7/2)²

Evaluate the exponent

1 + tan² θ = 7/4

Subtract 1 from both sides

tan² θ = 3/4

So, we have

tan θ = √3/2

Take the inverse of both sides

cot θ = 2/√3

Rationalize

cot θ = 2√3/3

Hence, the cotangent of the angle is 2√3/3

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Answer:

a. 396cm².

b. 414.69 km²

Step-by-step explanation:

a.

The surface area of a cuboid is the total area of all the faces of the cuboid. A cuboid is a six-sided object with each face being a rectangle. The formula for the surface area of a cuboid is:

Surface area = 2lw + 2lh + 2wh=2(lw+lh+wh)

where:

Here, l=7cm, w=6cm and h=12 cm

substituting value:

Surface area : 2(7*6+7*12+6*12)=396 cm²

Therefore, Surface area is 396cm².

b.

The surface area of a cylinder is the total area of all the faces of the cylinder. It can be calculated using the following formula:

Surface area = 2πr² + 2πrh=2πr(r+h)

Where:

In this case, we have:

r = 6 km

h = 5 km

Therefore, the surface area of the cylinder is:

Surface area =2π*6(6+5)=132π or 414.69 km²

Therefore, the surface area of the cylinder is approximately 414.69square kilometers.

Answer:

a)  396 cm²

c)  132π km²

Step-by-step explanation:

The formula for the surface area of a rectangular prism is:

[tex]\boxed{\begin{minipage}{5.5cm}\underline{Surface area of a rectangular prism}\\\\$SA=2(wl+hl+hw)$\\\\where:\\ \phantom{ww}$\bullet$ $w$ is the width.\\\phantom{ww}$\bullet$ $l$ is the length.\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the values of w, l and h into the formula:

[tex]\begin{aligned}\textsf{Surface area}&=2(6 \cdot 7+12 \cdot 7+12 \cdot 6)\\&=2(42+84+72)\\&=2(198)\\&=396\; \sf cm^2\end{aligned}[/tex]

Therefore, the surface area of the given rectangular prism is 396 cm².

[tex]\hrulefill[/tex]

The formula for the surface area of a cylinder is:

[tex]\boxed{\begin{minipage}{6.5cm}\underline{Surface area of a cylinder}\\\\$SA=2\pi rh+2\pi r^2$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius of the circular base.\\\phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

From inspection of the given diagram:

Substitute the values of r and h into the formula:

[tex]\begin{aligned}\textsf{Surface area}&=2 \pi (6)(5)+2 \pi (6)^2\\&=2 \pi (30)+2 \pi (36)\\&=60 \pi+72 \pi\\&=132 \pi \; \sf km^2\end{aligned}[/tex]

Therefore, the surface area of the given rectangular prism is 132π km².

A relation is a general term that defines the relationship between inputs and outputs, regardless of whether each input has a unique output is: True.

A relation is a general term that defines the relationship between inputs and outputs, regardless of whether each input has a unique output.

On the other hand, a function is a specific type of relation in which each input value is associated with exactly one output value. Therefore, all functions can be considered as relations, but not all relations qualify as functions.

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Answer:

y = 55 , y = 100

Step-by-step explanation:

9

the sum of the 3 angles in a triangle is 180°

sum the angles and equate to 180

y + 75 + 50 = 180

y + 125 = 180 ( subtract 125 from both sides )

y = 55

10

the sum of the exterior angles of a triangle is 360°

sum the exterior angles and equate to 360

y + 120 + 140 = 360

y + 260 = 360 ( subtract 260 from both sides )

y = 100

If Gavin contributes towards a Pension fund (7,5% of his Basic Salary), then his net salary will be reduced by the amount he contributed towards pension fund i.e. R652.5.

1.1 Name of employee is Gavin Pillay. 1.2 UIF stands for Unemployment Insurance Fund. UIF is a benefit to provide short-term relief to workers who lose their jobs, or can't work due to illness, maternity or adoption leave. 1.3 The UIF paid by Gavin is calculated correctly.

Firstly, we need to calculate Gross Salary of Gavin, which is given as: Basic Salary + Overtime + M = R8700 + R1200 + R3201 = R13101Then, calculate total deductions from gross salary.

Total Deductions = PAYE + UIF = R1200 + 87 = R1287Hence, Gavin's net salary = Gross salary - Total Deductions = R13101 - R1287 = R11814Therefore, UIF paid by Gavin is calculated correctly as R87.1.4 This payslip is for the month of May i.e.

31/05/2023.1.5 Basic salary of Gavin = R8700. Deduct 7.5% of his basic salary to calculate his pension fund contribution. Pension Fund Contribution = 7.5/100 x R8700 = R652.5.1.6

Total deductions of Gavin includes PAYE, UIF and pension fund contribution.

Therefore, Total Deductions = PAYE + UIF + Pension Fund Contribution = R1200 + R87 + R652.5 = R1939.5.1.7

Gavin's age can not be determined as his date of birth is not provided.1.8 Address of Gavin's workplace is 21 Halsted Road, Pietermaritzburg?

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The relationship between the area rate of coverage and the speed is approximately four times as large. This means that doubling the speed leads to approximately four times the rate of coverage, making option B

To determine the relationship between the area rate of coverage (acre/hour) and the speed (mph) based on the given chart, we can compare the values at different speeds.

According to the chart, as the speed increases, the area rate of coverage also increases. For example, at 3 mph, the rate is 5.96 acres per hour, while at 3.5 mph, it is 6.95 acres per hour. This pattern continues as the speed increases: 4 mph yields 7.95 acres per hour, 4.5 mph gives 8.94 acres per hour, and so on.

By comparing the values, we can observe that as the speed doubles, the area rate of coverage also approximately doubles. For instance, at 3 mph, the rate is 5.96 acres per hour, and at 6 mph, it is 11.92 acres per hour. This indicates that doubling the speed results in approximately twice the rate of coverage.

Similarly, as the speed quadruples, the area rate of coverage also approximately quadruples. For instance, at 3 mph, the rate is 5.96 acres per hour, and at 6.5 mph, it is 12.92 acres per hour. This suggests that quadrupling the speed results in approximately four times the rate of coverage.

Option B

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The length of BL is approximately 7.9 units.

In triangle ABL, we are given that AL = 12, TS = 7.75, and SL = 5.1. We need to find the length of BL.

Since TS is parallel to AL, we can use the concept of similar triangles to solve this problem.

By drawing a line parallel to BL through point T, we create two similar triangles: triangle TSL and triangle ABL. The corresponding sides of similar triangles are proportional.

Let's denote the length of BL as x. Then we can set up the following proportion:

TS/AL = SL/BL

Substituting the given values:

7.75/12 = 5.1/x

To solve for x, we can cross-multiply and solve the resulting equation:

7.75 * x = 12 * 5.1

7.75x = 61.2

Dividing both sides of the equation by 7.75:

x = 61.2 / 7.75

x ≈ 7.9

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The inequality value that would be able to show/x/ ≤ 5 is option A

In mathematics, inequality refers to a mathematical statement that expresses a relationship between two quantities, indicating that one quantity is greater than, less than, or not equal to the other.

The symbol that we have  is telling us that the value of the x would be less than  or it would be seen to be equal to five.

We can see that /x/ ≤ 5 would exclude the negative values and this can be seen in the option that is marked option A

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Jessica would have to drive 20 miles for the cost to rent a moving truck for one day to be the same at Rentals Plus and Speedy Move.

Let's assume the number of miles driven is represented by 'm'. The cost of renting a moving truck for one day at Rentals Plus can be calculated using the equation:

Cost at Rentals Plus = $5.24 + $4.46 * m

Similarly, the cost of renting a moving truck for one day at Speedy Move can be calculated using the equation:

Cost at Speedy Move = $85.24 + $0.46 * m

To find the number of miles that makes the costs equal, we can set up the equation:

$5.24 + $4.46 * m = $85.24 + $0.46 * m

By rearranging the equation, we can solve for 'm':

$4.46 * m - $0.46 * m = $85.24 - $5.24

$4 * m = $80

m = $80 / $4

m = 20

Therefore, Jessica would have to drive 20 miles for the cost to rent a moving truck for one day to be the same at Rentals Plus and Speedy Move.

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If you earn $1 every 500 seconds and work for a normal 8-hour day, you would have Earned $57.

To calculate the amount of money earned in a normal workday, we need to determine the total number of seconds in 8 hours and then divide it by the time it takes to earn $1.

There are 60 seconds in a minute and 60 minutes in an hour. Therefore, there are 60 x 60 = 3600 seconds in an hour.

To find the total number of seconds in 8 hours, we multiply the number of seconds in an hour by 8:

8 hours x 3600 seconds/hour = 28,800 seconds.

Next, we need to determine how many times $1 can be earned in 28,800 seconds. Given that $1 is earned every 500 seconds, we divide 28,800 by 500:

28,800 seconds ÷ 500 seconds = 57.6.

Since we cannot earn a fraction of a dollar, we round down to the nearest whole number. Therefore, you would have earned 57 dollars during the 8-hour workday.

In summary, if you earn $1 every 500 seconds and work for a normal 8-hour day, you would have earned $57.

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Answer:

Step-by-step explanation:

Year 1: $850 * 0.91 = $773.50

Year 2: $773.50 * 0.91 = $704.69

Year 3: $704.69 * 0.91 = $641.95

...

Year 34: (continue the pattern)

We can continue this calculation for each year, but to save time, we can use an exponential decay formula:

Remaining Amount = Initial Amount * (1 - rate)^years

Substituting the values:

Remaining Amount = $850 * (1 - 0.09)^34

Calculating this expression:

Remaining Amount ≈ $850 * (0.91)^34 ≈ $255.88

After 34 years, approximately $255.88 will be left with Sally.

In the first two sets of statements, the conclusion is not necessarily true. For example, a lion has fur and is an animal, but it does not have claws.

In the third set of statements, the conclusion is true. A rhombus is a type of square, so it has four congruent angles. In the fourth set of statements, the conclusion is not necessarily true. For example, the number 12 is divisible by 12 and 6, but it is not divisible by 3.

In the fifth set of statements, the conclusion is true. A number that is divisible by 12 is also divisible by 6 and 3. In the sixth set of statements, the conclusion is not necessarily true. For example, the number 9 is divisible by 9 and 3, but it is not divisible by 6.

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Answer:

Carole will need to save for about 4 weeks to afford the coat.

Step-by-step explanation:

$80 - $25 = $55

$55 / $15 = 3.67 (approximately)

Rounded up: 4

Hamilton’s Method:A large chair store needs to assign 12 buyers to its five stores, with the number of employees for each store given below:5 buyers to store 1;3 buyers to store 2;2 buyers to store 3;1 buyer to store 4;1 buyer to store 5

A chair store needs to assign 12 buyers to its 5 stores. The number of employees for each store is shown below:

Hamilton’s Method: A large chair store needs to assign 12 buyers to its five stores, with the number of employees for each store given below:5 buyers to store 1;3 buyers to store 2;2 buyers to store 3;1 buyer to store 4;1 buyer to store 5; The number of buyers assigned to each store is as follows:

Store 1 gets 5 buyers Store 2 gets 3 buyers store 3 gets 2 buyers store 4 gets 1 buyer Store 5 gets 1 buyer Jefferson’s Method: To find out how many buyers should be assigned to each store using Jefferson’s Method, the following steps should be taken:

Step 1: Find the total number of buyers to be assigned

Step 2: Divide each store’s employee by the total number of employees

Step 3: Multiply the result of Step 2 by the total number of buyers store 1 gets 5.400 buyers store 2 gets 3.240 buyers store 3 gets 1.920 buyers store 4 gets 0.720 buyers store 5 gets 0.720 buyers

Webster’s Method: To find out how many buyers should be assigned to each store using Webster’s Method, the following steps should be taken:

Step 1: Find the total number of buyers to be assigned

Step 2: Divide each store’s employee by the total number of employees

Step 3: Multiply the result of Step 2 by the total number of buyers step 4: Round up the result of Step 3 to the nearest integer Store 1 gets 5 buyers store 2 gets 3 buyers store 3 gets 2 buyers store 4 gets 1 buyerStore 5 gets 1 buyerHill-Huntington's Method: To find out how many buyers should be assigned to each store using Hill-Huntington's Method, the following steps should be taken:

Step 1: Find the total number of buyers to be assigned

Step 2: Divide each store’s employee by the square root of the total number of employees

Step 3: Multiply the result of Step 2 by the square root of the total number of buyers store 1 gets 5 buyers Store 2 gets 3 buyers Store 3 gets 2 buyers Store 4 gets 1 buyer Store 5 gets 1 buyer

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Answer: 9/4

Step-by-step explanation:

Reciprocal means flip the fraction

The reciprocal of 4/9  is  9/4

Answer: the rciprocal is 9/4

Step-by-step explanation:

The correct conclusion based on the given statements is:

If the time is between 8:00 a.m. and 6:00 p.m., then people may swim or dive into the pool.

A conditional statement, also known as an "if-then" statement, is a type of logical statement that consists of two parts: a condition (the "if" part) and a result (the "then" part).

It expresses a cause-and-effect relationship, stating that if a certain condition is true, then a specific result or action will follow.

"If the time is between 8:00 a.m. and 6:00 p.m., then people may swim or dive into the pool." This conclusion follows from the given information that if the time is within the specified range, the public swimming pool is open, and if the pool is open, people are allowed to swim or dive into it.

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The slope of line N is -3/4.

To find the y-intercept of line M and the slope of line N, we need to understand their equations, which are in slope-intercept form: y = mx + b,

where m represents the slope and b represents the y-intercept.Line M is given by the equation:

y = -3x + 7. Here, we can see that the slope, m, is -3, and the y-intercept, b, is 7.

Therefore, the y-intercept of line M is 7.Line N is given by the equation: 3x + 4y = 8.

To find the slope of this line, we can solve for y:4y = -3x + 8y = (-3/4)x + 2

Here, we can see that the slope, m, is -3/4.

Therefore, the slope of line N is -3/4.

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Answer:

11x +12y = 13

22x + 24y = 26

Step-by-step explanation:

Infinite many solutions means that if you did something to the equations, they would end up being the same line.  A line on top of a line intersect infinitely many times that's why infinite solutions.

11x +12y = 13

22x + 24y = 26

Look at the last pair of equations.  If i divided the second equation by 2(all of the terms):

[tex]\frac{22x + 24y = 26}{2}[/tex]

=11x +12y = 13

You get the same equation as the first.  So this is the system that has infinite solutions.

EG = (19/5)FG = (19/5)(550/43) = 418.Hence, EG is Equal to 418.

Let's assume the length of EF is x.

According to the given information, the ratio of EF / FG / GH is 4/5 / 2.

This can be written as EF / FG = 4/5 and FG / GH = 2.

We know that the sum of the ratios in a proportion is equal to 1.

So, (EF / FG) + (FG / GH) = 1.

Substituting the given values, we have (4/5) + 2 = 1.

Simplifying, we get (4 + 10) / 5 = 1.

This implies that EF is equal to 14/5 times FG.

Now, let's consider the length of EG.

EG is equal to the sum of EF and FG.

So, EG = EF + FG = (14/5)FG + FG = (19/5)FG.

We are also given that EH = 55.

EH is equal to the sum of EG and GH.

So, EH = EG + GH.

Substituting the values, we get 55 = (19/5)FG + GH.

Since FG / GH = 2, we can write GH = (1/2)FG.

Substituting this into the equation, we have 55 = (19/5)FG + (1/2)FG.

To solve for FG, we need to find a common denominator.

Multiplying 19/5 by 2/2, we get 38/10.

So, 55 = (38/10)FG + (1/2)FG.

Combining like terms, we have 55 = (38/10 + 5/10)FG = (43/10)FG.

To isolate FG, we divide both sides by (43/10):

55 / (43/10) = FG.

Simplifying, we get FG = 550 / 43

Therefore, EG = (19/5)FG = (19/5)(550/43) = 418.

Hence, EG is equal to 418.

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A repeating decimal is acceptable to use during calculations when you require a certain level of precision

A repeating decimal is acceptable to use during calculations when you require a certain level of precision, but it's important to keep in mind that the result will be an approximation rather than an exact value.

It is commonly used in situations where a rounded value is sufficient and the level of precision doesn't significantly impact the final result.

On the other hand, you should convert a repeating decimal to a fraction for calculations when you need an exact value or when further mathematical operations or comparisons are required.

Converting a repeating decimal to a fraction allows for precise calculations and eliminates any potential rounding errors associated with working with decimal approximations.

Converting a repeating decimal to a fraction involves identifying the repeating pattern and expressing it as a fraction. This allows for exact calculations and maintains the mathematical properties of fractions.

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The Wronskian of the set of functions {xln|xl|, x^2ln|xl|} is zero, indicating that the functions are linearly dependent.

To compute the Wronskian of a set of functions {y1, y2}, we arrange the functions as columns in a matrix and calculate the determinant of that matrix. In this case, we have the set of functions {y1, y2} = {xln|xl|, [tex]x^2[/tex]ln|xl|}.

Let's construct the matrix:

| xln|xl| x^2ln|xl| |

|---------------------|

| y1 y2 |

To calculate the Wronskian, we compute the determinant of this matrix:

W = | xln|xl| x^2ln|xl| |

|---------------------|

| y1 y2 |

Using the properties of determinants, we can expand this determinant along the first row:

W = (xln|xl|)(y2) - ([tex]x^2[/tex]ln|xl|)(y1)

Simplifying further, we have:

W = xy2ln|xl| - x^2y1ln|xl|

Now, substituting the given functions for y1 and y2, we have:

W = x(x^2ln|xl|)ln|xl| - x^2(xln|xl|)ln|xl|

Simplifying inside the parentheses:

W = x^3ln|xl|^2ln|xl| - x^3ln|xl|^2ln|xl|

Combining like terms:

W = 0

Therefore, the Wronskian of the set of functions {y1, y2} = {xln|xl|, x^2ln|xl|} is equal to zero. This implies that the functions are linearly dependent.

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Answer:

see explanation

Step-by-step explanation:

the sum of the interior angles of a polygon is

sum = 180° (n - 2) ← n is the number of sides

a hexagon has 6 sides , then

sum = 180° × (6 - 2) = 180° × 4 = 720°

12

the polygon has 5 sides , so

sum = 180° × (5 - 2) = 180° × 3 = 540°

sum the interior angles and equate to 540

y + 90 + 120 + 90 + 110 = 540

y + 410 = 540 ( subtract 410 from both sides )

y = 130

13

the polygon has 7 sides , so

sum = 180° × (7 - 2) = 180° × 5 = 900°

sum the interior angles and equate to 900

p + 90 + 141 + 130 + 136 + 123 + 140 = 900

p + 760 = 900 ( subtract 760 from both sides )

p = 140

Cubic measurements are used to measure the volume of a cubic box, while volume measurements are used to measure the capacity of a three-dimensional object, whether solid or liquid.

Cubic measures and their relationship with volume measures. The terms cubic measures and volume measures are used in mathematics to represent three-dimensional magnitudes. Cubic measures are the units of measurement of a three-dimensional volume in a cubic box.

On the other hand, volume measurements are the units of measurement of the capacity of an object, whether it is a solid or a liquid.

The relationship between cubic measurements and volume measurements is explained by the fact that both are used to measure the amount of space an object occupies. In most cases, volume measurements are expressed in units such as cubic centimeters, cubic meters, milliliters, liters, gallons, and ounces.

In solids, volume measurements can be expressed in units such as cubic feet, cubic inches, and cubic yards. In summary, the relationship between cubic measurements and volume measurements lies in the fact that both are used to measure occupied space. for an object.

Cubic measurements are used to measure the volume of a cubic box, while volume measurements are used to measure the capacity of a three-dimensional object, whether solid or liquid.

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The Wronskian of the set {x, sin(2x), cos(2x)} is det(W) = 8x + 16sin^2(2x) + 4cos^2(2x). The Wronskian of a set of functions is a determinant that can determine linear independence or dependence of the functions.

To compute the Wronskian of the set {x, sin(2x), cos(2x)}, we arrange the functions in a matrix as follows:

W = | x sin(2x) cos(2x) |

| x' sin(2x)' cos(2x)' |

| x'' sin(2x)'' cos(2x)'' |

Here, the derivatives of the functions with respect to x are taken in the second and third rows. Taking the derivatives, we get:

W = | x sin(2x) cos(2x) |

| 1 2cos(2x) -2sin(2x) |

| 0 -4sin(2x) -4cos(2x) |

To compute the determinant of W, we expand it along the first row:

det(W) = x(2cos(2x)(-4cos(2x)) - (-2sin(2x))(-4sin(2x)))

- sin(2x)(1(-4cos(2x)) - 0(-4sin(2x)))

+ cos(2x)(1(-4sin(2x)) - 0(-4cos(2x)))

Simplifying the determinant, we get:

det(W) = 8x + 16sin^2(2x) + 4cos^2(2x).

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We can conclude that `An(B-C) = AnB - AnC` using double inclusion proofs.

Given a universal set `U` and subsets `A`, `B` and `C`.To prove: `An(B-C) = AnB - AnC`.

Now, we can prove this statement by double inclusion proofs. Let `x` be an arbitrary element of `U`.

Proof of `An(B-C) ⊆ AnB - AnC`:

Suppose that `x ∈ An(B-C)`, then `x ∈ A` and `x ∈ B-C`.

By definition of `B-C`, we have `x ∈ B and x ∉ C`.

Therefore, `x ∈ B` and `x ∉ C` implies that `x ∈ B` and `x ∉ AnC`.

Therefore, `x ∈ AnB` and `x ∉ AnC` implies that `x ∈ AnB - AnC`.

Hence, we have shown that `An(B-C) ⊆ AnB - AnC`.

Proof of `AnB - AnC ⊆ An(B-C)`:

Suppose that `x ∈ AnB - AnC`, then `x ∈ AnB` and `x ∉ AnC`.

By definition of intersection, we have `x ∈ A and x ∈ B`.Now, since `x ∈ AnC`, we have `x ∈ A and x ∈ C`.Since `x ∈ A` is a common element, we can combine both sets `B` and `C` to get `B ∪ C`.

Now, since `x ∈ B ∪ C`, we can write this as `x ∈ B ∪ (U-C)` or `x ∈ B - C or x ∈ A ∩ (U-C)`.

Thus, `x ∈ An(B-C)`.Hence, we have shown that `AnB - AnC ⊆ An(B-C)`.

Therefore, we can conclude that `An(B-C) = AnB - AnC` using double inclusion proofs.

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