A snow making machine priced at $1800 is on sale for 25% off. The sales tax rate is 6.25%. What is the sale price including tax? If necessary, round your answer to the nearest cent.

Answer:

Step-by-step explanation:

Perimeter

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The probability that a randomly selected quartz timepiece will have a replacement time less than 13.3 years is approximately 0.015 with a mean of 17 years and a standard deviation of 1.7 years.

To find the probability that a randomly selected quartz timepiece will have a replacement time of less than 13.3 years, we need to use the standard normal distribution formula which is as follows:

[tex]Z =\frac{X -μ }{σ}[/tex]

Where Z is the standard score

X is the variable value

μ is the mean

σ is the standard deviation

Given that the mean (μ) of the replacement times for the quartz timepieces is 17 years, the standard deviation (σ) is 1.7 years, and the variable value (X) we are looking for is 13.3 years.

Substitute the values into the standard normal distribution formula to get:

[tex]Z = \frac{13.3-17}{1.7} = -2.17[/tex]

Looking at the standard normal distribution table, we can find the probability of the standard score Z = -2.17 to be 0.015.

Therefore, the probability that a randomly selected quartz timepiece will have a replacement time less than 13.3 years is approximately 0.015.

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An interval is a set of real numbers that contains all real numbers lying between any two numbers of the set.

For each calculation, the smallest interval in which the result, using true instead of rounded values, must lie is as follows:

(a) 1.1062+0.947 = 2.0532 ≤ true result ≤ 2.053

(b) 23.46 - 12.753 = 10.707 ≤ true result ≤ 10.708

(c) (2.747) (6.83) = 18.6181 ≤ true result ≤ 18.6182

(d) 8.473/0.064 = 132.3906 ≤ true result ≤ 132.3907

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If the area of the rectangle is 32 square units and its perimeter is 36 units, then the length and width of the rectangle will be given as 16 units and 2 units respectively.

Area is defined as the measure of a specific region on ground which is enclosed by a closed polygon figure. The area of a rectangle is given as the product of its length (l) and its width (b) . Perimeter on the other hand is the sum of all four sides of a rectangle and is given by the formula as follows:

Perimeter of rectangle = 2 (length + width)

Now its is given that Area= length x width

32 = l*b ... 1

36 = 2(l+b)  ... 2

Using equation 1, we get b = 32/l. Putting this value in equation 2, we get:

36 = 2 (32/l + l)

18 = 32/l + l

⇒ l^2 - 18l + 32 = 0

Solving this quadratic equation we get,

l = 16, 2

Thus the length and width of the rectangle will be equal to 16 units and 2 units respectively or vice versa.

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Refer to complete question below:

On a separate piece of paper, draw and label rectangle with an area of 32 sq. Unit and a perimeter of 36 units. Use numbers or words to show that you are correct.

By using this value of x in the formula we previously discovered, we can get the patio's area Patio's size is equal to x2 + 4x + 4 = ((1 + 7)/3)2 + 4((1 + 7)/3) + 4 = 4.72 square meters.

A square is a 2D shape with equal-sized sides on each side. The area would be length times width, which is equal to side  side because all the sides are equal. As a result, a square's area is side square.

Let's first find the area of the entire rectangle:

A = lw = (3x + 6)(2x + 4) = 6x² + 30x + 24

Area of patio = (x + 2)² = x² + 4x + 4

Area of herb garden = (2x + 2)(x + 4) = 2x² + 10x + 8

Area of flower garden = (3x + 4)(x + 4) = 3x² + 16x + 16

Sum of areas = x² + 4x + 4 + 2x² + 10x + 8 + 3x² + 16x + 16

= 6x² + 30x + 28

0.25(6x² + 30x + 24) = 6x² + 30x + 28

Simplifying and solving for x, we get:

1.5x² - x - 1 = 0

Using the quadratic formula, we find that:

x = (1 ± √7)/3

x = (1 + √7)/3

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The area in square meters of the patio is 850 square meters.

What is a rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees) with opposite sides being parallel and equal in length.

Let's start by calculating the total area of the rectangle:

Area of rectangle = length x width = 100m x 40m = 4000 square meters

Now, let's denote the width of the herb garden as x meters. Then, the length of the herb garden would be 10 meters.

The area of the herb garden would be:

Area of herb garden = length x width = 10m x x = 10x square meters

The area of the patio can be calculated as:

Area of patio = (100 - x) x (40 - 2x) square meters

(100 - x) is the length of the patio, and (40 - 2x) is the width of the patio, since the herb garden takes up x meters of the width.

The area of the flower garden can be calculated by subtracting the area of the rectangle, the herb garden, and the patio from each other:

Area of flower garden = 4000 - 10x - (100 - x) x (40 - 2x) square meters

Now, we need to find the value of x so that the sum of the areas of the patio, herb garden, and flower garden is 25% of the area of the entire rectangle. In other words:

Area of herb garden + Area of patio + Area of flower garden = 0.25 x Area of rectangle

10x + (100 - x) x (40 - 2x) + 4000 - 10x = 0.25 x 4000

Simplifying this equation, we get:

-2x^2 + 30x + 1000 = 1000

-2x^2 + 30x = 0

-2x(x - 15) = 0

Therefore, x = 0 or x = 15. Since x cannot be 0 (since the herb garden would have no width), the value of x must be 15 meters.

Now we can calculate the area of the patio:

Area of patio = (100 - x) x (40 - 2x) = (100 - 15) x (40 - 2(15)) = 850 square meters

Therefore, the area in square meters of the patio is 850 square meters.

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Answer: 25

Step-by-step explanation:

Answer:

25

Step-by-step explanation:

Compute P(x ≤ 15) = (15-10)/(20-10) = 5/10 = 0.5.

Compute P(12 ≤ x ≤ 18) = (18-12)/(20-10) = 6/10 = 0.6.

Compute E(x): The expected value of x is: E(x) = (a+b)/2 = (10+20)/2 = 15

Compute Var(x):The variance of x is: Var(x) = (b - a)^2/12 = (20 - 10)^2/12 = 100/12 = 8.33.

The probability density function is as follows: As the random variable x is uniformly distributed between 10 and 20. Thus, f(x) = 1/(20-10) = 1/10 for 10 ≤ x ≤ 20.Compute P(x ≤ 15):Thus, P(x ≤ 15) = (15-10)/(20-10) = 5/10 = 0.5.Compute P(12 ≤ x ≤ 18):Thus, P(12 ≤ x ≤ 18) = (18-12)/(20-10) = 6/10 = 0.6.Compute E(x):The expected value of x is: E(x) = (a+b)/2 = (10+20)/2 = 15.Compute Var(x):The variance of x is: Var(x) = (b - a)^2/12 = (20 - 10)^2/12 = 100/12 = 8.33.

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Wind speeds, represented by random variable X, have a lognormal distribution. The corresponding value of the standard normal random variable (Z) is associated with a wind speed of 14.35 is 25.5

Wind speeds represented by random variable X, in miles per hour, have a lognormal distribution. In other words, log(X) is normal.

If [tex]\mu = 4.8[/tex] and [tex]\sigma = 0.4[/tex], what value of Z (the standard normal rv) is associated with a wind speed of 15 miles per hour.

The value of Z (the standard normal rv) associated with a wind speed of 15 miles per hour.

The standard score (z) of a random variable X is calculated as follows:

[tex]z = \frac{(X - \mu)}{\sigma}[/tex]

Given: μ = 4.8, σ = 0.4

Let X be a wind speed 15 mph.

To find the standard normal rv Z associated with a wind speed of 15 miles per hour, we will use the formula for calculating the standard score (z):

[tex]z = (X - \mu) /\sigma \\z = (15 - 4.8) / 0.4\\z = 25.5[/tex]

Therefore, the value of Z (the standard normal rv) associated with a wind speed of 15 miles per hour is 25.5.

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They would each need to sell 5 bicycles to make the same amount of money and if they both sell 5 bicycles, they would each make $500.

What do you mean by finding the break-even point ?

The key concept used here is the idea of finding the break-even point between two scenarios. In this case, the break-even point is the number of bicycles that Jim and Tom each need to sell in order to make the same amount of money. This is found by setting their total earnings equal to each other and solving for the number of bicycles. Once the break-even point is found, the total earnings for that number of bicycles can be calculated by plugging it back into the original equations. This concept is commonly used in business and finance to determine the minimum level of sales needed to cover costs and make a profit.

Calculating the number of bicycle and money :

To make the same amount of money, Jim and Tom would have to each sell the same number of bicycles, let's call it "b".

So Jim would make a total of:

250 + 50b dollars

Tom would make a total of:

400 + 20b dollars

To find the value of "b" where they both make the same amount of money, we can set the two expressions equal to each other and solve for "b":

250 + 50b = 400 + 20b

30b = 150

b = 5

Therefore, they would each need to sell 5 bicycles to make the same amount of money.

To find out how much they would make, we can substitute "b=5" into either of the expressions above:

Jim:

250 + 50(5) = $500

Tom:

400 + 20(5) = $500

Therefore, if they both sell 5 bicycles, they would each make $500.

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According to the given coordinate the value of f(x) is -6, 0, 3, 6, 12.

An equation is a mathematical statement that indicates that two expressions are equal. It typically contains variables, which are symbols that represent unknown values, and constants, which are values that are known. Equations are used to describe relationships between quantities and to solve problems by finding the values of variables that satisfy the equation. For example, the equation 2x + 3 = 7 is a statement that the sum of two times x and 3 is equal to 7, and we can solve for x by subtracting 3 from both sides and dividing by 2 to obtain x = 2

Given value is f(x) = 3x

now value of x is given in the table.

If value of x is -2 then value of f('x) will be -6

If value of x is 0 then value of f('x) will be 0

If value of x is 1 then value of f('x) will be 3

If value of x is 2 then value of f('x) will be 6

If value of x is 4 then value of f('x) will be 12

Therefore, according to the given coordinate the value of f(x) is -6, 0, 3, 6, 12.

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The probability that at least 2 people believe that they have seen a ufo is 0.1937102445. The probability that exactly 1 person believes that he or she has seen a ufo is problem: P(X = 1) = 10C₁ (0.10) (0.90)⁹= 0.3874204890.

The probability that at least 2 people believe that they have seen a UFO would be 0.1937102445. For this we use the binomial distribution formula.

P(X ≥ 2) = 1 − P(X = 0) − P(X = 1)P(X = 0) = (9/10)¹⁰

P(X = 1) = 10C₁ (0.10) (0.90)⁹= 0.3874204890 (rounded to 10 decimal places)

P(X ≥ 2) = 1 − 0.3874204890 − 0.3486784401 = 0.1937102445 (rounded to 10 decimal places)

The probability that 2 or 3 people believe that they have seen a UFO would be 0.1937102445. Using the formula of binomial distribution again we can solve for the probability of this event.

P(2 ≤ X ≤ 3) = P(X = 2) + P(X = 3)P(X = 2) = 10C₂ (0.10)² (0.90)⁸= 0.1937102445 (rounded to 10 decimal places)

P(X = 3) = 10C₃ (0.10)³ (0.90)⁷= 0.0573956280 (rounded to 10 decimal places)

P(2 ≤ X ≤ 3) = 0.1937102445 + 0.0573956280 = 0.2511058725 (rounded to 10 decimal places)

The probability that exactly 1 person believes that he or she has seen a UFO would be 0.3874204890. Using the binomial distribution formula to solve this problem:

P(X = 1) = 10C₁ (0.10) (0.90)⁹= 0.3874204890 (rounded to 10 decimal places)

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Answer:

x = 20

Step-by-step explanation:

if a line is parallel to a side of a triangle and intersects the other two sides, it divides those sides proportionally.

QR is parallel to ST and intersects the other two sides of the triangle, then

[tex]\frac{PQ}{QS}[/tex] = [tex]\frac{PR}{RT}[/tex] ( substitute values )

[tex]\frac{x}{45-x}[/tex] = [tex]\frac{16}{30-16}[/tex]

[tex]\frac{x}{45-x}[/tex] = [tex]\frac{16}{20}[/tex] ( cross-  multiply )

20x = 16(45 - x)

20x = 720 - 16x ( add 16x to both sides )

36x = 720 ( divide both sides by 36 )

x = 20

To determine the probability of a continuous distribution we use the integral to determine it and for the normal distribution the integral is not so simple, for that reason it is simpler to use range values from tables.

Probability can be determined from a continuous distribution in the following way:To compute the probability of a given interval for a continuous random variable, the area under the curve over the interval is determined. Integrals are used to calculate this area under the curve, which can be done either numerically or analytically using probability density functions.

For some distributions, such as the uniform distribution, calculating the area under the curve is straightforward. However, for other distributions, such as the normal distribution, it can be more difficult to calculate the integral analytically.

The normal distribution is a continuous probability distribution that is frequently used in statistics. It is defined by its probability density function, which is a bell-shaped curve with a mean and a standard deviation.

Calculating the area under the curve for the normal distribution requires the use of integrals. Integrals are difficult to solve analytically for the normal distribution because the probability density function is not simple. However, it is relatively simple to calculate the probability for a given range of values using standard statistical tables or computer software.

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Answer:

$2300

Step-by-step explanation:

$1610 is 7/10 the original cost. Multiplying by 10/7 reverses that, and we get the starting cost of $2300

Hope this helps!

Answer:

To simplify the expression (-3+1)+(-4-i), we can perform the addition operation within the parentheses first:

(-3+1)+(-4-i) = -2 + (-4-i)

Next, we can simplify the addition of -2 and -4 by adding their numerical values:

-2 + (-4-i) = -6 - i

Therefore, (-3+1)+(-4-i) simplifies to -6-i.

Step-by-step explanation:

If the volume of a right triangular prism is 72 cubic feet, the area of the base is 2 square feet and the length of DF is approximately 2.83 feet.

To solve the problem, we can use the formula for the volume of a right triangular prism, which is:

Volume = (1/2) x base x height x length

where base is the area of the triangular base, height is the height of the prism, and length is the length of the prism.

We are given that the volume is 72 cubic feet and the height is 9 feet. Therefore, we can write:

72 = (1/2) x base x 9 x length

Simplifying this equation, we get:

base x length = 16

We are also given that the base is an isosceles right triangle. This means that the two legs of the triangle are equal, and the hypotenuse is equal to the length of one leg times the square root of 2.

Let's call the length of one leg of the triangle DF. Then, we can write:

base = (1/2) x DF x DF

Substituting this expression for base into the equation we derived earlier, we get:

(1/2) x DF x DF x length = 16

Simplifying this equation, we get:

DF x DF x length = 32

We know that the hypotenuse of the triangle is DF times the square root of 2. Since the hypotenuse is also one of the edges of the base of the prism, we can set it equal to the length of the prism:

DF x √(2) = length

Substituting this expression for length into the equation we derived earlier, we get:

DF x DF x DF x sqrt(2) = 32

Simplifying this equation, we get:

DF^3 = 16

Taking the cube root of both sides, we get:

DF = 2

Therefore, the area of the base is:

base = (1/2) x DF x DF = 2 square feet

And the length of DF is:

DF x √(2) = 2 x √(2) feet = approximately 2.83 feet.

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The mean, median, and mean absolute deviation MAD of the data are 15%, 10%, and 8%.

To find the mean, median, and mean absolute deviation (MAD) of the data, we need to first convert all the fractions to percentages:

Whitney: 16%

Chen: 8%

Bjorn: 6%

Dustin: 2%

Philip: 12%

Maria: 40%

a) Mean:

To find the mean, we add up all the percentages and divide by the total number of friends (6):

Mean = (16 + 8 + 6 + 2 + 12 + 40) / 6 = 15%

Therefore, the mean is 15%.

b) Median:

To find the median, we need to arrange the data in order from smallest to largest:

2%, 6%, 8%, 12%, 16%, 40%

Since there are six values, the median is the average of the two middle values: (8 + 12) / 2 = 10%

Therefore, the median is 10%.

c) Mean Absolute Deviation (MAD):

To find the MAD, we first need to find the absolute deviation of each value from the mean:

Whitney: |16 - 15| = 1%

Chen: |8 - 15| = 7%

Bjorn: |6 - 15| = 9%

Dustin: |2 - 15| = 13%

Philip: |12 - 15| = 3%

Maria: |40 - 15| = 25%

Next, we find the average of these absolute deviations:

MAD = (1 + 7 + 9 + 13 + 3 + 25) / 6 = 8%

Therefore, the mean absolute deviation is 8%.

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Answer:

The ratio of red to blue squares in the given set of 2 reds and 18 blues can be written as:

Red:Blue = 2:18

To simplify the ratio, we can divide both the numerator and denominator by the greatest common factor (GCF) of 2 and 18, which is 2. Dividing both terms by 2, we get:

Red:Blue = 1:9

Therefore, the ratio of red to blue squares in its simplest form is 1:9.

Answer:

We know that A receives R500 less than C, so we can write:

4x = 9x - 500

Solving for x, we get:

5x = 500

x = 100

Now we can calculate the amounts received by each person:

A = 4x = 4(100) = R400

B = 7x = 7(100) = R700

C = 9x = 9(100) = R900

To check our answer, we can verify that the ratios of the amounts received by A, B, and C are indeed 4:7:9:

A:B = 400:700 = 4:7

B:C = 700:900 = 7:9

Therefore, the total amount divided is:

400 + 700 + 900 = R2000

So the amount that is divided is R2000.

Step-by-step explanation:

The total amount of money divided is R2000.

Ratio is described as the comparison of two quantities to determine how many times one obtains the other. The proportion can be expressed as a fraction or as a sign: between two integers.

We are given that;

The ratio of A, B and C= 4:7:9

Now,

Let's start by assigning variables to the unknowns in the problem. Let's call the total amount of money "T". Then, if A receives 4x, B receives 7x, and C receives 9x, where "x" is some constant, we can write:

4x + 500 = C's share

We can also write an equation to represent the fact that the three shares add up to the total amount:

4x + 7x + 9x = T

Simplifying this equation, we get:

20x = T

Now we can substitute the first equation into the second equation and solve for x:

4x + 7x + (4x + 500) = 20x

15x + 500 = 20x

500 = 5x

x = 100

Now we can find the individual shares by multiplying x by the appropriate ratio factor:

A's share = 4x = 400

B's share = 7x = 700

C's share = 9x = 900

Finally, we can check that these add up to the total amount:

400 + 700 + 900 = 2000

Therefore, by the given ratio the answer will be R2000.

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We can construct the 95% confidence interval for the true proportion. To do this, we need to calculate the margin of error, which is equal to the critical value (1.96) multiplied by the standard error (0.014). This equals 0.028.

The 95% confidence interval is then the sample proportion (0.38) plus or minus the margin of error (0.028). This is [tex](0.38 - 0.028, 0.38 + 0.028) = (0.352, 0.408).[/tex]

The test statistic in this case is the Z-statistic, as we are assuming that the underlying population is normally distributed. To conduct the hypothesis test, we must first state the null and alternative hypotheses.

Null Hypothesis (H0): The proportion of students at the school who fear public speaking is equal to or greater than 40%.
Alternative Hypothesis (H1): The proportion of students at the school who fear public speaking is less than 40%.

We must then calculate the test statistic, which is the Z-statistic in this case. To do this, we need to first calculate the sample proportion, which is the number of students who fear public speaking (137) divided by the total number of students surveyed (361). This equals 0.38. We then need to calculate the standard error of the sample proportion (SE), which is the square root of [tex](pq/n)[/tex], where p is the sample proportion (0.38) and q is the complement of the sample proportion (1-0.38 = 0.62). SE = [tex](0.38 x 0.62)/361 = 0.014.[/tex] The Z-statistic is then calculated as the difference between the sample proportion (0.38) and the population proportion (0.40) divided by the standard error [tex](0.014). Z = (0.38 – 0.40)/0.014 = -0.14.[/tex]

To conclude, we can use the Z-statistic and 95% confidence interval to test the hypothesis that the proportion of students at the school who fear public speaking is less than 40%. The Z-statistic of -0.14 falls within the critical region and the 95% confidence interval does not include 0.40, suggesting that the proportion of students at the school who fear public speaking is indeed less than 40%.

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On solving the question we can say that Therefore, the solutions to the inequality given inequality are: x < 4 or x > 6.

An inequality in mathematics is a relationship between two expressions or values ​​that are not equal. Imbalance therefore leads to inequality. An inequality establishes a connection between two values ​​that are not equal in mathematics. Equality is different from inequality. The inequality sign () is most commonly used when two values ​​are not equal. Various inequalities are used to contrast values, no matter how small or large. Many simple inequalities can be solved by changing both sides until only variables remain. But many things contribute to inequality.

two inequalities

4x - 6 < 10

4x < 16

x < 4

2x - 4 > 8

2x > 12

x > 6

Therefore, the solutions to the given inequality are:

x < 4 or x > 6.

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Answer:

9

Step-by-step explanation:

To find the answer to this question, we have to find 3/4 of 12

To do this, we need to do 12 divide by 4, then that answer multiplied by 3...

This means that she took 9 drinks!

Hope this helps, have a lovely day! :)

Answer:

9

Step-by-step explanation:

[tex] multiply \: \\ = \frac{3}{4} \times 12[/tex]

[tex] = 9[/tex]

Therefore, sally took 9 drink for her party.

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Answer:

Correct choices

[tex]\csc (\theta) = - \dfrac{5}{3} \quad \quad \text{2nd option}\\\\\cot(\theta) = \dfrac{4}{3} \quad \quad \text{3rd option}\\\\\cos(\theta) = -\dfrac{4}{5} \quad \quad \text{4th option}\\\\[/tex]

Step-by-step explanation:

[tex]\text{If \;$ \tan\theta = \dfrac{3}{4} $}} \\\\\text{then }\\\theta = \tan^{-1} \left(\dfrac{3}{4}\right)\\\\= 36.87^\circ \text{ in Q1}\\[/tex]

But since tan θ is periodic it will also be 3/4 in Q3 which is 180° + 36.87 = 216.87°

The equation in standard form for the circle with center (5,0) passing through (5, 9/2) is 4x² + 4y² - 40x + 19 = 0

Given that

Center = (5, 0)

Point on the circle = (5. 9/2)

The equation of a circle can be expressed as

(x - a)² + (y - b)² = r²

Where

Center = (a, b)

Radius = r

So, we have

(x - 5)² + (y - 0)² = r²

Calculating the radius, we have

(5 - 5)² + (9/2 - 0)² = r²

Evaluate

r = 9/2

So, we have

(x - 5)² + (y - 0)² = (9/2)²

Expand

x² - 10x + 25 + y² = 81/4

Multiply through by 4

4x² - 40x + 100 + 4y² = 81

So, we have

4x² + 4y² - 40x + 19 = 0

Hence, the equation is 4x² + 4y² - 40x + 19 = 0

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The correct answer is A. a-c+b-d=0. This is because when two sets of numbers are both negative, the result of subtracting the larger number from the smaller number will always be negative.

Subtraction involves taking one number or value away from another. It is one of the four basic operations in mathematics, along with addition, multiplication, and division.

When subtracting a from c and b from d, the result of either subtraction will always be a negative number. When the two negative numbers are added together, the result will always be 0.

The other options are not always true. In option B, ac > bd, this is not always true because when a, b, c, and d are all negative, it is possible for the result of ac to be less than the result of bd. In option C, a+c>b+d, this is not always true because when both sets of numbers are negative, it is possible for the result of a+c to be less than b+d. Finally, in option D, a/d < b/c, this is not always true because when both sets of numbers are negative, it is possible for the result of a/d to be greater than b/c.

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The correct answer is A. a-c+b-d=0 because the expression is always true when a, b, c and d are all less than zero.

Expression is a combination of symbols and operators that evaluate to a single value. It could be a mathematical equation, an arithmetic expression, a logical expression, or a combination of these.

This is because the expression is equivalent to (a-c)+(b-d)=0, which is always true when a, b, c and d are all less than zero.

This can be proven through a simple calculation.

Let us assume that the values of a, b, c and d are -1, -3, 8 and -4 respectively.

Substituting these values into the expression gives us

(-1-3)+(8-4)=0, which is clearly true.

Therefore, A. a-c+b-d=0 is the correct answer as the expression is always true when a, b, c and d are all less than zero.

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a) The perimeter of square is 16 cm

b) The perimeter of rectangle is 18 cm

c) The perimeter of triangle is 27 cm

a) The perimeter of a square with side length 4 cm can be found by adding the length of all four sides. Since all sides of a square are equal, the perimeter is 4 times the length of a side. Therefore, the perimeter of a square of side 4 cm is:

Perimeter = 4 x 4 cm = 16 cm

b) The perimeter of a rectangle with length 5 cm and breadth 4 cm can be found by adding twice the length and twice the breadth of the rectangle. Therefore, the perimeter of a rectangle of length 5 cm and breadth 4 cm is:

Perimeter = 2 x (length + breadth)

Perimeter = 2 x (5 cm + 4 cm)

Perimeter = 2 x 9 cm

Perimeter = 18 cm

c) The perimeter of a triangle with sides 11 cm, 7 cm, and 9 cm can be found by adding the length of all three sides. Therefore, the perimeter of a triangle with sides 11 cm, 7 cm, and 9 cm is:

Perimeter = 11 cm + 7 cm + 9 cm

Perimeter = 27 cm

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The answers are 24.50, 8.50, 23.50 respectively.

Given that we are to forecast for week 3, find the error for week 4, and make a final prediction for week 7. We are to use the moving average method with k = 2.The calculation of the moving average is shown belowWeek Time Series Moving average Error 1 30 _____ _____ 2 19 _____ 5.5 3 30 24.50 -6.50 4 16 24.50 8.50 5 21 23.00 -2.00 6 25 18.50 6.50 7 Prediction -> 23.50 -2.50The forecast for week 3 is 24.50, error for week 4 is 8.50 and final prediction for week 7 is 23.50. Thus, the answers are 24.50, 8.50, 23.50 respectively.

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Eventually, the differential equation's general solution is:

y = y_h + y_p

y = c1e^(-2x) + c2e^(-x) - (1/6)e^(-x) + (2/3)

In the context of differential equations, the homogeneous solution of a differential equation is a solution that satisfies the equation when the right-hand side is equal to zero.

To find the particular solution of y'' + 3y' + 2y = 4e^x using the variation of parameters method, we first find the homogeneous solution of the differential equation by setting the right-hand side to zero:

y'' + 3y' + 2y = 0

The characteristic equation is r^2 + 3r + 2 = 0, which factors as (r + 2)(r + 1) = 0. Therefore, the solutions are y_h = c1e^(-2x) + c2e^(-x), where c1 and c2 are constants.

Next, we find the Wronskian of the homogeneous solution:

W(y1, y2) = |e^(-2x) e^(-x) | = e^(-3x)

To find the particular solution, we assume that it has the form y_p = u1(x)e^(-2x) + u2(x)e^(-x), where u1(x) and u2(x) are unknown functions to be determined.

We then find y_p' and y_p'':

[tex]y_p' = u1'(x)e^(-2x) + u2'(x)e^(-x) - 2u1(x)e^(-2x) - u2(x)e^(-x)y_p'' = u1''(x)e^(-2x) + u2''(x)e^(-x) - 4u1'(x)e^(-2x) - 2u2'(x)e^(-x) + 4u1(x)e^(-2x) + u2(x)e^(-x)u1''(x)e^(-2x) + u2''(x)e^(-x) + u1'(x)e^(-2x) + u2'(x)e^(-x) - 4u1'(x)e^(-2x) - 2u2'(x)e^(-x) + 4u1(x)e^(-2x) + u2(x)e^(-x) = 4e^x[/tex]

Simplifying and grouping terms, we get:

[tex]u1''(x)e^(-2x) - 3u1'(x)e^(-2x) + u2''(x)e^(-x) - u2'(x)e^(-x) = 4e^x[/tex]

To solve for u1(x) and u2(x), we use the method of undetermined coefficients and assume that they are both linear combinations of the exponential function and its derivative:

u1(x) = A(x)e^x

u2(x) = B(x)e^(2x)

Substituting these expressions into the previous equation and solving for A(x) and B(x), we get:

A(x) = -e^x/6

B(x) = 2e^x/3

Therefore, the particular solution is:

[tex]y_p = (-e^x/6)e^(-2x) + (2e^x/3)e^(-x)y_p = (-1/6)e^(-x) + (2/3)[/tex]

Eventually, the differential equation's general solution is:

y = y_h + y_p

y = c1e^(-2x) + c2e^(-x) - (1/6)e^(-x) + (2/3)

Therefore, the particular solution of the given differential equation y′′+3y′+2y=4ex is

[tex]y(x)=c_1e^{-x} + c_2e^{-2x} - 4 + 2e^{x}.[/tex]

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The value of the unknown lengths in these similar triangles is FH is 6.67 units and EG is 27 units.

A triangle is a polygon with three sides and three angles. It is a two-dimensional shape that is commonly studied in mathematics, geometry, and other fields. The sum of the angles in a triangle is always 180 degrees, and the lengths of the sides can vary. Triangles can be classified based on the lengths of their sides and the measures of their angles. Common types of triangles include equilateral, isosceles, scalene, acute, right, and obtuse triangles. Triangles have many important properties and are used in various applications, including construction, engineering, and physics.

Here,

1. Let x be the length of FH. We have:

AB/EF = BD/FH

12/8 = 10/x

Cross-multiplying, we get:

12x = 80

x = 80/12

x ≈ 6.67

Therefore, FH ≈ 6.67.

2. Let y be the length of EG. We have:

AC/BD = FH/EG

15/9 = 5/y

Cross-multiplying, we get:

5y = 135

y = 135/5

y ≈ 27

Therefore, EG ≈ 27.

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