Two dice are thrown together. Let A be the event ''getting 6 on the first die'' and B be the event '' getting 2 on the second die'' Are the event A and B independent?

The polynomial function with the given roots 3+i, 2, and -4 is f(x) = (x - (3+i))(x - 2)(x + 4).

To find a polynomial function with the given roots, we use the concept of zero-product property. The roots are 3+i, 2, and -4. To construct the polynomial, we form factors for each root: (x - (3+i)), (x - 2), and (x + 4). Since complex roots occur in conjugate pairs, we write (x - (3+i)) as (x - 3 - i).

Multiplying these factors together, we obtain f(x) = (x - 3 - i)(x - 2)(x + 4). To simplify the expression, we can expand the polynomial by distributing and combining like terms.

In this case, expanding the polynomial function is not necessary since we have achieved the desired form with factors. The polynomial f(x) = (x - (3+i))(x - 2)(x + 4) represents a function whose roots are 3+i, 2, and -4.

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The expected gain made by each policy sold is $1.10. The variance of the probability distribution is $101.29. X represents the random variable denoting the expected returns under death or no death scenarios.


To calculate the expected gain, we multiply the probability of death (.001) by the payout ($10,000) and subtract the annual premium ($290). Thus, the expected gain is (0.001 * $10,000) - $290 = $1.10.

To calculate the variance, we need to calculate the squared difference between the actual gain in each scenario (death or no death) and the expected gain. In the death scenario, the gain is $10,000 - $290 = $9,710, and in the no-death scenario, the gain is -$290 (due to the premium payment). The squared differences are (9,710 – 1.10)^2 = 94,413.69 and (-290 – 1.10)^2 = 7,981.69, respectively.

Finally, we multiply each squared difference by its corresponding probability (0.001 for death, 0.999 for no death) and sum them up to obtain the variance. The variance is (0.001 * 94,413.69) + (0.999 * 7,981.69) = 101.29.

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The transformation that could map one triangle to the other is a reflection.

A reflection is a transformation that flips an object over a line, called the line of reflection.

When the parallelogram is folded along its diagonal, the line of reflection is the diagonal itself.

By folding the parallelogram, one triangle is reflected onto the other, resulting in congruent triangles.

The line of reflection serves as the mirror, reflecting the shape of one triangle onto the other.

To visualize this, imagine folding a sheet of paper in such a way that one vertex of the parallelogram coincides with another vertex.

The folded side represents the line of reflection, and the two resulting triangles are congruent due to the reflection.

It's important to note that translation, rotation, and a combination of reflection and translation are not applicable in this case.

Translation involves sliding an object without any flipping or rotating, so it cannot transform one triangle into the other.

Rotation also involves turning an object around a fixed point, and the reflection line is not a fixed point.

Finally, a combination of reflection and translation would imply both a flip and a slide, which is not possible in this scenario.

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Answer:

its option D. rotation and translation

Step-by-step explanation:

The solutions to the equation 0.5x² = 15 are x ≈ 5.48 and x ≈ -5.48.

To solve the equation 0.5x² = 15, we need to isolate x. We can do this by performing algebraic operations on both sides of the equation.

First, let's multiply both sides by 2 to eliminate the coefficient of 0.5:

2 * (0.5x²) = 2 * 15

This simplifies to:

x² = 30

Now, to solve for x, we take the square root of both sides of the equation:

√(x²) = √30

Since we're taking the square root, we have to consider both the positive and negative roots:

x = ±√30

Therefore, the solutions to the equation 0.5x² = 15 are x = √30 and x = -√30. This means that there are two possible values for x that satisfy the equation.

In decimal form, the approximate values for the square root of 30 are:

√30 ≈ 5.48

Thus, the solutions to the equation 0.5x² = 15 are x ≈ 5.48 and x ≈ -5.48.

These are the values of x that make the equation true when substituted back into the original equation 0.5x² = 15.

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The probability of selecting a red marble first and then a blue marble, without replacement, is 4/15.

To find the probability of selecting a red marble first and then a blue marble without replacement, we need to consider the outcomes of both selections.

Given that the jar contains four blue marbles and two red marbles, let's calculate the probabilities for each event:

Event 1: Selecting a red marble

The probability of selecting a red marble on the first draw is given by:

P(red) = Number of red marbles / Total number of marbles

P(red) = 2 / 6 = 1/3

After the first draw, there are now five marbles left in the jar, with one red marble remaining.

Event 2: Selecting a blue marble

The probability of selecting a blue marble on the second draw, without replacement, is given by:

P(blue) = Number of blue marbles / Total number of marbles after the first draw

P(blue) = 4 / 5 = 4/5

To find the probability of both events occurring (selecting a red marble first and then a blue marble), we multiply the individual probabilities:

P(red and then blue) = P(red) * P(blue)

P(red and then blue) = (1/3) * (4/5)

P(red and then blue) = 4/15

Therefore, the probability of selecting a red marble first and then a blue marble, without replacement, is 4/15.

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The waiter earned a total of $180.

Here, we have,

To find the total amount of money the waiter earned, we need to calculate the sum of his base salary and the tips he received.

Given information:

Base salary per hour: $2.50

Tip amount: $150

Number of hours worked: 12

First, let's calculate the base salary earned by multiplying the hourly rate by the number of hours worked:

Base salary = $2.50/hour * 12 hours = $30

Next, we can calculate the total amount earned by adding the base salary to the tip amount:

Total amount earned = Base salary + Tip amount = $30 + $150 = $180

Therefore, the waiter earned a total of $180.

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The simplified value of (7√²)√² is 14.

To simplify the expression (7√²)√² using the properties of irrational exponents, we can apply the following rules:

Rule 1: [tex](a^m)^n = a^{(m*n)[/tex]

Rule 2: √a² = |a|

Using Rule 1, we can rewrite (7√²)√² as 7(√²)².

Then, applying Rule 2, we simplify (√²)² as |√²|.

The square root of a square (√a^2) always results in the absolute value of a.

In this case, √² equals |2|, which is simply 2.

Substituting this value back into the expression,

7(√²)² = 7(2) = 14.

Therefore, the simplified form of (7√²)√² is 14.

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The correct answer is A. 15.203%.The nominal rate of interest convertible monthly, at which the accumulated value of $1000 at the end of 11 years is $4000, can be determined using the formula for compound interest.

To find the nominal rate of interest convertible monthly, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

where A is the accumulated value, P is the principal amount, r is the nominal interest rate, n is the number of compounding periods per year, and t is the number of years.

In this case, we have:

A = $4000

P = $1000

n = 12 (since it is compounded monthly)

t = 11

Substituting these values into the formula, we get:

$4000 = $1000(1 + r/12)^(12*11)

To solve for r, we need to isolate it in the equation. However, this involves a complex calculation that cannot be easily solved algebraically. Therefore, we can use numerical methods or financial calculators to find the value of r.

Using these methods, we find that the nominal rate of interest convertible monthly is approximately 15.203% (rounded to three decimal places). Therefore, option A is the correct answer.

It's important to note that in real-world scenarios, interest rates are typically expressed as annual rates. However, the question specifically asks for the nominal rate of interest convertible monthly, which is why the answer is given as a monthly percentage.

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To find the mean, median, and mode of the prices of the flat-screen TVs, we can use the given set of prices: 1200, 999, 1499, 895, 695, 1100, 1300, and 695.

1. Mean: The mean is obtained by summing up all the prices and dividing by the total number of prices.

Sum of prices = 1200 + 999 + 1499 + 895 + 695 + 1100 + 1300 + 695 = 7383 Mean = 7383 / 8 = 922.875

2. Median: The median is the middle value when the prices are arranged in ascending or descending order.  Arranging the prices in ascending order: 695, 695, 895, 999, 1100, 1200, 1300, 149 The median is the average of the two middle values, which are 999 and 1100. Median = (999 + 1100) / 2 = 1099.5

3. Mode: The mode is the value that appears most frequently in the set of prices.  In this case, the mode is 695, as it appears twice, more than any other value. Therefore, the mean is 922.875, the median is 1099.5, and the mode is 695 for the prices of the flat-screen TVs.

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The congruence of triangles is reflexive, we need to show that any triangle is congruent to itself is proved. This establishes the reflexive property of congruence of triangles.

To prove that congruence of triangles is reflexive, we need to show that any triangle is congruent to itself.

Proof: Consider triangle ABC.

We need to prove that triangle ABC is congruent to triangle ABC.

By definition, two triangles are congruent if their corresponding sides and angles are equal.

In triangle ABC, all sides and angles are equal to themselves.

Side AB is equal to side AB.

Side BC is equal to side BC.

Side CA is equal to side CA.

Angle ABC is equal to angle ABC.

. Angle BCA is equal to angle BCA.

Angle CAB is equal to angle CAB.

Therefore, all corresponding sides and angles of triangle ABC are equal to themselves.

Hence, triangle ABC is congruent to itself.

This proves that congruence of triangles is reflexive.

In this flow proof, we start by considering triangle ABC and aim to prove that it is congruent to itself. We use the definition of congruence, which states that two triangles are congruent if their corresponding sides and angles are equal.

Since all sides and angles of triangle ABC are equal to themselves, we conclude that triangle ABC is congruent to itself. This establishes the reflexive property of congruence of triangles.

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It should take approximately 19.8 hours for a population of 8000 bacteria to shrink to 500 when the rate of decline is 0.14.

To determine how long it should take for a population of 8000 bacteria to shrink to 500 using the given formula H = 1/r (ln P - ln A), we need to substitute the values into the equation.

Given:

r = 0.14 (rate of decline)

P = 8000 (initial bacteria population)

A = 500 (reduced bacteria population)

Substituting these values into the equation, we have:

H = 1/0.14 (ln 8000 - ln 500)

Now we can simplify the expression:

H = 1/0.14 (ln (8000/500))

H = 1/0.14 (ln 16)

H = 1/0.14 (2.7725887)

H ≈ 19.8

Therefore, it should take approximately 19.8 hours for a population of 8000 bacteria to shrink to 500 when the rate of decline is 0.14.

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(a) The 90% confidence interval for the mean weight of Allens Party Mix Lollies,is approximately 180.38 grams to 182.02 grams.(b) Convenience sampling was used in this scenario, where inspectors conveniently selected 30 packets of Allens Party Mix Lollies from a shop. (c) It is not possible to retain 90% confidence while reducing the margin of error in the confidence interval. (d)To estimate the mean weight of Allens Party Mix Lollies within ±1 gram with 90% confidence, a sample size of approximately 55 packets would be required.(e) The probability that the mean weight exceeds 182 grams is approximately 25.14%.

(a)To construct the confidence interval, we use the formula for the confidence interval, which incorporates the sample mean, standard deviation, and critical value. By calculating the values, we can determine a range within which we are 90% confident that the true mean weight of the packets falls.

(b)Convenience sampling involves selecting items or individuals that are readily available and easily accessible. While it can be quick and convenient, it may lead to a non-representative sample. In this case, a random or systematic sampling method would have been a better choice to ensure a more representative sample.

(c) The margin of error in a confidence interval is influenced by the standard deviation, critical value, and sample size. To reduce the margin of error and have a narrower confidence interval, a larger sample size would be required. This trade-off between confidence level, margin of error, and sample size is inherent in statistical analysis.

(d): To determine the required sample size, we use the formula for sample size calculation, which takes into account the desired margin of error, standard deviation, and confidence level. By plugging in the given values, we can calculate the sample size needed to achieve the desired level of confidence.

(e) By assuming a normal distribution for the sample mean, we can calculate the Z-score and use a Z-table or calculator to find the probability that the mean weight exceeds a certain value. In this case, we find the probability that the Z-score is greater than 0.67, indicating the likelihood of the mean weight exceeding 182 grams.

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To determine the number of miles David drove the truck, we need to subtract the base fee and divide the remaining amount by the additional charge per mile.

Let's denote the number of miles driven by 'm'. The equation can be set up as follows:

$209.57 - $14.99 = $0.94 * m

Simplifying the equation:

$194.58 = $0.94 * m

To solve for 'm', we divide both sides of the equation by $0.94:

m = $194.58 / $0.94

m ≈ 206.7

Therefore, David drove the truck for approximately 206.7 miles. Since it's not possible to drive a fraction of a mile, we can assume that David drove either 206 or 207 miles, depending on the rounding convention used.

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The possible values for (\overline{ab}) are: 12, 13, 21, 23, 31, 32.

And the possible values for (\overline{cd}) are: 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34.

Given the digits 1, 2, 3, 1, 2, 3, and 44 arranged to form two two-digit numbers ((\overline{ab}) and (\overline{cd})), we need to determine the possible values for (\overline{ab}) and (\overline{cd}).

To find the possible values, we need to consider the given digits and their arrangement.

We have the following digits: 1, 2, 3, 1, 2, 3, and 44.

Since we are forming two two-digit numbers, (\overline{ab}) and (\overline{cd}), we can assign the digits in the following way:

(\overline{ab}): The tens digit is represented by a, and the ones digit is represented by b.

(\overline{cd}): The tens digit is represented by c, and the ones digit is represented by d.

To find the possible values for (\overline{ab}) and (\overline{cd}), we need to consider the available digits and their arrangement.

From the given digits, we have 1, 2, 3, 1, 2, 3, and 44.

To form two two-digit numbers, we need to select the appropriate digits for each place value.

The tens digit for (\overline{ab}) (represented by a) can be chosen from {1, 2, 3}.

The ones digit for (\overline{ab}) (represented by b) can also be chosen from {1, 2, 3}.

Similarly, the tens digit for (\overline{cd}) (represented by c) can also be chosen from {1, 2, 3}.

The ones digit for (\overline{cd}) (represented by d) can be chosen from {1, 2, 3, 4}.

Since the problem states that the numbers are randomly arranged, we need to consider all possible combinations of digits.

Now, let's determine the possible values for (\overline{ab}) and (\overline{cd}):

Possible values for (\overline{ab}):

(\overline{12}), (\overline{13}), (\overline{21}), (\overline{23}), (\overline{31}), (\overline{32})

Possible values for (\overline{cd}):

(\overline{11}), (\overline{12}), (\overline{13}), (\overline{14}), (\overline{21}), (\overline{22}), (\overline{23}), (\overline{24}), (\overline{31}), (\overline{32}), (\overline{33}), (\overline{34})

Please note that the number 44 counts as a single digit since it represents a two-digit number itself.

Therefore, the possible values for (\overline{ab}) are: 12, 13, 21, 23, 31, 32.

And the possible values for (\overline{cd}) are: 11, 12, 13, 14, 21, 22, 23, 24, 31, 32, 33, 34.

These are the possible combinations of two two-digit numbers that can be formed using the given digits.

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The forecasted demand for the week of October 12, based on the 3-week moving average, is 16 pints. This estimate provides an approximation of the expected demand for type A blood at Damascus Hospital during that week, considering recent trends in usage.

To calculate the forecasted demand for the week of October 12, we employ a 3-week moving average. This approach involves taking the average of the number of pints used over the previous three weeks.

Let's refer to the table provided to determine the moving average:

Week Number of Pints

Sept 1 15

Sept 8 18

Sept 15 12

Sept 22 20

Sept 29 16

Oct 6 14

To calculate the moving average, we sum the number of pints used over the three most recent weeks and divide it by 3:

Moving Average = (12 + 20 + 16) / 3 = 16

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To draw the image of quadrilateral ABCD under a translation by 1 unit to the right and 4 units up, we will move each point of the quadrilateral in the specified direction.

Let's assume the coordinates of the original quadrilateral ABCD are as follows:

A(x₁, y₁), B(x₂, y₂), C(x₃, y₃), D(x₄, y₄)

To perform the translation, we will add the given values to the x-coordinates and y-coordinates of each point:

A'(x₁ + 1, y₁ + 4), B'(x₂ + 1, y₂ + 4), C'(x₃ + 1, y₃ + 4), D'(x₄ + 1, y₄ + 4)

Now, plot the original quadrilateral ABCD and then move each point to its corresponding new position.

For example, if point A had coordinates (2, 3), after the translation, it will move to (2 + 1, 3 + 4) = (3, 7). Similarly, you can calculate the new coordinates for points B, C, and D using the same process.

Once you have the new coordinates for each point, connect them to form the image of the quadrilateral ABCD under the translation.

The new quadrilateral A'B'C'D' will be a shifted version of the original quadrilateral, 1 unit to the right and 4 units up.

It's important to note that the scale and proportions of the quadrilateral will remain the same after the translation. Only its position in the coordinate plane will change.

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The L1 norm of vector A is greater than or equal to the L1 norm of vector B.

Basically, if the L2 norm of vector A is greater than the L2 norm of vector B, it is indeed always true that the L1 norm of vector A is greater than or equal to the L1 norm of vector B. The Lp norm is defined as follows:

[tex]||x||_p = (|x_1|^p + |x_2|^p + ... + |x_n|^p)^(1/p),[/tex]

where x = [x₁, x₂, ..., xₙ] is a vector.

For the L2 norm (p = 2), the formula is:

[tex]||x||_2 = \sqrt(|x_1|^2 + |x_2|^2 + ... + |x_n|^2).[/tex]

For the L1 norm (p = 1), the formula is:

[tex]||x||₁ = |x_1| + |x_2| + ... + |x_n|.[/tex]

If ||A||₂ > ||B||₂, it implies that:

[tex]\sqrt(|A_1|^2 + |A_2|^2 + ... + |A_n|^2) > \sqrt(|B_1|^2 + |B_2|^2 + ... + |B_n|^2).[/tex]

Squaring both sides of the inequality, we get:

[tex]|A_1|^2 + |A_2|^2 + ... + |A_n|^2 > |B_1|^2 + |B_2|^2 + ... + |B_n|^2.[/tex]

Since the squares of the magnitudes are positive, we can conclude that:

[tex]|A_1| + |A_2| + ... + |A_n| > |B_1| + |B_2| + ... + |B_n|.[/tex]

Therefore, the L1 norm of vector A is greater than or equal to the L1 norm of vector B.

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The value of 9i(13i) simplifies to -117.

The correct option is (F).

To simplify the expression 9i(13i), we can apply the rules of complex number multiplication:

9i(13i) = 9i x 13i

Using the property i² = -1, we can simplify further:

9i x 13i = 9 x (-1) * 13 = -117

So the expression 9i(13i) simplifies to -117.

Therefore, the value of 9i(13i) simplifies to -117.

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The best way to remedy Ask customers throughout the day on both weekdays and weekends.

The cause of the bias in the survey is likely sampling bias. This means that the sample of 110 customers may not be representative of the entire population of food court shoppers. The bias can arise if the sample is not selected randomly or if certain groups of shoppers are overrepresented or underrepresented in the sample.

To remedy the sampling bias, the best way would be to  Ask customers throughout the day on both weekdays and weekends.

By surveying customers at different times, including both weekdays and weekends, a more diverse and representative sample can be obtained. This will help to capture the preferences of a wider range of food court shoppers and reduce the bias introduced by sampling only weekday mornings.

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The manager of a shopping mall wishes to expand the number of shops available in the food court. He has a market researcher survey the first 110 customers who come into the food court during weekday mamings in distin what types of food the shoppers would like to see added to the food court,

Which of the following is the best way to remedy this problem?

OA cisne the sample size so that more people respond to the question OB Ask customers throughout the day on both weekdays and hands

OC Reward the question so that it is balanced

Answer:

Step-by-step explanation:

The square root of a number cannot be negative. So

     [tex]x-1\geq 0[/tex]

           [tex]x\geq 1[/tex]

So the solution is

     [tex]1\leq x\leq[/tex] ∞

A. It is not possible to design a triangular prism mailer that can accommodate a poster with a maximum rolled diameter of 6 inches and a length of almost 38 inches.

B. A triangular prism has two triangular bases and three rectangular faces connecting them.

In order to design a mailer that can hold a poster with a maximum rolled diameter of 6 inches, the triangular bases of the prism would need to have a larger diameter than 6 inches.

However, since a triangular prism has two triangular bases, it is not possible for the prism to have a circular cross-section that would accommodate a rolled poster with a diameter larger than its own base dimensions.

The triangular prism's net, which is a two-dimensional representation that can be folded to create a three-dimensional shape, would consist of two triangles and three rectangles.

However, regardless of how these shapes are arranged or folded, it is not possible to create a triangular prism mailer that can hold a poster with a maximum rolled diameter of 6 inches and a length of almost 38 inches.

Different packaging solutions or shapes would need to be considered to meet the requirements of the poster's dimensions.

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The polynomial -2x³ - 7x⁴ + x³ can be written in standard form as -7x⁴ - x³ - 2x³. It is a 4th-degree polynomial and has three terms.

To write the polynomial -2x³ - 7x⁴ + x³ in standard form, we rearrange the terms in descending order of the degree of the variable. Doing so, we get -7x⁴ - x³ - 2x³.

The highest degree of the variable, x, in the polynomial is 4, making it a 4th-degree polynomial.

The number of terms in the polynomial is determined by counting the separate algebraic expressions separated by addition or subtraction signs. In this case, we have three terms: -7x⁴, -x³, and -2x³.

Therefore, the polynomial -2x³ - 7x⁴ + x³ can be classified as a 4th-degree polynomial with three terms.

In summary, the given polynomial -2x³ - 7x⁴ + x³ is written in standard form as -7x⁴ - x³ - 2x³. It is a 4th-degree polynomial with three terms.

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The question asks to find the value of Q U given that Q T is equal to 11 meters. However, without any context or additional information provided, it is not possible to determine the specific meaning or calculation of Q U.

The variables Q T and Q U could represent any quantities or variables in a given scenario, and their relationship or formula is not specified.To accurately determine the value of Q U, it is essential to have more information about the context or the relationship between Q T and Q U.

Without such information, it is not possible to generate a meaningful answer or perform any calculations. Therefore, the value of Q U remains unknown until further clarification or context is provided.

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Answer:

A

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given slope m = 3 , then

y = 3x + c ← is the partial equation

to find c substitute (- 3, - 5 ) into the partial equation

- 5 = 3(- 3) + c = - 9 + c ( add 9 to both sides )

4 = c

y = 3x + 4 ← equation of line

The equation is:

y = 3x + 4

Work/explanation:

First, we will write the equation in point slope:

[tex]\Large\pmb{y-y_1=m(x-x_1)}[/tex]

where m = slope;

(x₁, y₁) is a point on the line

Plug in the data:

[tex]\large\begin{gathered}\sf{y-(-5)=3(x-(-3)}\\\sf{y+5=3(x+3)}\\\sf{y+5=3x+9}\\\sf{y=3x+9-5}\\\sf{y=3x+4}\end{gathered}[/tex]

Hence, the equation is y = 3x + 4.

The period of the periodic function y = -3sin(x) is 2π/3 radians or 120 degrees. A suitable window for graphing the function is Xmin = 0, Xmax = 4π/3, Ymin = -3, Ymax = 3.

The given function is y = -3sin(x). To identify the period of the function, we need to determine the values of x for which sin(x) repeats itself. Since the sine function has a period of 2π, the period of y = -3sin(x) is:

T = 2π / |(-3)| = 2π / 3

Therefore, the period of the function is 2π/3 radians or 120 degrees.

To evaluate the function at 90 degrees, we need to convert 90 degrees to radians:

90° = π/2 radians

Substituting this value into the function, we get:

y = -3sin(π/2) = -3(1) = -3

Therefore, the value of the function at 90 degrees is -3.

To plot the graph of the function and verify the period, we can use a window of:

Xmin = 0

Xmax = 4π/3

Ymin = -3

Ymax = 3

This window covers one and a third cycles of the function, which is enough to verify the period.

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When two lines intersects in a plane, we get Vertical Angles Linear Pair Adjacent Angles

Vertical Angles: The angles opposite each other at the intersection point are called vertical angles.

Linear Pair: When two adjacent angles are supplementary, i.e. the sum of their adjacent angle is [tex]180^0[/tex] Is called a linear pair. In other words, if angle C and angle D are a linear pair, then the sum of the measures of angle C and angle D is 180 degrees.

Adjacent Angles: Any two angles or rays  having a common arm between them is called an adjacent angle.  Adjacent angles are formed by the intersection of two lines Adjacent angles do not have a specific relationship in terms of their measures unless they are vertical angles or form a linear pair.

When two lines intersects in a plane, we get Vertical Angles Linear Pair Adjacent Angles.

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The simplified form is 5x√3.

To simplify the expression in (a) by multiplying by √3 instead of √75:

Let's assume the expression in (a) is represented by the variable x.

So, we get,

x * √75

To simplify this, we can rewrite this √75 into this √(25 * 3) because 25 is a perfect square:

x * √(25 * 3)

Using the properties of square roots, we can  separate the square root into two separate square roots as follows:

x * (√25 * √3)

Since √25 is equal to 5, we can simplify this equation further:

x * (5 * √3)

Finally, when we can multiply 5 and √3 together we get,

x * 5√3

Therefore, the simplified expression by multiplying by √3 instead of √75 is 5x√3.

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Leg-Leg Congruence (LL):** If the legs of two right triangles are congruent, then the triangles are congruent.

Hypotenuse-Leg Congruence (HL):** If the hypotenuse and a leg of two right triangles are congruent, then the triangles are congruent.

The LL congruence rule is based on the fact that the legs of a right triangle are opposite acute angles, and since all acute angles in a right triangle are congruent, then the corresponding legs of two congruent right triangles must also be congruent.

The HL congruence rule is based on the Pythagorean Theorem. If the hypotenuse and a leg of two right triangles are congruent, then the other leg of each triangle must also be congruent, since the Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Note that we can omit the A for any right angle since we know that all right triangles contain a right angle and all right angles are congruent.

Here are some examples of how to use the LL and HL congruence rules:

* If the legs of two right triangles are 3 cm and 4 cm, respectively, then the triangles are congruent by the LL congruence rule.

* If the hypotenuse and a leg of two right triangles are 5 cm and 3 cm, respectively, then the triangles are congruent by the HL congruence rule.

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The second equation, "b² - c² = a²," is equivalent to the first equation, "a² = b² - c²."

In the given statement, "a² = b² - c²," we can rearrange it to "b² - c² = a²" using the symmetric property of equality.

The property that justifies the statement is the symmetric property of equality.

According to this property, if two expressions or equations are equal, then they can be written in reverse order and still be equal.

This property allows us to switch the positions of the terms on either side of the equation without changing its validity.

Therefore, the second equation, "b² - c² = a²," is equivalent to the first equation, "a² = b² - c²."

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The solutions to the equation x² - 3x - 8 = 0 are:

x = (3 + √41)/2

x = (3 - √41)/2

To solve the equation x² - 3x - 8 = 0 by completing the square, follow these steps:

Step 1: Move the constant term to the other side of the equation:

x² - 3x = 8

Step 2: Take half of the coefficient of the x-term (-3), square it, and add it to both sides of the equation:

x² - 3x + (3/2)² = 8 + (3/2)²

x² - 3x + 9/4 = 8 + 9/4

Step 3: Simplify the right side of the equation:

x² - 3x + 9/4 = 32/4 + 9/4

x² - 3x + 9/4 = 41/4

Step 4: Rewrite the left side of the equation as a perfect square:

(x - 3/2)² = 41/4

Step 5: Take the square root of both sides:

√[(x - 3/2)²] = ±√(41/4)

Step 6: Solve for x:

x - 3/2 = ±√(41/4)

x = 3/2 ± √(41/4)

Simplifying further, we have:

x = (3 ± √41)/2

Therefore, the solutions to the equation x² - 3x - 8 = 0 are:

x = (3 + √41)/2

x = (3 - √41)/2

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