How can I use the compound angle formula to write sin 165degrees as surds?

The solutions of the system of equations are option (A) (3, 7) and (-3, -5)

To solve the system of equations, use the concept of substitution, which involves solving one equation for one variable and then substituting that expression into the other equation to eliminate one variable and solve for the other variable.

In this case, solve equation (1) for y in terms of x and substitute that expression into equation (2), which allowed us to solve for x. Then we used the values of x to find the corresponding values of y.

Here we have

y = 2x + 1 --- (1)

y = x²+ 2x -8 --- (2)

From (1) and (2)

=>  x²+ 2x - 8 = 2x + 1

Subtract 2x + 1 from both sides

=>  x²+ 2x - 8 - 2x - 1 =  2x + 1 - 2x - 1  

=> x² - 9 = 0

Now add 9 on both sides

=> x² - 9 = 0 + 9

=> x² = 9

=> x = √9  

=> x = ± 3

From (1)

At x = 3

=> y = 2(3) + 1 = 7

At x = - 3

=> y = 2(-3) + 1 = - 5

Therefore,

The solutions of the system of equations are option (A) (3, 7) and (-3, -5)

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The perimeter of the quadrilateral is 8 units. Therefore, the answer to the question is 8 units.

The perimeter of a quadrilateral is the sum of the lengths of its sides. To calculate the perimeter, we need to calculate the distances between the points of the quadrilateral.

To draw a quadrilateral with the given points on the coordinate plane, we can draw four lines connecting the given points. The first line will connect the points (–5, –2) and (–4, –2), which is a horizontal line with a length of 1 unit. The second line will connect the points (–4, –2) and (–4, 1), which is a vertical line with a length of 3 units. The third line will connect the points (–4, 1) and (–5, 1), which is a horizontal line with a length of 1 unit. The fourth line will connect the points (–5, 1) and (–5, –2), which is a vertical line with a length of 3 units.

The total length of the sides of the quadrilateral is calculated by adding the lengths of the four lines. Thus, the perimeter of the quadrilateral is 8 units.

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Answer:

Every day, the chef consumes 5443.20 grams of butter, which is calculated using the conversion coefficients 16 oz/1 pound and 28.35 grams/1oz.

A restaurant chef uses 12 pounds of butter every day, as specified in the question.

We need to figure out how much butter the chef uses each day in grams.

Applying the conversion parameters provided, 16 oz/1 lb and 28.35 grams/1oz

According to the data provided, the needed solution is as follows: 12 lb 16 oz/1 lb 28.35 g/1 ounce 5443.20 grams

As a result, the chef consumes 5443.20 grams of butter every day.

Step-by-step explanation:

Brainliest pls

The derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.

Using the product rule of differentiation, we can find the derivative of the function f(x) = x⋅5x as follows:

f'(x) = (x)'(5x) + x(5x)'

where (x)' and (5x)' are the derivatives of x and 5x with respect to x, respectively.

(x)' = 1 (the derivative of x with respect to x is 1)

(5x)' = 5 (the derivative of 5x with respect to x is 5)

Substituting these values, we get:

f'(x) = 1⋅5x + x⋅5

Simplifying further, we get:

f'(x) = 5x + 5x

Therefore, f'(x) = 10x.

In conclusion, the derivative of f(x) = x⋅5x is f'(x) = 10x, which means that the rate of change of the function at any point x is 10 times the value of x at that point.

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The solution to the inequality f(x) ≤ 0 is the interval [-6, 2]. In other words, the values of x that satisfy the inequality are those that lie between -6 and 2, inclusive.

To solve the inequality f(x) ≤ 0, we need to find the values of x for which the function f(x) is less than or equal to zero.

We start by factoring the quadratic expression f(x) = x^2 + 4x - 12:

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

Setting this expression to zero, we get:

(x + 6)(x - 2) = 0

This gives us two solutions: x = -6 and x = 2.

Now, we need to determine the sign of f(x) in the intervals between these two solutions. We can use a sign chart to do this:

x f(x)

-∞ +

-6 0

2 0

+∞ +

From the sign chart, we see that f(x) is positive for x < -6 and for x > 2, and it is negative for -6 < x < 2.

To summarize, the solution to the inequality f(x) ≤ 0 for the function f(x) = x^2 + 4x - 12 is the interval [-6, 2].

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Using multiple principle, The different area codes that are possible are 144

The multiplication principle is a counting principle that states the total number of possible outcomes of a series of independent events is equal to the product of the number of outcomes for each event. It is used in probability theory and other areas of mathematics and science.

According to the question

Under the original plan for area codes in 1945, the first digit could be any number from 2 through 9, the second digit was either 0 or 1, and the third digit could be any number except 0.

Therefore, the number of possible area codes can be calculated as follows:

For the first digit, there are 8 possibilities (2, 3, 4, 5, 6, 7, 8, or 9). Here 9 is inclusive.

For the second digit, there are 2 possibilities (0 or 1).

For the third digit, there are 9 possibilities (any digit except 0).

Using the multiplication principle, the total number of possible area codes can be determined by multiplying the number of choices for each digit:

8 × 2 × 9 = 144

So there are 144 different area codes possible according to this plan.

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Answer: 1

Step-by-step explanation:

because it not zero and not 4

Answer:

Since 1 000 paperclips = 1 kilogram

5,600 paperclips x 0.001 kilograms per paperclip = 5.6 kilograms

Answer: 5.6 kilograms

Step-by-step explanation:

5,600 grams = 5.6 kilograms

Let a sample space be partitioned into three mutually exclusive and exhaustive events, B1, B2, and, B3.We have to complete the following probability table, prior probabilities, conditional probabilities, joint probabilities, and posterior probabilities, and also we have to round our answers to 2 decimal places.

Given information:Probability of B1, P(B1) = 0.11 Probability of A given B1, P(A | B1) = 0.45 Probability of A intersection B1, P(A ∩ B1) = ?

Probability of B1 given A, P(B1 | A) = ?     Probability of B2, P(B2) = ?     Probability of A given B2, P(A | B2) = 0.62, Probability of A intersection B2,  P(A ∩ B2) = ?     Probability of B2 given A, P(B2 | A) = ?

Probability of B3, P(B3) = 0.38.    Probability of A given B3, P(A | B3) = 0.85.   Probability of A intersection B3,

P(A ∩ B3) = ?

Probability of B3 given A, P(B3 | A) = ?

Total probability of A, P(A) = ?

Total probability of sample space = 1.     Let's complete the probability table:Prior probabilities, Conditional probabilities, Joint probabilities, Posterior probabilities      P(B1) = 0.11P(A | B1) = 0.45P(A ∩ B1) = P(A | B1) * P(B1) / P(A)P(B1 | A) = P(A | B1) * P(B1) / P(A)P(B2) = 1 - (P(B1) + P(B3)) = 1 - (0.11 + 0.38) = 0.51P(A | B2) = 0.62P(A ∩ B2) = P(A | B2) * P(B2) / P(A)P(B2 | A) = P(A | B2) * P(B2) / P(A)P(B3) = 0.38P(A | B3) = 0.85P(A ∩ B3) = P(A | B3) * P(B3) / P(A)P(B3 | A) = P(A | B3) * P(B3) / P(A)Total = 1P(A) = P(A ∩ B1) + P(A ∩ B2) + P(A ∩ B3) = 0.11 * 0.45 + 0.51 * 0.62 + 0.38 * 0.85 = 0.6579.  So, the probability table is as follows:Prior probabilities,Conditional probabilities,Joint probabilities,Posterior probabilities

P(B1) = 0.11P(A | B1) = 0.45P(A ∩ B1) = 0.0495P(B1 | A) = 0.3419P(B2) = 0.51P(A | B2) = 0.62P(A ∩ B2) = 0.3162P(B2 | A) = 0.4857P(B3) = 0.38P(A | B3) = 0.85P(A ∩ B3) = 0.3217P(B3 | A) = 0.1724Total = 1P(A) = 0.6579

Hence, the completed probability table is as shown above.

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In linear equation, The number of suitcases are 503.

What in mathematics is a linear equation?

A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.  

                                   Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

Equation

y = 1.98x + 7.97

Where,

x = Number of passengers

y = Number of suitcases

Finding the number of suitcases:

y = 1.98x + 7.97

y = 1.98(250) + 7.97

y = 495 + 7.97

y = 502.97 ≈ 503

Hence,

The number of suitcases are 503.

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The expected number of pairs of professors in conflict is given by (m choose 2) * 1/n.

It can be calculated using the principle of linearity of expectation. We can first calculate the probability that any two professors pick the same time slot, which is 1/n. Then, we can count the number of pairs of professors, which is given by the binomial coefficient (m choose 2) = m(m-1)/2. Therefore, the expected number of pairs of professors in conflict is:

Expected number of pairs in conflict = (m choose 2) * 1/n

This formula holds when the selection of time slots by each professor is independent of the choices made by all other professors. Note that this formula assumes that each professor selects only one time slot, and does not consider the possibility of a professor selecting multiple time slots. If professors are allowed to select multiple time slots, then the formula would need to be modified accordingly.

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The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.

WHAT IS AXIS ?

An axis is a straight line around which an object rotates or is symmetrical. In geometry, it is a reference line that is used to measure distances, angles, and other geometric properties of objects. In a two-dimensional plane, there are two axes: the x-axis and the y-axis. The x-axis is the horizontal line that runs from left to right, while the y-axis is the vertical line that runs from bottom to top. Together, the x-axis and y-axis form a coordinate system that is used to plot points and graph functions. In three-dimensional space, there is also a z-axis that runs perpendicular to the x-axis and y-axis

To perform a dilation with a scale factor of 1/2 around the origin, we need to multiply the coordinates of each point by 1/2. This gives us:

A' = (1/2)A = (1/2)(-2,-5) = (-1,-2.5)

B' = (1/2)B = (1/2)(0,3) = (0,1.5)

C' = (1/2)C = (1/2)(6,-3) = (3,-1.5)

The new coordinates of points A, B, and C after the dilation are (-1,-2.5), (0,1.5), and (3,-1.5), respectively.

To reflect each point over the y-axis, we need to change the sign of its x-coordinate, while leaving the y-coordinate unchanged. This gives us:

A" = (-x,y) = (-(-1),-2.5) = (1,-2.5)

B" = (-x,y) = (-0,1.5) = (0,1.5)

C" = (-x,y) = (-3,-1.5) = (-3,-1.5)

The new ordered pairs of points A', B', and C' after the reflection over the y-axis are (1,-2.5), (0,1.5), and (-3,-1.5), respectively.

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After figuring out the given issue, we discovered that the original square's area and the dilated square's area differ by 32 square centimeters.

Square Area = Side x Side. Hence, Side2 square units are equivalent to the area of the square. and four side units make up a square's perimeter.

The formula for calculating the area of a initial square with sides that are 2 centimeters long is:

A = s²

A = 2²

A = 4 square centimeters

The revised side length of this square after dilation by a scale factor of 3 is:

s' = 3s

s' = 3(2)

s' = 6 centimeters

Calculations for the dilated square's area are as follows:

A' = s'²

A' = 6²

A' = 36 square centimeters

The area of a dilated square differs from the size of the original square by:

A' - A = 36 - 4

A' - A = 32 square centimeters

As a result, the original square's area and the dilated square's area differ by 32 square centimeters.

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In conclusion for a. If the system is homogeneous , every solution is trivial true. b. If there exists a trivial solution, the system is homogeneous false. c. If there exists a nontrivial solution, there is no trivial solution false.

How to solve?

a. If the system is homogeneous, every solution is trivial.

This statement is true. A homogeneous system of linear equations has the form Ax = 0, where A is the coefficient matrix and x is the vector of variables. The trivial solution is always x = 0, which satisfies the equation. Any other solution would require a nonzero x vector, but then Ax would be nonzero, contradicting the fact that it equals zero in a homogeneous system.

b. If there exists a trivial solution, the system is homogeneous.

This statement is false. A system of linear equations can have a trivial solution (i.e., all variables are equal to zero) without being homogeneous. For example, the system

x + y = 0

2x + 2y = 0

has a trivial solution (x = 0, y = 0) but is not homogeneous.

c. If there exists a nontrivial solution, there is no trivial solution.

This statement is false. A homogeneous system of linear equations can have both trivial and nontrivial solutions. For example, the system

x + y = 0

2x + 2y = 0

has both a trivial solution (x = 0, y = 0) and a nontrivial solution (x = 1, y = -1).

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Answer:

n=20

Sum of age of men = 50n

when 2 left,

(50n-55-63)/n-2=50-1

50n-55-63=49n-98

50n-55=49n-35

50n-49n=-35+55

n=20

When x=10, the value of y in the equation 3x+4y=34 is y=1.

To find the value of y when x=10 in the equation 3x+4y=34, we can substitute x=10 into the equation and solve for y:

3x+4y=34

3(10) + 4y = 34

30 + 4y = 34

4y = 34 - 30

4y = 4

y = 1

An equation is a mathematical statement that shows the relationship between two or more variables using symbols and numbers. It is a statement that asserts the equality of two expressions. An equation typically consists of two sides, separated by an equals sign (=). Each side of the equation may contain variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Solving an equation involves manipulating the expressions on both sides of the equation to isolate the variable and find its value. Equations are used in various branches of mathematics, physics, engineering, economics, and other sciences to model and solve problems. For example, the equation "2x + 5 = 11" has two sides, left-hand side (2x + 5) and right-hand side (11), separated by an equals sign.

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Answer:

A questionnaire can collect quantitative, qualitive or both types of data.

Step-by-step explanation:

Answer:

Categorical data

Step-by-step explanation:

Data that relates to certain categories e.g males, females or any types of car

The results showed that the mean difference in assessed value between the two pawn shops was $20.

Value is the worth or importance that something has to an individual or group. It is often determined by ideas such as usefulness, importance, or desirability. Value can also be determined by material or monetary worth, such as the amount of money someone is willing to pay for an item or service.

The research team conducted a hypothesis test to determine if the difference in the assessed values was statistically significant. The null hypothesis was that there was no difference in the assessed values between the two pawn shops. The alternative hypothesis was that there was a statistically significant difference in the assessed values between the two pawn shops. The researchers used a two-tailed t-test to determine if the difference in the assessed values was statistically significant.

The t-test results showed that the difference in assessed values between the two pawn shops was statistically significant, with a p-value of 0.0001. This indicated that the difference in assessed values between the two pawn shops was not due to chance. The research team concluded that the two pawn shops were not assessing jewelry of the same value.

The findings of this research are important for the jewelry industry. This research showed that two pawn shops were assessing jewelry differently, indicating that there is a need for a standardized approach to assess jewelry value. The findings of this study can help improve the accuracy of jewelry valuations and help ensure that consumers are paying a fair price for their jewelry. It can also help jewelers, pawn shops, and other jewelry buyers and sellers determine a fair value for their jewelry purchases.

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The Given statement "If a distribution is normal, then it is not possible to randomly select a value that is more than 4 standard deviations from the mean"  is a true statement.

A normal distribution is a bell-shaped curve that represents how data is spread out around an average value. About 99.99% of the data in a normal distribution falls within 4 standard deviations from the average.

This means that it is very rare to find a data point that is more than 4 standard deviations away from the average. Therefore, if a distribution is normal, it is not possible to randomly select a value that is more than 4 standard deviation from the mean.

The answer is true.

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The equation of the line y = (5/3)x + (10/3) represents the line passing through the point (3, 5) that cuts off the smallest area from the first quadrant.

To find the equation of the line that passes through the point (3, 5) and cuts off the least area from the first quadrant, we need to consider the slope of the line.

Any line passing through the point (3, 5) can be written in a point-slope form as:

y - 5 = m(x - 3)

where m is the slope of the line. We want to find the slope that minimizes the area cut off by the line.

Consider a line passing through the origin with slope m. The area cut off by this line in the first quadrant is given by:

A(m) = (1/2)(3)(m*3) = (9/2)m

Note- that the area cut off by the line passing through (3, 5) with slope m is equal to the area cut off by the line passing through the origin with slope m plus the area of the triangle formed by the point (3, 5), the origin, and the point where the line intersects the y-axis. The y-intercept of the line passing through (3, 5) with slope m is given by:

y - 5 = m(x - 3)

y = mx - 3m + 5

Setting x = 0, we get:

y = -3m + 5

The coordinates of the point where the line intersects the y-axis are (0, -3m + 5), and the area of the triangle is:

(1/2)(3)(|-3m + 5 - 0|) ⇒ (3/2)|-3m + 5|

Therefore, the total area cut off by the line passing through (3, 5) with slope m is:

A(m) = (9/2)m + (3/2)|-3m + 5|

To find the slope that minimizes this expression, we need to consider two cases:

Case 1: -3m + 5 ≥ 0, i.e., m ≤ 5/3

In this case, the expression simplifies to:

A(m) = (9/2)m + (9/2)m - (3/2)(5)

= (9m - (15/2)

This expression is minimized when m = 5/3, which is within the range of possible slopes.

Case 2: -3m + 5 < 0, i.e., m > 5/3

In this case, the expression simplifies to:

A(m) = (9/2)m - (9/2)m + (3/2)(5)

= (15/2)

This expression is minimized when m = 5/3, which is again within the range of possible slopes.

Therefore, the line passing through (3, 5) with slope m = 5/3 cuts off the least area from the first quadrant. The equation of the line is:

y - 5 = (5/3)(x - 3)

Simplifying, we get:

y = (5/3)x + (10/3)

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The system of linear equations to find the cost of an apple and the cost of an apricot is as follows:

9x + 6y = 8.50

3x + 2y = 7.40

Jeremiah bought 9 apples and 6 apricots for $8.50 yesterday. He bought 3 apples and 2 apricots for $7.40 today.

The system of linear equation to find the cost of an apple and the cost of an apricot can be represented as follows:

Therefore, system of equation can be solved using different method such as elimination method, substitution method and graphical method.

The linear equation is as follows:

9x + 6y = 8.50

3x + 2y = 7.40

where

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For the first question, the indefinite integral of [tex]16x³ - 15x² + 6[/tex] is [tex]4x⁴ - 5x³ + 6x + C[/tex], where C is the constant of integration.

For the second question, the indefinite integral of 7cos(x) + 5sin(x) is [tex]7sin(x) + 5cos(x) + C,[/tex]where C is the constant of integration.

1)To check the result, differentiate the indefinite integral using the power rule and product rule: [tex](16x³ - 15x² + 6)' = 64x² - 30x + 6.[/tex]

2)To check the result, differentiate the indefinite integral using the sum rule and product rule: [tex](7sin(x) + 5cos(x))' = 7cos(x) - 5sin(x).[/tex]

In summary, to find the indefinite integral of a function, use the power rule and product rule to integrate the individual terms. To check the result, differentiate the indefinite integral.

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The exponential distribution is a probability distribution that models the time between events in a Poisson process, where events occur randomly and independently at a constant average rate.

It is commonly used in reliability theory, queuing theory, and other fields to model the failure or waiting times of systems.

(a) The mean time to decay for Cobalt-60 is 5.27 years.

(b) The standard deviation of the decay time is 2.6355 years.

(c) The 99th percentile is 13.6825 years.

(d) The mean time of the experiment is 15.8175 years and the standard deviation is 4.86788 years.

Note: The answers are calculated based on the exponential distribution of radioactive decay with a half-life of 5.27 years for Cobalt-60.

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The probability of randomly drawing a penny is 6/24 or 1/4, since there are 6 pennies out of a total of 24 coins.

How to solve and What is Probability?

The probability of randomly drawing a quarter is 10/24 or 5/12, since there are 10 quarters out of a total of 24 coins. The probability of randomly drawing a coin that is not a penny is 18/24 or 3/4, since there are 18 coins that are not pennies out of a total of 24 coins.

Probability is the branch of mathematics that deals with measuring the likelihood or chance of an event or outcome occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

Probability theory is used to make predictions and informed decisions based on available data in various fields, including statistics, finance, engineering, and science.

It involves understanding and analyzing random events, and determining the likelihood of specific outcomes. Probability is an essential tool for decision-making in various applications, such as risk analysis, game theory, and quality control.

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First before two trains were [tex]990[/tex] kilometers apart, it will require [tex]5.5[/tex] hours.

Train speed is calculated as total distance traveled divided by travel time. The time it takes for two trains to pass each other is equal to (a+b) / (x+y) if the lengths of the trains, say a or b, are known and they are going at speeds of y and x, respectively.

Typically, a locomotive fueled by electricity or diesel powers trains. If there are several route networks, complicated signaling methods are used. One of the quickest forms of land transportation is rail.

[tex]distance = rate * time[/tex]

distance between trains [tex]= (85 km/h) * t + (95 km/h) * t[/tex]

distance between trains [tex]= (85 + 95) km/h * t[/tex]

distance between trains [tex]= 180 km/h * t[/tex]

Now, we can set up an equation to solve for the time it takes for the trains to be [tex]990[/tex] km apart:

[tex]180 km/h * t = 990 km[/tex]

[tex]t = 990 km / 180 km/h[/tex]

[tex]t = 5.5[/tex] hours

Therefore, it will take [tex]5.5[/tex] hours before the two trains are [tex]990[/tex] km apart.

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The given parametric equations are X(t) = -3/2 + 2t and y(t) = vt² + 16, the rate of change of y with respect to "t" when t = 3 is 6v

We have the parametric equations that are X(t) = -3/2 + 2t and y(t) = vt² + 16.

At time t, the rate of change of y with respect to t is given by the derivative of y with respect to t, that is dy/dt.

So, y(t) = vt² + 16

Differentiating with respect to t, we get

⇒ dy/dt = 2vt.

Now, t = 3 gives us,

y(3) = v(3)² + 16 ⇒ 9v + 16.

Therefore, the rate of change of y with respect to t at t = 3 is

dy/dt ⇒ 2vt ⇒ 2v(3) ⇒ 6v.

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In the following code, you can add another array declaration that creates an array of 5 doubles called prices and another array of 5 Strings called names and corresponding System.out.println commands.

public class Test1 {

public static void main(String[] args) {

// Array example

int[] highScores = new int[10];

// Add an array of 5 doubles called prices.

double[] prices = new double[5];

// Add an array of 5 Strings called names.

String[] names = new String[5];

System.out.println("Array highScores declared with size " + highScores.length);

// Print out the length of the new arrays.

System.out.println("Array prices declared with size " + prices.length);

System.out.println("Array names declared with size " + names.length);

}

}

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The new price of the television is 15/16 of the original price

Percentages are essentially fractions where the denominator is 100. To show that a number is a percent, we use the percent symbol (%) beside the number.

Represent the original price by x

x × 25/100 = x/4%

The new price will be

x - x/4

= (4x- x)/4

= 3x/4

The new price is now increased by 25%

25/100 × 3x/4

= 1/4 × 3x/4

= 3x/16

the new price

3x/16 + 3x/4

= (12x + 3x)/16

= 15x/16

Therefore the new price is 15/16 of the original price.

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Answer:

(-1,1)

(0,2

(1,3)

(2,4)