Simplify. Remove all perfect squares from inside the square root. 18

The slope of the line through (-2,3) and (1,-3) is -2.

Y = mx + b, where m is the line's slope and b is the y-intercept, is the slope-intercept form of a linear equation (the point where the line intersects the y-axis). This form is helpful because, given a line's equation, it enables us to quickly determine the slope and y-intercept. By charting the y-intercept and utilising the slope to identify additional points on the line, we can also use this technique to graph a line. We can also create an equation for a line given its slope and y-intercept using this approach.

The slope of the line is given as:

slope = (y2 - y1) / (x2 - x1)

Substituting the value of the given coordinates we have:

slope = (-3 - 3) / (1 - (-2))

slope = -6 / 3

slope = -2

Hence, the slope of the line through (-2,3) and (1,-3) is -2.

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The equation to  the given graph is y= (1/2) * 2ˣ. So correct option is B.

A parabola is a type of curve that results from the intersection of a cone with a plane that is parallel to one of its sides. It is a conic section, which means it is a geometric shape that can be formed by cutting a cone with a plane.

The shape of a parabola is similar to that of a U-shaped curve, but it is symmetric around a vertical line called the axis of symmetry. The point where the axis of symmetry intersects the parabola is called the vertex.

The equation of a parabola can be written in different forms, but the most common form is:

y = ax^2 + bx + c

where "a", "b", and "c" are constants that determine the shape, position, and orientation of the parabola. If "a" is positive, the parabola opens upward, and if "a" is negative, it opens downward.

We can observe that the given graph passes through the points (-1,1), (0,2), (1,4), (2,8). This suggests that the graph is a parabola that opens upwards.

We can use the general form of a quadratic equation to write the equation of the parabola in the form y = ax² + bx + c, where a, b, and c are constants. Since the parabola opens upwards, we know that the coefficient of the x² term is positive.

To find the values of a, b, and c, we can substitute the coordinates of the given points into the equation and solve the resulting system of equations.

Using the point (-1,1), we get:

1 = a(-1)² + b(-1) + c

1 = a - b + c

Using the point (0,2), we get:

2 = a(0)² + b(0) + c

2 = c

Using the point (1,4), we get:

4 = a(1)² + b(1) + c

4 = a + b + c

Using the point (2,8), we get:

8 = a(2)² + b(2) + c

8 = 4a + 2b + c

Substituting c = 2 from the second equation into the third and fourth equations, we get:

4 = a + b + 2

8 = 4a + 2b + 2

Simplifying these equations, we get:

a + b = 2

2a + b = 3

Solving for a and b, we get:

a = 1

b = 1

Therefore, the equation of the parabola is y = x² + x + 2.

To check which of the given options matches this equation, we can simplify the equation:

y = x² + x + 2

y = (x + 1/2)² + 7/4

This shows that the given parabola is a vertical shift of the standard parabola y = x² by 7/4 units upward.

Out of the given options, the only equation that matches this form is option B, y = (1/2) * 2ˣ. Therefore, the answer is B) y= (1/2) * 2ˣ.

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Answer:

-2xy-z^2

Step-by-step explanation:

combine like terms

x^2-2xy-x^2-z^2

-2xy-z^2

Answer: 30 album pages

Step-by-step explanation:

cards = 10

teams = 24

total cards = 240

240 / 8 = 30

With given quadratic equation for time, it will take the rock approximately 4.47 seconds to reach the river.

What exactly are quadratic equations?

Quadratic equations are equations of the second degree, meaning they involve terms raised to the power of two. These equations are written in the form of ax² + bx + c = 0, where x is the unknown variable, and a, b, and c are constants (numbers).

The graph of a quadratic equation is a curve called a parabola, which can be either upwards or downwards depending on the sign of the coefficient a. Quadratic equations can have one, two, or zero real solutions, depending on the discriminant b² - 4ac.

Now,

In this equation, h represents the height of the rock above the river at time t in seconds. When the rock hits the river, its height h will be zero. So we can set h to zero and solve for t:

h = -16t² + 320

0 = -16t² + 320 (substituting h = 0)

16t² = 320

t² = 20

t = ±√20

Since time cannot be negative, we take the positive square root:

t = √20 ≈ 4.47 seconds

Therefore, it will take the rock approximately 4.47 seconds to reach the river.

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The percentage of the student body that is at least 20 years old is 50%.

The ages of students at a university are normally distributed with a mean of 20. To find the percentage of the student body that is at least 20 years old solution to the is as follows: As the Mean of ages of students at a university, μ = 20. Age is normally distributed so it will be divided into two groups  Therefore, the mean age is equal to the median age. As the age is normally distributed, 50% of the population will have an age greater than or equal to 20 years old. Thus, the percentage of the student body that is at least 20 years old is 50%. Hence, the required percentage of the student body that is at least 20 years old is 50%.

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Therefore, the length of the perpendicular is 1 cm. Therefore, the length of the perpendicular is approximately 1.335 cm (rounded to three decimal places).

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is used to solve problems related to geometry, physics, engineering, and many other fields. Trigonometry is based on the study of the six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions describe the ratios of the lengths of the sides of a right triangle, and can be used to calculate the unknown side lengths or angles of a triangle.

Here,

a. Given that the hypotenuse is 2 cm and the angle is 30 degrees, we can use trigonometry to find the length of the perpendicular. Let's call the length of the perpendicular x. Using the trigonometric ratio for sine (sin), we have:

sin(30°) = opposite/hypotenuse

sin(30°) = x/2

Multiplying both sides by 2:

2*sin(30°) = x

We know that sin(30°) is equal to 0.5, so we can substitute that in:

2*0.5 = x

x = 1

b. Given that the base is 3 cm and the angle is 24 degrees, we can use trigonometry to find the length of the perpendicular.

Let's call the length of the perpendicular x.

Using the trigonometric ratio for tangent (tan), we have:

tan(24°) = opposite/adjacent

tan(24°) = x/3

Multiplying both sides by 3:

3*tan(24°) = x

We may use a calculator to find the value of tan(24°), which is approximately 0.445:

3*0.445 = 1.335

c. cos 72°=x/9

0.3091*9=x

x=2.78 units

d.  cos 23°=5.9/x

x=5.9/0.9205

x=6.409 units

e. sin 36°=3.6/x

x=3.6/0.5877

x=6.1255 units

f. cos 48°=4.3/x

x=4.3/0.6691

x=6.4226 units

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Therefore, the value of θ for the acute angle in a right triangle is 46°.We can use the fact that the sine and cosine of complementary angles are equal to find θ.

To find the value of θ for the acute angle in a right triangle, we need to use one of the trigonometric ratios. In this case, we are given the sine of θ and the cosine of 44°.  Recall that the sine of an angle θ is defined as the ratio of the opposite side to the hypotenuse in a right triangle:

sin(θ) = opposite / hypotenuse

And the cosine of an angle φ is defined as the ratio of the adjacent side to the hypotenuse:

cos(φ) = adjacent / hypotenuse

In a right triangle, the two acute angles are complementary, which means that their sum is 90°. That is,θ + 90° = 90°

θ = 90° - 44°.Now we can use the given equation sin(θ) = cos(44°) and substitute θ: sin(90° - 44°) = cos(44°) => sin(46°) = cos(44°)

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The width of the bookshelf will be 6 inches when the area and length are 216 in and  36 inches.

What is the width?

Width is a dimension of a geometrical figure that tells about the structure of the figure. In general, width is used to measure the object.

What is formulae of width?

Width is obtained by dividing area and length.

                               W=A/L

substitute values, 216/36= 6inches.

The width is gonna be 6 because all you have to do is divide 216 and 36

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Answer: 23

Step-by-step explanation:

The 15 equations and 15 unknowns are part of a system of linear equations. These equations can be used to find the values of the 15 unknowns. The purpose of each set of equations is to determine the unknowns in the system. The equations are derived using algebraic methods such as substitution, elimination, and graphing.

The 15 equations are:

1. x + y = 15

2. x - y = 5

3. 2x + 4y = 60

4. x + 2y = 30

5. 3x + y = 45

6. 3x - y = 35

7. x + 4y = 40

8. 4x + y = 55

9. x - 4y = 5

10. 2x - y = 15

11. 3x - 2y = 20

12. 5x + y = 55

13. x - 5y = -5

14. 2x + 5y = 55

15. 4x - 3y = 25

The 15 unknowns are:

1. x

2. y

3. x + y

4. x - y

5. 2x + 4y

6. x + 2y

7. 3x + y

8. 3x - y

9. x + 4y

10. 4x + y

11. x - 4y

12. 2x - y

13. 3x - 2y

14. 5x + y

15. x - 5y

Each equation is derived using algebraic methods such as substitution, elimination, and graphing. By substituting the known values into the equations, the unknowns can be determined. Elimination is used to get rid of terms from the equations, making it easier to solve for the unknowns. Graphing is used to plot out the equations and to help visualize the relationships between the unknowns and the equations.

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the baker needs 15 gallons of milk to make 80 chocolate pies for the community festival. To find out how much milk is needed to make 80 chocolate pies, we need to first determine the total amount of milk required for one chocolate pie. We know that the baker needs 3 cups of milk to make one chocolate pie.

To convert cups to gallons, we need to divide by 16 since there are 16 cups in a gallon. So, one chocolate pie requires 3/16 gallons of milk.

To find out how much milk is needed to make 80 chocolate pies, we can multiply the amount of milk required for one pie by the number of pies. This gives us:

(3/16) gallons of milk per pie x 80 pies = 15 gallons of milk

Therefore, the baker needs 15 gallons of milk to make 80 chocolate pies for the community festival.

It's important to make sure we convert units correctly when solving these types of problems. In this case, we had to convert cups to gallons to make sure we were using the same unit of measurement for both the amount of milk needed for one pie and the total amount needed for 80 pies. By following these steps, we can accurately determine the amount of ingredients needed to fulfill an order, which is important for bakers and other food service professionals.

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Therefore , the solution of the given problem of function comes out to be  1, -7/3x, 35/27x², -455/162x³, 4368/6561x⁴.

The midterm exam will include questions on design, geometry, numbers, each division, as well as actual and hypothetical geographic places. A piece of work describes the connections between variable elements that cooperate to produce the same outcome. A utility is a structure made up of numerous unique parts that work together to create unique outcomes for each input.

Here,

This is how the Maclaurin sequence is given:

=> F(x) =  [tex]f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + f''''(0)x^4/4![/tex]+ ...

where the values of the function and its derivatives at the value of x are f(0), f'(0), f''(0), f'''(0), and f''''(0).

Let's determine f(x)'s first four derivatives:

=> f'(x) = (-7/3)(1+x) where f(x) = [tex](1+x)(-7/3)^(-10/3)[/tex]

=> f''(x) = [tex](70/9)(1+x)^(-13/3)[/tex]

=> f'''(x) = [tex](-910/27)(1+x)^(-16/3)[/tex]

=> f''''(x) = [tex](21840/81)(1+x)^(-19/3)[/tex]

Now, we can add these to the formula for the Maclaurin series:

=> f(x) = [tex]f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + f''''(0)x^4/4![/tex]+ ...

=> f(x) = [tex]f(0) + (-7/3)*f(0)*x + (70/9)*f(0)x^2/2! + (-910/27)f(0)x^3/3! + (21840/81)f(0)x^4/4![/tex] + ...

=> f(x) = f(0)[tex](1 - 7/3x + 35/27x^2 - 455/162x^3 + 4368/6561x^4 + ...)[/tex]

=> f(x) = [tex]f(0)(1 - 7/3x + 35/27x^2 - 455/162x^3 + 4368/6561*x^4)[/tex]

=> f(0) = [tex](1+0)^(-7/3) = 1[/tex]

Replacing the incomplete episodes with:

=> f(x) = 1 – 7 – 3 – 35 – 27 – 455 – 162 – 4368 – 6561 x

The elements of the Maclaurin series of (1+x)(-7/3) up to degree 4 are as follows:

=> 1, -7/3x, 35/27x², -455/162x³, 4368/6561x⁴

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If Sally started with 2 feet of cloth, the cloth she has remaining is [tex]\frac{23}{21}[/tex] feet.

Algebra is a branch of mathematics that deals with mathematical symbols and the rules for manipulating these symbols to solve equations and study mathematical structures. It involves the use of letters and symbols to represent numbers and quantities in equations, and the manipulation of these equations to solve for unknown values or to simplify expressions.

Algebra includes topics such as arithmetic operations with real numbers, solving linear and quadratic equations, graphing lines and curves, working with functions, and manipulating algebraic expressions. It also involves the use of mathematical reasoning and problem-solving skills to solve complex problems and real-world applications.

Algebra plays an important role in many areas of mathematics, science, engineering, economics, and finance. It provides a powerful tool for modeling and analyzing real-world situations, as well as for understanding and predicting mathematical patterns and relationships.

[tex]2- \frac{1}{3}-\frac{4}{7} \\\frac{5}{3}- \frac{4}{7}\\\frac{35-12}{21} \\\frac{23}{21}[/tex]

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The complete question is:

Beginning the e-commerce development process with a mobile presence rather than a desktop website is referred to as Mobile First Design.Mobile First Design is a strategy for designing responsive websites that prioritize mobile devices over desktop computers. This means that when designing a site, the mobile experience comes first, followed by the tablet and desktop experiences.

Mobile First Design has become a popular design strategy due to the increasing use of mobile devices to access the internet. It acknowledges the fact that mobile devices have smaller screens, slower internet connections, and limited processing power, which means that websites must be designed to work well on these devices

Mobile First Design includes a variety of design strategies to optimize sites for mobile devices, including responsive design, adaptive design, and progressive enhancement. Responsive design involves using flexible grids, images, and media queries to create a website that responds to the screen size of the device it's viewed on.Adaptive design uses a combination of flexible and fixed elements to create a website that adapts to different screen sizes.

Progressive enhancement involves designing a website with basic functionality that works well on any device, and then adding additional features for more capable devices as they become available.

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The linear function rule to model the cost y for any number of samples x is (y = x + 2.25) and this can be determined by forming the linear equation.

An international food festival charges for admission and for each sample of food.

Admission and 6 samples cost $8.25.

Admission and 8 samples cost $10.25.

The following steps can be used in order to determine the linear function rule to model the cost y for any number of samples x:

Step 1 - The linear equation is formed in order to determine the linear function.

Step 2 - The linear equation that represents the situation "admission and 6 samples cost $8.25" is:

8.25 - 6c = d   --- (1)

Step 3 - The linear equation that represents the situation "admission and 8 samples cost $10.25" is:

10.25 = 8c + d    --- (2)

Step 4 - Substitute the value of 'd' in equation (2).

10.25 = 8c + 8.25 - 6c

2 = 2c

c = 1

Step 5 - Substitute the value of 'c' in the equation (1).

d = 8.25 - 6

d = 2.25

Step 6 - So, the linear function rule to model the cost y for any number of samples x is:

y = x + 2.25.

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Answer:

Step-by-step explanation:

[tex]\frac{2}{3}[/tex] +[tex]\frac{3}{4}[/tex]

=[tex]\frac{17}{12}[/tex]

The amount or fraction of work Jerry gets done in 1 hour is 1/3 when it takes him 3 hours to clean all the tank in his section and it take Hart x hours to clean all the tanks in his section.

We know that it takes jerry 3 hours to clean all the tank in his section and it take Hart x hours to clean all the tanks in his section,

when they work together, they can clean the tanks in 1 hour.

now let work be - 1

therefore,

work x Jerry = work x Hart

1×3 = 1×x

3 = 1x

3 = x

therefore in 1 hour work done by Jerry will be 1/3.

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You have a bag of 10 marbles. Three of them are red, five are blue, and two are green. The probability of randomly picking a green or blue marble out of the bag is 0.7 or 70%.

The given information is that there is a bag of 10 marbles, among which three are red, five are blue, and two are green. We are required to find out the probability of randomly picking a green or blue marble out of the bag.

So, the probability of randomly picking a green or blue marble out of the bag can be determined using the formula of probability.

The formula is:

P(E) = Number of Favorable Outcomes / Total Number of Possible Outcomes

Where,

P(E) is the probability of the event and E is the event.

For the given problem,

the event is picking a green or blue marble from the bag.

The total number of possible outcomes is the total number of marbles in the bag, which is 10.

Number of Favorable Outcomes

There are five blue marbles and two green marbles in the bag.

So, the number of favorable outcomes is

5 + 2 = 7.

P(E) = Number of Favorable Outcomes / Total Number of Possible Outcomes

P(E) = 7 / 10

P(E) = 0.7

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From the given information provided, the percentage of children who travel by bicycle is 15%.

According to the given information, out of 20 children, 3 of them travel to school by bicycle. To calculate the percentage of children who travel to school by bicycle, we need to divide the number of children who travel by bicycle by the total number of children and multiply by 100.

Percentage of children who travel to school by bicycle = (Number of children who travel by bicycle / Total number of children) x 100%

= (3/20) x 100%

= 15%

Therefore, the percentage of children who travel to school by bicycle is 15%.

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The probability that you catch two random fish from the same species is 0.375, or 37.5%.

This is because the total number of fish in the lake is 2,000, so if you catch two random fish there is a 1 in 2,000 chance of catching two of the same species.

To find the probability, take the number of fish of one species (800 walleye, 800 bass, 400 perch) divided by 2,000.

In this case it would be 800 divided by 2,000 which is 0.4. Multiply 0.4 by itself, which is 0.16, then multiply 0.16 by 100 which gives you 0.375, or 37.5%.

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Yes, the number of linearly independent columns in a matrix A is equal to the number of linearly independent rows in A.

This is because the definition of linear independence states that no linear combination of a set of vectors can equal the zero vector, so if one set of vectors is linearly independent, the other set of vectors must also be linearly independent.

To explain further, the number of linearly independent vectors in a matrix A is determined by the rank of the matrix. The rank of a matrix is the number of linearly independent rows and columns in the matrix.

So if the rank of A is n, then the number of linearly independent columns and rows of A must be n. This is because if there are n linearly independent columns in the matrix, the number of linearly independent rows must also be n.

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In the following question, Yes, there is sufficient evidence to support the claim that the computer-aided instruction (CAI) technique performs differently than the traditional method.

To determine this, a two-tailed hypothesis test can be performed using the sample data given. The null hypothesis is that the computer-aided instruction technique has the same mean as the traditional method (101 hours). The alternative hypothesis is that the computer-aided instruction technique has a different mean than the traditional method (101 hours). With a sample mean of 100 hours, a sample size of 140 students, a known population standard deviation of 6 hours, and a level of significance of 0.01, the hypothesis test is conducted and results in a p-value of less than 0.01. This suggests that there is sufficient evidence to reject the null hypothesis and support the alternative hypothesis, that the computer-aided instruction technique performs differently than the traditional method. The conclusion is that the computer-aided instruction technique has a different mean than the traditional method.

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There is sufficient evidence to support the claim that the new license training method using computer-aided instruction performs differently than the traditional method. Therefore, the researcher can proceed with the new method.

Hypothesis testing is a technique used to make inferences about the population based on the sample data. It is divided into two types, one-tailed and two-tailed. A two-tailed test involves testing for the difference between the two population means, while a one-tailed test involves testing for a higher or lower population mean. A level of significance is predetermined to accept or reject the null hypothesis.
Given, using traditional methods, it takes 101 hours to receive a basic flying license. A new license training method using computer-aided instruction (CAI) has been proposed. A researcher used the technique with 140 students and observed that they had a mean of 100 hours. Assume the standard deviation is known to be 6.
We have to determine if there is sufficient evidence to support the claim that the technique performs differently than the traditional method at a level of significance of 0.01.
Null Hypothesis H₀: μ₁ = μ₂  (The new technique and traditional method do not perform differently.)
Alternate Hypothesis H₁: μ₁ ≠ μ₂  (The new technique and traditional method perform differently.)
The test statistic for testing the difference in two population means with known standard deviation is given by:
z = (x₁ - x₂) / (σ/√n)
Where,
x₁ = sample mean of the new technique
x₂ = sample mean of the traditional method
σ = known standard deviation
n = sample size
z = test statistic
z = (100 - 101) / (6 / √140)
z = -2.624
The corresponding p-value can be obtained from the Z-table.
p-value = P(z < -2.624) + P(z > 2.624)
p-value = 0.0088 + 0.0088
p-value = 0.0176
The level of significance α = 0.01
Since p-value < α, we can reject the null hypothesis.
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Answer:

the area of the photograph when it is put into the frame is 80 square inches, and the area of the frame is 88 square inches.

Step-by-step explanation:

Answer: 36 inches(2)

Step-by-step explanation:

lets use the two given points to form a system of equations and solve for the values of a and b.

Starting with the given equation:

y = ab

At point (2, 60):

60 = ab^(2)

At point (4, 960):

960 = ab^(4)

Now we can solve for a and b using algebraic manipulation.

First, we can divide the second equation by the first equation:

960/60 = (ab^(4))/(ab^(2))

16 = b^(2)

Taking the square root of both sides, we get:

b = 4

Substituting b = 4 into either of the two equations, we can solve for a:

60 = a(4^(2))

60 = 16a

a = 60/16

a = 15/4

Therefore, the values of a and b are:

a = 15/4

b = 4

So the exponential function is:

y = (15/4) * 4^x

"another way, same answer "

So there are 200 different ways a doctor can prescribe the asthma drug to a patient, taking into account the various options for form, manufacturer, and strength.

This illustrates the importance of carefully considering all the available options and selecting the most appropriate prescription for each patient's unique needs.The number of different ways a doctor can prescribe the asthma drug to a patient depends on the number of options available for each of the three factors: form, manufacturer, and strength.

For each of the three factors, there are 10 different options. Therefore, the total number of different ways the doctor can prescribe the drug is the product of the number of options for each factor:

10 options for form x 10 options for manufacturer x 2 options for strength (regular or extra) = 200 total options.

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The clock's minute hand is longer than the hour hand.

We know that the face of a clock is divided into 12 equal parts, and the radius of the clock face is 6 inches. We also know that the hands of the clock will form a central angle. The accurate statements about the clock are:

The length of each hour's section is π inches, or 3.14 inches.

The central angle of each hour section is 30°.

The clock's minute hand is longer than the hour hand.

Since there are 12 divisions, 360°/12 = 30°. Therefore, the central angle of each hour section is 30°.

The length of each hour's section can be calculated as the circumference of the clock face (2πr) divided by 12. Therefore, the length of each hour's section is (2π × 6)/12 = π inches, or approximately 3.14 inches.

The clock's minute hand is longer than the hour hand because it needs to travel a greater distance in the same amount of time.

The minute hand travels 360° in 60 minutes, while the hour hand travels 360° in 12 hours. Therefore, the minute hand must be longer to cover more distance in the same amount of time.

Therefore, the accurate statements about the clock are:

The length of each hour's section is π inches, or 3.14 inches.

The central angle of each hour section is 30°.

The clock's minute hand is longer than the hour hand.

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The coordinates of point Q in the parallelogram is (2b + 2a,2c)

The coordinates are given as:

P(2b, 2c) Q(?, ?) S(0, 0) R(2a, 0)

The parallelogram form of the object indicates that the distances PQ and RS are equal.

The distance RS is calculated as:

RS = ( 2a - 0, 0)

This gives

RS = (2a, 0)

Coordinate Q is calculated as:

Q = P + RS

This gives

Q = (2b,2c) + (2a,0)

Evaluate the sum

Q = (2b + 2a,2c)

Hence, the coordinates of point Q is (2b + 2a,2c)

Midpoint of SQ=(b+ a ,c)

Midpoint of PR=(a+ b, c)

hence, SQ and PR bisect each other.

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Answer: Always true.

Step-by-step explanation: Use algebra. They end up being the same statements, which means x could be whatever and the equation would still be true.