If 0° < θ < 90° or 180° < θ < 270°, explain why sin 2θ is always positive.

A 99% confidence interval for an IQ test conducted for a sample of 101 sixth graders, where the sample mean was 104.2, with a standard deviation of 12 is 104.2 ± 3.076.

A 99% confidence interval can be calculated using the formula:

sample mean ± (critical value) x (standard deviation of the mean/ √(sample size))

[tex]sample\ mean\± (critical\ value) ( \frac{standard\ deviation\ of\ the\ mean}{\sqrt{sample\ size} } )[/tex]

To find the critical value, we can use the standard normal distribution table. For a 99% confidence interval, the critical value for a two-tailed test is approximately 2.576.

So, for this problem where sample mean = 104.2 and standard deviation of mean = 12, the 99% confidence interval would be:

104.2 ± 2.576 x (12 / √101)

≈ 104.2 ± 3.076

The standard deviation of the mean is the standard deviation of the sample divided by the square root of the sample size.

This gives us a range of approximately (101.124, 107.276). So we can say that there is a 99% probability that the true population mean IQ score of sixth graders is between 101.122 and 107.276.

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Dilation is a type of transformation in geometry that changes the size of a shape without changing its orientation. When a shape is dilated by 1 2, it means that the shape is multiplied by a factor of 1 2.

This means that the shape is made half of its original size. The shape will be reduced to one-half of its original area and its perimeter will be halved as well. When a shape is dilated by a factor greater than 1, it will be expanded; when a shape is dilated by a factor less than 1, it will be reduced.

When a shape is dilated by 1 2, the center of dilatation is the same as the original shape's center of dilation. The scale factor of the dilation is 1 2, so the coordinates of the center of dilation will not change.

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In linear equation , The fifth son is younger than first son will be 31/3. And The age of the fifth son will be 1 and 5/12.

What in mathematics is a linear equation?

A linear equation is a first-order (linear) term plus a constant in the algebraic form y=mx+b, where m is the slope and b is the y-intercept.  

                                              The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.

In a family, there are five sons. The oldest son is 11 and 3/4 years old, and every next younger brother is younger than the next by 2 and 7/12 years.

Then the fifth son is younger than first son will be

⇒ 4(2 + 7/12)

⇒ 4(31/12)

⇒ (31/3)

The age of the fifth son will be

⇒ (11 + 3/4) – 4(2 + 7/12)

⇒ (47/4) – 4(31/12)

⇒ (47/4) – (31/3)

⇒ 17/12

⇒ 1 + 5/12

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Answer: If Caleb bought 5/4 of the 12 grams of tea for his mom, we can use the following equation to find out how many grams he got his mom:

5/4 * 12 grams = (5/4) * 12 = 15 grams

So Caleb bought 15 grams of tea for his mom

Step-by-step explanation:

Answer:

x=6

Step-by-step explanation:

10x+20=80

10x=60

x=6

There is a significant association if is between 0.85 and 1. There is a moderate connection if lies between 0.5 and 0.85. If, then there is a weak association between the two variables.

The strength of the linear link between two separate variables, x and y, is shown by correlation coefficients.

A positive relationship is shown by a linear correlation coefficient greater than zero.

Indicating a negative association is a value less than zero. The two variables x and y do not have any relationship when their values are 0, to sum up.

The correlation coefficient (r2) is a statistic that expresses how closely two variables' movements are related to one another.

To determine the linear relationship between two variables, the Pearson product-moment correlation, the most popular correlation coefficient, is used.

However, this correlation value might not necessarily be a reliable indicator of reliance in a non-linear connection.

Hence, There is a significant association if is between 0.85 and 1. There is a moderate connection if lies between 0.5 and 0.85. If, then there is a weak association between the two variables.

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If the model train is 57 cm long, then 266 m is the length of real train.

In the given question, a model train has a scale of 3 cm : 14 m.

If the model train is 57 cm long, then we have to find how long the real train is.

The scale of model train is 3 cm :  14 m.

Now firstly convert m in cm. As we know that 1 m = 100 cm

Now, the scale of model train is 3 cm :  1400 cm.

The length of the model train is 57 cm.

So in the scale of the model train 3 cm = 57 cm

Now the scale of the model train 1 cm = 19 cm

Now find the length of the real train by multiplying the 1400  by 19 because in the scale of 1 cm is equal to the 19 cm.

So, the length of the real train = 1400 × 19

The length of the real train = 26600 cm

The length of the real train = 266 m

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The new velocity of the object is -4.04 m/s

Impulse is the change in momentum of an object which a force causes. It is  equal to the magnitude of the force multiplied by the length of time the force is applied.

Mathematically:

Impulse (I) = change in momentum = force(F) × time (t)

Also,

Ft = m(v-u)

I = m(v-u)

where m is the mass of the object, v is the new velocity and u is the initial velocity

From the given information; m = 148kg, u = -3.5m/s, I = -80 kg m/s:

I = m(v-u)

-80 = 148(v - (-3.5))

-80 = 148(v +3.5)

-80 = 148v + 518

148v = -80-518

148v = -598

v = -598/148

v = -4.04 m/s

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The behavior of the graph relative to the x axis can be described by the equation of the line.

The equation of a line is y = mx + b, where m is the slope and b is the y-intercept. The slope of a line represents the rate of change of the graph with respect to the x axis; a positive slope indicates the graph is increasing, while a negative slope indicates it is decreasing. The y-intercept is the point at which the line crosses the y axis. For example, if the equation of the line is y = 2x + 4, then the slope of the line is 2, meaning the graph increases by 2 units every time the x axis increases by 1 unit, and the y-intercept is 4, meaning the line crosses the y axis at the point (0, 4). By analyzing the equation of the line, you can determine the behavior of the graph relative to the x axis.

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The instantaneous rate of change of 2/(t-3) when the variable is equal to -1, is:

f'(-1) =  -1/8

Here we have the function:

f(t) = 2/(t - 3)

To find the instantaneous rate of change at t = -1, we need to differentiate the function with respect to t, and then evaluate it at t = -1

First, the derivate of the function is:

f'(t) = df(t)/dt = (-1)*2/(t - 3)²

f'(t) = -2/(t - 3)²

Now to get the instantaneous rate of change, we evaluate the above function in t = -1 we will get:

f'(-1) = -2/(-1 - 3)² = -2/16 = -1/8

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Replace -0.5 with the fraction -1/2 if needed.

======================================================

Explanation:

The legs of any right triangle are always perpendicular. They meet up to form a right angle, aka 90 degree angle.

The line y = 2x+4 goes through (2,8). This can be confirmed with a graph. Or note how plugging x = 2 leads to y = 8.

Therefore, y = 2x+4 forms one of the legs of this triangle.

The slope here is 2 = 2/1; the negative reciprocal is -1/2 = -0.5, which is the slope of the perpendicular line. We want this line to go through (2,8)

So,

y = mx+b

8 = -0.5*2+b

8 = -1+b

8+1 = b

9 = b

b = 9

The equation of the perpendicular line is y = -0.5x+9

Replace -0.5 with the fraction -1/2 if needed.

Check out the graph below.

So, on solving the question the inequality equation, that will be formed here is  3x3+2y>1 and 4xx-y< -3

If not, shade the other side of the plane. Graph each of the system's inequalities similarly. The place where all of the system's solutions intersect is where the system of inequalities' solution is located.

One solution pair exists for each independent system (x,y). There is just one possible location: the intersection of the two lines. There is no way to fix an inconsistent system. The two lines are parallel and will never cross one another, as you can see. There are countless ways to solve a dependent system.

If the inequality is true when the values of x and y are substituted, then an ordered pair (x,y) is the solution to the linear inequality. As a result, if the absolute value expression is set to a negative number, there is no solution.

the inequality equation, that will be formed here is

3x3+2y>1

and 4xx-y< -3

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17 ≥ 6x is the inequality which can be represented for the given variable x.

If not, shade the other side of the plane. Graph each of the system's inequalities similarly. The place where all of the system's solutions intersect is where the system of inequalities' solution is located.

One solution pair exists for each independent system (x, y). There is just one possible location: the intersection of the two lines. There is no way to fix an inconsistent system. The two lines are parallel and will never cross one another, as you can see. There are countless ways to solve a dependent system.

If the inequality is true when the values of x and y are substituted, then an ordered pair (x,y) is the solution to the linear inequality. As a result, if the absolute value expression is set to a negative number, there is no solution.

You must pay $8.50 or less for the copies and each copy is $0.30

Let the number of copies be x and we have the inequality,

8.50 ≥ 0.30x

Divide 5 and multiply 10 from both sides

17 ≥ 6x

Thus, 17 ≥ 6x is the inequality which can be represented for the given variable x.

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The constant of proportionality for respective problems is 2/7, 15, 2.75, 13, 4/3.

A) In the first block,

Using points we can find the equation of line ,

Using the formula (y - y₀) = m (x - x₀)

where m is the slope of the equation

The equation of line becomes 2x = 7y

⇒ y = 2/7x

⇒ 2/7 is the constant of proportionality

B) From the graph given,

Again Using the formula (y - y₀) = m (x - x₀)

If we consider two points,

(2,30) and (4,60)

The equation of line becomes

15x = y

Here, the constant of proportionality is 15.

C) It is given the equation is y = 2.75x

The constant of proportionality here is 2.75

D) It is given that,

The price of 5 games is $65.00

The price of 1 game will be 65 / 5

= $13.00

The equation becomes

y = 13x

So, the constant of proportionality becomes 13.

E) Let us consider 2 points (3,12) and (6,24)

The equation of line using the formula (y - y₀) = m (x - x₀) becomes 4x = 3y

⇒ y = 4/3 x

The constant of proportionality becomes 4/3.

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The answer is 72 boys

The unitary method is a method in which you find the value of a unit and then the value of a required number of units.

Given here: The ratio of boys to girls is 4:3 and total students as 168

thus there are 96 boys and 72 girls and also 3/4 th of the boys study french

which is equivalent to 72 boys

Therefore the number of boys that study spanish is equal to 96-72=24

Hence, 24 boys study spanish.

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The function for the given compound interest will be A = 3000(1+6/4)  and A = 5000(1+7.2)  .

What is compound interest?

The interest on savings that is computed on both the initial principal and the interest accrued over time is known as compound interest. Compound interest is computed by multiplying the starting principal amount by one and the annual interest rate is raised to the number of compound periods minus one. The final step is to deduct the initial loan principal from the calculated value.

When the amount compounds quarterly, it means that the amount compounds 4 times in a year. i.e., n = 4. We use this fact to derive the quarterly compound interest formula. Thus, the quarterly compound interest formula is:

$3000 deposit that earns 6% annual interest compounded quarterly.

A = P (1 + r/4)4

A = 3000(1+6/4)

$5000 deposit that earns 7.2% annual interest compounded monthly.

A = P (1 + r)

A = 5000(1+7.2)  

Hence the function for the given compound interest will be A = 3000(1+6/4)  and A = 5000(1+7.2)  .

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The image of (−8,12) after a dilation is (-2, 3).

A scale factor is a numerical value that can be used to alter the size of any geometric figure or object in relation to its original size. It is used to find the missing length, area, or volume of an enlarged or reduced figure as well as to draw the enlarged or reduced shape of any given figure. It should be remembered that the scale factor only affects how big a figure is, not how it looks.

Given:

Point (-8, 12)

scale factor = 1/4

After applying the scale factor then image

= (-8/4, 12/4)

= (-2, 3)

Hence, the image is (-2, 3).

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Answer:

114

Explanation:

18.6 rounds up to 19

6.5 rounds down to 6

19 x 6 = 114

[tex]b^2-4ac[/tex] when a is -4. b is -6, and c is 5

Substitute for variables with given values:
[tex]-6^2-4(-4)(5)[/tex]

[tex]=\fbox{116}[/tex]

To make the given expression x² + 4x + ___ a perfect square ( x + 2)² we must add 4 to the given expression.

As given in the question,

Given expression is equal to :

x² + 4x + ___

Using the formula of perfect square we have :

( a + b )² = a² + 2ab + b²

Here in the given expression x² + 4x + ___

a = x

b = 2

This implies that :

x² + 4x + ___

= x² + 2 ( 2 ) ( x ) + ( 2 )²

= x² + 2 ( 2 ) ( x ) + 4

= ( x + 2 )²

We must add '4' to the given expression.

Therefore, '4' must be added to the given expression x² + 4x + ___ to make it perfect square ( x + 2 )².

The above question is incomplete , the complete question is :

What must be added to the given expression x² + 4x + ___  to make it perfect square ?

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The equations can be completed by;

1)  y^8 - 1/y^8

2) 1/y^8 + 1/y^2

We have to know that there are laws that can be used when we are dealing with exponents. Two of the laws that we need to know here are the law of multiplication and the law of division.

Note that;

a^m/a^n = a^m - n (division law)

a^m * a^n = a^ m + n (multiplication law)

Having these laws at the back of our minds;

y^5 y^3 - (y^-2)^4 = y^8 - y^-8

= y^8 - 1/y^8

And;

(y^-2)^4 + y^3/y^5

y^-8 + y^-2

1/y^8 + 1/y^2

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Answer:

Slope = 1.25

Y-intercept = -1.5

Step-by-step explanation:

Slope:

[tex]\displaystyle m=\frac{y_2-y_1}{x_2-x_1}=\frac{11-(-14)}{10-(-10)}=\frac{11+14}{10+10}=\frac{25}{20}=1.25[/tex]

Y-intercept:

[tex]y=mx+b\\\rightarrow y=1.25x+b\\\rightarrow 11=1.25(10)+b\\\rightarrow 11=12.5+b\\\rightarrow -1.5=b[/tex]

Final Equation:

[tex]y=1.25x-1.5[/tex]

The Total number of smoothies after calculating are 100 .

What is ratio ?

ratio can be defined as the Given value divided by the total value, is called as ratio.

Given ,

Sally's Smoothies makes two flavors: strawberry and banana. The ratio of strawberry

smoothies to banana smoothies is 3:2.

If Sally makes 60 strawberry smoothies, The number of total smoothies does she make are.

let the banana smoothie be x .

so,

3/2 = 60/x

x = 2* 60 / 3

x = 2*20

x = 40

Total number of smoothies are = 60+40 = 100

Hence , The Total number of smoothies are 100 .

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Answer:

I don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and don't know what to say to someone who lost a lot of weight and height for all the help and their kids the the other one of it and don't

Answer:

The answer is (d) y=3x+2

Inserting the given points in the given equation yields the answer. To make the process faster, insert the zero first then you will get y=2. You have to choose whether B and D. Then insert the x values and you will obtain the correct answer which is valid for the given points x and y.

Answer: 64

Step-by-step explanation:

(47 + 39) - 22

= 86 - 22

= 64

While writing the coordinates in the coordinate plane the coordinate that comes first coordinate is represented by x-coordinate.

As given in the question,

In the coordinate plane:

Therefore, the first coordinate in the coordinate plane is represented by x-coordinate.

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This figure has Line, Point and Rotational Symmetry.

If a figure can be rotated by an angle between 0° and 360° so that the image coincides with the preimage, it has rotational symmetry. The smallest angle for which the figure can be rotated to coincide with itself is the angle of rotational symmetry.

A line of symmetry is a line that divides two equal and symmetrical parts of a shape or object. This line is also known as the axis of symmetry or the mirror line because it divides the figure symmetrically and the divided parts appear to be mirror reflections of each other.

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