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The average rate of change of the trigonometric function f(x) = 3 · sin 2x on the interval [0, π / 4] is 12 / π. (Correct choice: D)

In this question we need to determine the average rate of change of a trigonometric function on the interval [0, π / 4]. Graphically speaking, the average rate of change is represented by a line secant to the curve of the function, whose definition is shown below:

m = [f(π / 4) - f(0)] / (π / 4 - 0)

If we know that f(x) = 3 · sin 2x, then the average rate of change is:

m = [3 · sin (π / 2) - 3 · sin 0] / (π / 4 - 0)

m = (3 - 0) / (π / 4)

m = 12 / π

The average rate of change of the function is 12 / π.

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The number of lines that can run through any two of these points is 66

What are Collinear Points?

Points that are on the same straight line are referred to as collinear points. They must lie on the same straight line, even if they don't have to be co-planar. The Latin terms "col" and "linear," where "col" means "together" and "linear" means "in the same line," are the source of the English word "colinear." Collinearity is the state of being shared by two points.

A group of three or more points that are located along the same straight line is called a collinear point. On separate planes, collinear points may exist, but not on different lines. Collinearity is the quality of two points being parallel to one another. Any three or more points are only considered to be collinear if they are situated along the same straight line.

12 points are given, not three of which are collinear.

∴ Number of lines formed = ¹² C ₂

= 12!/ 10!⋅2!

=66

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Answer:

x+9=2x-4

Step-by-step explanation:

it just has to be

age=x

9 years later =+9

x+9

will be is =

x+9=

4 less = -4

double of current age =2x

=x+9=2x-4

Answer:

The current age is X and 9 years later the age will be X+9

Step-by-step explanation:

And it was stated that in 9 years time the age will be 4 less than double of the current age

And the current age is X, double of the current age is 2×X= 2x and it was stated that the double of the age will be 4 less than the person age in 9 years time so the equation statement is

X + 9 = 2x + 4

Andy spent $8.28 on 23 feet wire and Emily spent $16.2 on 15 yards wire.

In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.

From the table, we have (110, 3.30) and (145, 4.35).

Here, slope (m)=(y2-y1)/(x2-x1)

m=(4.35-3.30)/(145-110)

= 1.05/35

= 0.03

Substitute, m=0.03 and (x, y)=(110, 3.30) in y=mx+c we get

3.30=0.03(110)+c

3.30=3.3+c

c=0

Substitute, m=0.03 and c=0 in y=mx+c we get

y=0.03x

Andy needs 23 feet of the wire.

So, 23 feet =23×12

= 276 inches

Cost of 276 inches wire is y=0.03×276=$8.28

Emily needs 15 yards of the wire.

So, 15 yards =15×36

= 540 inches

Cost of 540 inches wire is y=0.03×540=$16.2

Therefore, Andy spent $8.28 on 23 feet wire and Emily spent $16.2 on 15 yards wire.

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It will take approximately 12.3 seconds for the apples to fall to the ground.

A strong, invisible force called gravity pulls everything on Earth down. Gravity keeps you and all the things that are around you from floating up into the sky. An example of gravity in action is when you jump from the ground. Gravity is the force that pulls you back down.

To find the time taken to fall to the ground

Using the formula;

[tex]H = \frac{1}{2} gt^{2}[/tex]

where H = 500 feet

Converting feet to meters

H = 152.4 meters

152.4 = [tex]\frac{1}{2} (9.8)t^{2}[/tex]

152.4 = [tex]4.9t^{2}[/tex]

[tex]t^{2} = 152.4[/tex]

[tex]t = \sqrt{152.4}[/tex]

t = 12.3seconds

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The proof of the congruent sides RU and TS is given below.

What is congruent triangles?

In any orientation, two triangles are said to be congruent if their three sides and three angles are equal.

Therefore, if two triangles have the same number of sides on each side, they are said to be congruent. Additionally, we can determine that they are congruent if we have a side, an angle between the sides, and then another side that is congruent, in that order: side, angle, side.

Let the given figure is a rectangle RSTU.

Since the all four angles of rectangle are of 90 degree.

So, each angle of rectangle RSTU is 90 degree.

Let, US divides the angle RUT and angle RST and form two new triangles, ΔSRU and ΔSTU.

Both the triangles having same measure of angles and sides.

⇒  ΔSRU ≅ ΔSTU

As the triangles are congruent so their sides are congruent.

Hence, RU is congruent to TS.

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The exact length of the curve: y = ln 1 − x^2 , 0 ≤ x ≤ 1 6 will be[tex]$$\int \frac{d x}{a^2-x^2}=\frac{1}{2 a} \ln \left|\frac{a+x}{a-x}\right|+c$$[/tex]

[tex]$$\begin{aligned}y & =\ln \left(1-x^2\right), \quad 0 \leq x \leq \frac{1}{6} \\\frac{d y}{d x} & =\frac{1}{1-x^2} \cdot \frac{d}{d x}\left(1-x^2\right) \quad\left\{\frac{d}{d u} \ln u=\frac{1}{u}\right\} . \\& =\frac{-2 x}{1-x^2}\end{aligned}$$[/tex]

Now, the length of the curve is given by

[tex]$$\begin{aligned}L & =\int_a^b \sqrt{1+\left(\frac{d y}{d x}\right)^2} d x \\& =\int_0^{1 / 6} \sqrt{1+\left(\frac{-2 x}{\left.1-x^2\right)^2}\right.} d x \\& =\int_0^{1 / 6} \frac{\sqrt{\left(1-x^2\right)^2+4 x^2}}{\left(1-x^2\right)} d x \\& =\int_0^{1 / 6} \frac{\sqrt{1+x^4-2 x^2+4 x^2}}{\left(1-x^2\right)} d x\end{aligned}$$[/tex]

[tex]=\int_0^{1 / 6} \frac{\sqrt{1+x^4+2 x^2}}{1-x^2} d x\\\\=\int_0^{1 / 6} \frac{\sqrt{\left(1+x^2\right)^2}}{1-x^2} d x\\\\=\int_0^{1 / 6} \frac{1+x^2}{1-x^2} d x[/tex]

Now, since the degree of Numerator & degree of Denominator are the same,

So, we will perform a long division process

[tex]$$\begin{aligned}& \therefore \frac{1+x^2}{1-x^2}=-1+\frac{2}{1-x^2} \quad\left\{\begin{array}{l}1-x^2 \frac{-1}{\frac{x^2+1}{x^2+1}} \\\frac{2}{2}\end{array}\right. \\& \therefore L=\int_0^{1 / 6}\left(-1+\frac{2}{1-x^2}\right) d x \\&\end{aligned}$$[/tex]

[tex]$$\begin{aligned}& =\left[-x+2 \cdot \frac{1}{(271} \ln \left|\frac{1+x}{1-x}\right|\right]_0^{1 / 6} \\& =\left(-\frac{1}{6}+\ln \left|\frac{1+1 / 6}{1-1 / 6}\right|\right)-\left(0+\ln \left|\frac{1+0}{1-0}\right|\right) \\& =-\frac{1}{6}+\ln \left(\frac{7 / 6}{5 / 6} \cdot-(0+0)\right.\end{aligned}$$[/tex]

[tex]L=\frac{-1}{6}+\ln \left(\frac{7}{5}\right)\\[/tex]

L=0.1698 approximately 0.17 units

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The complete solution will be;

Jeremy cannot find the cost by adding the percent's.

The 25% off of $60 is $45,

The 25% off of $60 and then subtracting 10% from the cost is $39.

The two total cost are not equal.  

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

Given that;

Jeremy bought a $60 pair of sneakers on sale for 15%.

And, Then he used a coupon for additional 10% off.

Now,

Let the cost of the sneakers = x

So, We can formulate;

⇒ x - 15% of x - 10% of x = 60

Solve for x as;

⇒ x - 15x/100 - 10x/100 = 60

⇒ x - 25x/100 = 60

⇒ x - x/4 = 60

⇒ 3x = 60 × 4

⇒ x = 80

Thus, The cost of the sneakers = $80

Now,

The 25% off of $60 is,

⇒ 60 - 25% of 60

⇒ 60 - 25 × 60 / 100

⇒ 60 - 15

⇒ $45

And, The 25% off of $60 and then subtracting 10% from the cost.

⇒ 60 - 25% of 60 - 10% of 60

⇒ 60 - 15 - 6

⇒ 45 - 6

⇒ $39

By adding both percent we get;

⇒ $45 + $39

⇒ $84

Thus, The complete solution will be;

''Jeremy cannot find the cost by adding the percent's.

The 25% off of $60 is $45,

The 25% off of $60 and then subtracting 10% from the cost is $39.

The two total cost are not equal.''

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The length of the slide is 10 feet.

From the provided information, the height of the slide ladder is 8 feet and the distance between the bottom of the slide and the ladder is 6 feet. The three sides form a right-angle triangle.

Pythagoras theorem postulates that in a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides which are perpendicular and base. Hence,

Hypothenuse^2 = Base^2 + Perpendicular^2

The length of the slide using Pythagoras theorem is where length of the slide is the hypothenuse of the triangle,

Length^2 = 8^2 + 6^2 = 100

Length = √100 = 10 feet

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Answer:

I x I > 6

Step-by-step explanation:

The triangle LMO and LNO are, if LM ≅ ON, ∠ M ≅ ∠ N concurrent by SAS congruence.

Three interconnected vertices make up one of the basic geometric shapes denoted by the symbol Δ.

Given:

LM ≅ ON

∠ M ≅ ∠ N

As you can see, the side LO is common in both the triangle,

So In the concurrency SAS, the two sides of a triangle are two to be equal, and an angle is to be equal.

Therefore, the triangle LMO and LNO are, if LM ≅ ON, ∠ M ≅ ∠ N concurrent by SAS congruence.

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The student's grade on a 4-point scale (with A = 4) is a random variable X with a discrete probability distribution.

The probabilities distributions are classified as discrete or continuous depending on the values assumed by their input variables, as follows:

The students grades have these possible outcomes:

As there are only certain values, this is a discrete distribution.

The problem asks to classify the distribution of the student's grade as discrete or continuous.

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Answer:

B

F

Step-by-step explanation:

this is vector, and its to be moving from point to a point through line , and you have a direction of it's move , so it will be from T to S ( magnitude) and the arrow is the direction <--

In the given word "WISCONSIN" there are exactly 9 letters, but some are repeated, so we can rearrage the letters in the word in 9!/8 ways.

When the order of the arrangements counts, a permutation is a mathematical approach that establishes the total number of alternative arrangements in a collection. Choosing only a few items from a collection of options in a specific sequence is a common task in arithmetic problems. Permutations are commonly mistaken with combinations, a different mathematical concept. The sequence in which the components are selected does not, however, affect combinations. In other words, whereas ab and be are equivalent in combinations, they are deemed to be distinct arrangements in permutations.

The formula for calculating the permutation is:

p(n,k) = n! /(n-k)!

where , n - Total number of elements in a set

             k - Number of selected elements

In the above question , the number of ways to rearrange 9 letters would be 9! but, since there are 2Ss , 2Ns and 2Is so the number of ways to rearrange 9-letter strings will be:

9!/(2!*2!*2!) = 9!/8

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The displacement of the student after taking mentioned path is 4 blocks.

Lets understand the displacement before heading to calculations. So, displacement is the shortest distance between initial and final points. The distance is the total path covered. In this problem, we see that 3 blocks covered in South direction were equated by 3 blocks covered in North direction.

So, the displacement is the distance covered in West direction. Based on this, the displacement of the student is 4 blocks in treading mentioned path.

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The graph represents the piecewise function is [tex]y=\frac{5 x}{4}+\frac{15}{4}[/tex].

What is graph?

A graph is a structure that resembles a set of items in discrete mathematics, more especially in graph theory, in which certain pairs of the objects are conceptually "connected." The items are represented by mathematical abstractions known as vertices, and each pair of connected vertices is referred to as an edge.

Given,

Graph representing a piecewise function:

A piecewise function is a function that is defined by different formulas or functions for each given interval. It’s also in the name: piece. The function is defined by pieces of functions for each part of the domain.

So,

Using the formula:

[tex]y=\frac{5 x}{4}+\frac{15}{4}[/tex]

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The values of the x and y are:

I) x = 20 and y = 10√2.

J) x = 16.97 and y = 33.94.

K) x = 9.89 and y = 24.49.

L) x =  35.75 and y = 123.84.

What is the 45-45-90 rule and 30-60-90 rule?

One of these right triangles is known as a 45-45-90 triangle because its angles are 45 degrees, 45 degrees, and 90 degrees. This is a right triangle that is isosceles. The other triangle is known as a 30-60-90 triangle because its angles are 30 degrees, 60 degrees, and 90 degrees.

We have,

special right triangle, some are the 45-45-90 rule and some are the 30-60-90 rule.

in the 45-45-90 leg of the triangle is equal to a, then:

The second leg is also equal to a;

The hypotenuse is a√2;

The area is equal to a²/2; and

The perimeter equals a(2 + √2).

Assume that the shorter leg of a 30 60 90 triangle is equal to a. Then:

The second leg is equal to a√3;

The hypotenuse is 2a;

I) in the first triangle:

The second leg is also equal to y = 10√2;

The hypotenuse is y√2:

= y√2

= (10√2)√2

= 20

so, x = 20 and y = 10√2.

J) in the second is of 30-60-90 triangle:

x√3 = 12√6

x = 12√6 / √3

x = 16.97

y = 2x = 16.97 * 2

y = 33.94

so, x = 16.97 and y = 33.94.

K) the third diagram is of a 30-60-90 triangle:

2x = 14√2

x = 14√2 / 2

x = 9.89  = 10

z = x√3 = 10√3

y = z√2 = 10√3 * √2

y = 24.49

so, x = 9.89 and y = 24.49.

L) the fourth diagram is of a 30-60-90 triangle:

z = 16√10 * √2

= 71.5

2x = 71.5

x =  35.75

y = x√3

y = 123.84

so, x =  35.75 and y = 123.84.

Hence, the values of the x and y are:

I) x = 20 and y = 10√2.

J) x = 16.97 and y = 33.94.

K) x = 9.89 and y = 24.49.

L) x =  35.75 and y = 123.84.

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a = 5 cm

b = 12 cm

c = ?

c = √{ a² + b }

c = √{ 5² + 12² }

c = √{ 25 + 144 }

c = 13 cm

____

The height of a triangle when the area is 16 inches h=4

The area of the triangle(A) is given by:

A=1/2bh---(1)

where,

b is the base

h is the height

As per the statement:

The area of a triangular block is 16 square inches.

A=16 in

It is given:  If the base of the triangle is twice the height

⇒ b=2h---(2)

Substitute these in [1] we have;

A=1/2bh

16=1/2(2h)h

h^2=16

h=4in

Substitute h =4 in [2] we have;

b=2h

b=2(4)

b=8in.

Therefore, the base and height of the triangle are 8 inches and 4 inches.

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Answer:

y = 2/3 x + 6

Step-by-step explanation:

I do not see a point C, so I guess you get to make your own point C.  I will make it at (06,)

The equation of the line seen is

y = 4/6x + 5  If you start at A and count up, you will get 4 unit and then count right you will be 6 units.  This is the slope 4/6.  the 5 is where the line crosses the x axis (0,5)

We can simplify  4/6 to 2/3

y = 2/3 x + 5

When lines are parallel, they have the same slope, but different y intercept.  

Since I made C (0,6)  The equation of this parallel line would be

y = 2/3 x + 6

Draw a line parallel to AB but make sure that it goes through the point (0,6)

Answer: 60 degrees

Step-by-step explanation:

Make the hexagon into a bunch of equilateral triangles, and split the rhombus into two equilateral triangles.

Answer:

----------------------------

Sides of the given rhombus are parallel to sides of the hexagon.

If you extend sides of the rhombus they will make diagonals connecting opposite vertices..

These diagonals are angle bisectors and hence the angle x if half the interior angle of the hexagon:

Answer:

Isolate the radical, then raise each side of the equation to the power of its index a=0

Step-by-step explanation:

a + b is not identity

The distance between City A (north latitude 45°28′ N) and City B (latitude 39°52' N) is 286 miles.

What is latitude?

Latitude is the measurement of distance north or south of the Equator. It is measured with 180 imaginary lines that form circles around Earth east-west, parallel to the Equator. These lines are known as parallels.

The measure of the central angel is 45°28' - 39°52' = 5.8'

8' = 8 ° / 60

   = 0.133333

θ = 4° + 0.133333

  = 4.133333° × [tex]\frac{\pi }{180}[/tex]

  = 4.133333° × 0.017453

  ≈ 0.072140 radian

now , θ  ≈ 0.072140 radian and r = 3960 miles

S  = θ r

   = 3960 × 0.072140

   ≈ 285.6744

S = 286 miles

Therefore, the distance between City A (north latitude 45°28′ N) and City B (latitude 39°52' N) is 286 miles.

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Answer:

we would expect the mean of the sample to fall below 66 about 5% of the time

Step-by-step explanation:

To answer this question, we can use the 68-95-99.7 rule, which states that in a normal distribution, approximately 68% of the values fall within one standard deviation of the mean, 95% fall within two standard deviations, and 99.7% fall within three standard deviations.

In this case, the mean of the sample is 70 and the standard deviation is 10, so the values that fall within one standard deviation of the mean are between 60 and 80. The values that fall below 66 are outside of this range, so we need to determine what percentage of the values fall within two standard deviations of the mean.

The values that fall within two standard deviations of the mean are between 50 and 90, so the percentage of values that fall below 66 is approximately (100 - 95) = 5%.

Therefore, we would expect the mean of the sample to fall below 66 about 5% of the time.

The lines of the functions f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7 have slopes that are negative reciprocals and are said to be perpendicular lines.

The general formula for the equation of a line in slope intercept form is;

y = mx + c

where;

m is slope

c is y-intercept

We are given the linear equations as; f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7

Now, when two lines are parallel, their slopes will be the same and equal. However, when two lines are perpendicular, their slopes will be a negative reciprocal of each other.

In our given functions, we see that the slope of g(x) is a negative reciprocal of the slope of f(x). Thus, both lines are concluded to be perpendicular to each other.

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The complete question is;

Consider the functions f(x) = ²/₃x - 5 and g(x) = -³/₂x + 7. what do the slopes indicate about the lines?

The material remained for the blanket after trimming off by Keith is 2.5 yards.

Simple subtraction and multiplication can provide the result. Beginning will be by the multiplication. Firstly finding the exact amount of material trimmed = 3×1/6

Performing multiplication and division on Right Hand Side of the equation

Trimmed material = 1/2 or 0.5

Now, we need to subtract the trimmed material from overall amount of material

Remaining material = 3 - 0.5

Performing subtraction on Right Hand Side of the equation

Remaining material = 2.5 yards

Hence, 2.5 yards of material remained.

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