What is an equation for the line that passes through the points (4,8) and (-2,-7)?

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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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:

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 <--

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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a = 5 cm

b = 12 cm

c = ?

c = √{ a² + b }

c = √{ 5² + 12² }

c = √{ 25 + 144 }

c = 13 cm

____

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:  y = 2x - 8.

Step-by-step explanation:

To find the equation of a line parallel to the given line, we need to find the slope of that line. The slope of the line 2x + y - 7 = 0, this can be found by taking the coefficient of the x term and the y term and dividing them. In this case, the slope is 2/1 = 2.

Now that we know the slope of the line we want to find, we can use the slope-intercept form of the line, y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis). Since the line we are looking for is parallel to the given line, it will have the same slope, so we can use the same value for m.

The line x + y - 4 = 0 and 2x – y = 8 intersect at the point (4,0). We can use this point to find the value of b in the equation y = 2x + b. Substituting the values from the intersection point, we get 0 = 2(4) + b, which simplifies to b = -8.

Therefore, the equation of the line that is parallel to the given line and passes through the intersection of the two lines is y = 2x - 8.

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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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 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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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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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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Answer:

1/20 pound of sand in each vase

Step-by-step explanation:

Raj made 1/4 and divided it into 5 vases.

1/4 ÷ 5

= 1/4 × 1/5

= 1/20

Each of the 5 vases had 1/20 lb. of sand.

Answer:

To find the exact values of the sine and cosine of 2 theta, we can use the trigonometric identities that relate these functions to the sine and cosine of theta.Since theta lies in quadrant IV, we know that the cosine of theta is negative. We can use the identity cos(theta) = sin(pi/2 - theta) to find the sine of theta:sin(theta) = sin(pi/2 - theta)

= -sin(theta)Solving this equation for sin(theta), we find that sin(theta) = -2/5.To find the cosine of 2 theta, we can use the identity cos(2 theta) = 1 - 2sin^2(theta):cos(2 theta) = 1 - 2sin^2(theta)

= 1 - 2(-2/5)^2

= 1 + 8/25

= 9/25To find the sine of 2 theta, we can use the identity sin(2 theta) = 2sin(theta)cos(theta):sin(2 theta) = 2sin(theta)cos(theta)

= 2(-2/5)(-cos(theta))

= 4/5cos(theta)Thus, the exact values of the sine and cosine of 2 theta are sin(2 theta) = 4/5cos(theta) and cos(2 theta) = 9/25.

Step-by-step explanation:

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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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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Answer:

I x I > 6

Step-by-step explanation:

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 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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Answer:

To find the sum of 30 and 40 by factoring out the greatest common factor, first you need to find the greatest common factor (GCF) of both numbers. The GCF of 30 and 40 is 10. Then, you can divide both numbers by 10 to get 3 and 4 respectively. Finally, add the numbers together to get the sum of 7. Therefore, the sum of 30 and 40 by factoring out the greatest common factor is 7.

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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Dilation and rotation are the a series of transformations, which transformation would make two figures similar as opposed to congruent.

Rotations, reflections and translations are known as rigid transformations; this means they do not change the size or shape of a figure, they simply move it. These rigid transformations preserve congruence.

Rotations is defined as the motion of an object around a center or an axis.

Dilation, however, are not rigid transformations, since they change the size of a shape. Dilation would not change the shape, just the size; the angle measures would be the same, and the ratio of corresponding sides would be equal to the scale factor used in the dilation.

This transformation produces an image that is the same as the original shape. But there is a difference in the size of the shape. A dilation should either stretch or shrink the original shape.

This would give us a similar, but not congruent, figure.

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In a series of transformations, which transformation would make two figures similar as opposed to congruent?

A) Rotation and translation

B) Dilation and rotation

C) Reflection and rotation

D) Translation and reflection

Answer:

29

Step-by-step explanation:

First, you will calculate 3 cubed, which is 27, and then multiply 1 and 2. That will result in 27+2 which will be 29

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 value of x in the solution set of the inequality: - 2(3x+2) > -8x + 6 is x>5.

- 2(3x+2) > -8x + 6

Let's solve the inequality by isolating the variable x.

or, - 6x -4 > -8x + 6

Add 4 on both sides

or, - 6x - 4 + 4 > -8x + 6 + 4

or, -6x > -8x + 10

Add 8x on both sides

or, -6x + 8x > -8x +8x +10

or, 2x > 10

Divide both sides by 2

or, 2x/2 > 10/2

0r, x > 5

Hence, the value of x in the solution set of - 2(3x+2) > -8x + 6 is x>5.

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The complete question is :

What value of x is in the solution set of the inequality -2(3x+2) > -8x + 6?

Answer:

5

Step-by-step explanation: