Find the linear approximation of the function f(x) = √1-x at a = 0 and use it to approximate the numbers √0.9 and √0.99. (Round your answers to four decimal places.)L(x) =√0.9 ≈√0.99 ≈

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 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:

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 kids' judgments of the significance of excellent nutrition in the food given may vary based on the students' ages and, consequently, on school level, making this variable an useful one to stratify the survey on.

Each sample has an equal chance of being picked when using the random sampling sampling approach. It is intended that a sample picked at random will be a fair reflection of the entire population. Two different approaches to sampling exist: Random selection is a key component of probability sampling, which enables you to draw robust statistical conclusions about the entire group. Non-probability sampling incorporates non-random selection based on practicality or other factors, making it simple to gather data.

The student's degree of education is one potential stratification factor. Depending on the kids' ages and, thus, on their academic ability, the students' judgments of the significance of nutritious meals may vary. In contrast to middle school and high school pupils, elementary school children may place different values on the food that is offered.

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It will take 11 years for the investment to exceed £7000 if £5200 is invested over a year at a compound interest rate of 2.8%.

The addition of interest to the principal of a loan or deposit is known as compound interest, sometimes known as interest on principal and interest. It occurs when interest is reinvested, added to the lent capital instead of being paid out, or the borrower is required to pay it, resulting in the next period's interest being generated on the principal amount plus any accumulated interest.

In contrast to simple interest, which does not compound since past interest is not added to the current period's principal, compound interest allows for the accumulation of interest from prior periods.

We solve this problem by using compound interest.

Given,

£5200 is invested over a year at a compound interest rate of 2.8%.

£7000. 1.028 = 5 200 x 5200 × 1.028¹¹ = 7045.76 2.

So, number of years=11

Therefore, It will take 11 years for the investment to exceed £7000.

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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.

On integrating we will find that the area enclosed by the curves given is   64 / 3.

First we will plot a graph ,

The graph for x = y will be a straight line passing through the origin and the graph for  4x + y2 = 12 will be a parabola.

Integration of  4x + y2 = 12

= ∫4/2x² + y² dx = C

= 2x² + y² x - C

On putting the value of x and y we get ,

the area as 64 / 3.

In integration we generally add various parts to form one single object. It is a mathematical concept and comes under the branch of calculus.

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Answer:

I x I > 6

Step-by-step explanation:

Since the spacecraft returned to planet Earth and slowed until a parachute could be opened for a landing at sea, a statement which best explains how the spacecraft slowed down for a safe landing is that: C. the parachute created an upward force to resist the downward force of gravity.

In Mathematics, the force of gravity is also referred to weight and it can be defined as the force that is acting on a physical object or body, especially due to the effect of gravity.

Newton's Third Law of Motion states that for every action there is an equal but opposite reaction. This ultimately implies that, in every interaction, there is a pair of forces acting on the two interacting objects such as between a spacecraft and a parachute.

In this context, we can reasonably infer and logically deduce that the parachute would create an upward force in order to resist the downward force of gravity that is acting on it, in accordance with Newton's Third Law of Motion.

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(3xy^2)^3×(-4x^2y)/(2x^2y^2)^2

= -108x^5y^7/4x^4y^4

= -108xy^3/4

Answer: -27xy^3

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 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 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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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.

Answer:

  QR = 20

Step-by-step explanation:

You want the measure of side QR of isosceles triangle PQR with congruent sides PQ and QR marked 5x and 3x+8, respectively.

The sides of the isosceles triangle are congruent, so we have ...

  PQ = QR

  5x = 3x +8

  2x = 8 . . . . . . . subtract 3x

  x = 4 . . . . . . . divide by 2

  QR = 3x +8 = 3(4) +8 = 20 . . . . . . find the length of QR

The measure of QR is 20 units.

__

Additional comment

You will recognize that the segment QS is both the altitude and median of the triangle. That means the triangle is isosceles, so symmetrical about QS.

If you need a formal proof, you can observe that QS is congruent to itself, so the two right triangles are congruent by LL (or SAS).

Examining the triangles, the length of side QT is 2.5

This is a term used in geometry to mean that the respective sides of the triangles are proportional and the corresponding angles of the triangles are congruent

Hence assuming the corresponding angles of the triangle are congruent then the side should be in proportions

The solution is worked out using sides SQ and QR, then ST and TU

(12 + QT) / 18 = 12 / 6(QT)

let QT = x

(12 + x) / 18 = 12 / 6x

cross multiplying

6x(12 + x) = 18 * 12

72x + 6x² = 216

6x² + 72x - 216

x = 2.5 OR -14.4

take the positive value x = 2.5

The length of QT is 2.5

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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

The df would equal in the chi-square statistic for this test is found as 4.

The formula for the chi-square (X²) statistic is;

(X²)  = ∑(O - E)²/E

For the given question-

The answer to the query is;

We would determine whether two qualities, A and B, under examination are independent or not using the chi-square test.

Taking that as a given, the null hypothesis H0: The two factors are unrelated.

Now, the degrees of freedom in the independence test using chi-square is,

v = (r-1) (s-1) (s-1).

r = 3 and s = 3.

The degree of freedom are;

ν =  (r-1) (s-1) = (3-1) (3-1) = 2×4

ν  = 4

Consequently, let test statistic be,

(X²)  = ∑(O - E)²/E

Consequently, df would equal in the chi-square statistic for this test is found as 4.

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When the original point is mirrored in the x-axis, it becomes (-3,4).

The horizontal and vertical distances from the two reference axes are used to define the location of a point on a coordinate plane. The x-value and y-value are often denoted as (x,y). To find the coordinates of a point in a coordinate system, do the inverse. Start at the point and trace a vertical line up or down to the x-axis. There you have your x-coordinate. Then repeat the process, but this time follow a horizontal line to determine the y-coordinate.

Here,

The reflected point= (-3,-4)

A point reflected in the x-axis followed by the y-axis,

=(-3,4)

The original point is (-3,4) when point reflected in the x-axis followed by the y-axis.

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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:

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 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:

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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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.

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:

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:

The equation intersects the x or y axis a total ot three times:

(-1.5,0), (5,0), and (-15,0).

Step-by-step explanation:

I will assume that the "all points of intersection" refers to the intersection of the line with the x and y axes.  One can find those points by either a mathematical or graphing approach.

    We and to find all points that have a zero for either the x or y coordinate:  (0,y) and (x,0)

(x,0):  Solve  p(x)=(2x+3)(x-5) for p(x) = 0:

  0=(2x+3)(x-5)

 (2x+3)=0 and (x-5)=0

 (2x+3)=0

2x=-3

x= - 1.5    (-1.5,0)

and

  (x-5)=0

X=5     (5,0)

--

(0,y):  Solve  p(x)=(2x+3)(x-5) for p(x)  for x=0:

p(x) = (2x+3)(x-5)

p(x) = (2*0+3)(0-5)

p(x) = (3)*(-5)

p(x) = -15       (-15,0)

See the attached graph.  The points of intersection can be located, as shown.  They match the values obtained mathematically, but a free graphing utility (DESMOS) makes graphing a pleasant chore.