There may be many polynomial equations with integer coefficient which have the square root of 8 - 1 and the square root of 8 + 1 as its roots. Construct an equation of smallest degree with integer coefficients that has the square root of 8 - 1 and the square root 8 + 1 as its roots

The expression for the area under the graph of f(x) = √x as a limit of Riemann sums is A = lim [n -> ∞ ] Ax [f(x₁) + f(x₂) + ... + f(xₙ)]

For a continuous function f(x), the area A of the region S that lies under the graph of the function can be expressed as the limit of the sum of the areas of approximating rectangles. Each rectangle has a width of Ax, where A is the interval over which we want to find the area and x is a point within that interval. The height of each rectangle is f(x).

Using this definition, we can find an expression for the area under the graph of f(x) = √x, where 1 ≤ x ≤ 13. We start by dividing the interval [1, 13] into n subintervals, each of width Ax = (13-1)/n = 12/n. We can label the endpoints of the subintervals as x₁, x₂, ..., xₙ+1, where x₁ = 1 and xₙ+1 = 13.

Next, we approximate the area under the curve using n rectangles. The height of the ith rectangle is f(xi), where xi is any point in the ith subinterval. The width of each rectangle is Ax, so the area of the ith rectangle is f(xi) Ax. The total area of the rectangles is then the sum of the areas of each rectangle:

f(x₁) Ax + f(x₂) Ax + ... + f(xₙ) Ax

We can simplify this expression by factoring out the common factor of Ax:

Ax [f(x₁) + f(x₂) + ... + f(xₙ)]

Taking the limit as n approaches infinity, we obtain:

A = lim [n -> ∞] Ax [f(x₁) + f(x₂) + ... + f(xₙ)]

To evaluate the limit, we need to find a formula for the sum of f(xi) for i = 1 to n.

This is a sum of square roots, which can be approximated using numerical methods or evaluated exactly using calculus techniques such as the definite integral.

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Answer:

[tex]\displaystyle \lim_{n \to \infty}\sum^n_{i=1}\left(\dfrac{12}{n}\right)\sqrt[7]{1+\dfrac{12i}{n}}[/tex]

Step-by-step explanation:

The Riemann sum is a method by which we can approximate the area under a curve using a series of rectangles.

The area under the curve of f(x) on the interval [a, b] is represented by:

[tex]\boxed{\begin{minipage}{6cm}$\displaystyle \int^b_af(x)\; \text{d}x=\lim_{n \to \infty}\sum^n_{i=1}f(x_i) \cdot \Delta x$\\\\\\where $\Delta x=\dfrac{b-a}{n}$ and $x_i=a+\Delta x \cdot i&\\\end{minipage}}[/tex]

Δx is the width of each rectangle.

[tex]f(x_i)[/tex] is the height of each rectangle.

[tex]x_i[/tex] is the right endpoint of each rectangle.

Therefore:

[tex]\begin{aligned}\displaystyle \int^b_af(x)\; \text{d}x&=\lim_{n \to \infty}\sum^n_{i=1}f(x_i) \cdot \Delta x\\\\&=\lim_{n \to \infty}\sum^n_{i=1}f\left(a+\left(\dfrac{b-a}{n}\right)i\right) \cdot \left(\dfrac{b-a}{n}\right)\end{aligned}[/tex]

The given interval is [1, 13]. Therefore, a = 1 and b = 13:

[tex]\implies \Delta x=\dfrac{13-1}{n}=\dfrac{12}{n}[/tex]

As f(x) = ⁷√x then:

[tex]\implies f(x_i)=f(a+i \Delta x)=\sqrt[7]{a+\Delta x \cdot i}=\sqrt[7]{1+\dfrac{12i}{n}}[/tex]

Substituting into the summation formula:

[tex]\begin{aligned}\displaystyle \int^{13}_1 \sqrt[7]{x}\; \text{d}x&=\lim_{n \to \infty}\sum^n_{i=1}\sqrt[7]{1+\left(\dfrac{13-1}{n}\right)i} \cdot \left(\dfrac{13-1}{n}\right)\\\\&=\lim_{n \to \infty}\sum^n_{i=1}\left(\dfrac{12}{n}\right)\sqrt[7]{1+\dfrac{12i}{n}}\\\\\end{aligned}[/tex]

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Answer:

4:5 or 4/5 or .8 or 80%

Step-by-step explanation:

divide both by 3. then convert to percent or decimal

Answer:

-m = x is the answer hope it helps

The value of variable 'v' in the expression ( v - 2 ) / (2v² + 10v ) + 1 / ( 2v + 10 ) = 1 / 2 is equal to -4.

The expression is equal to,

( v - 2 ) / (2v² + 10v ) + 1 / ( 2v + 10 ) = 1 / 2

Simplify the expression we have,

⇒ ( v - 2 ) / 2v ( v + 5 ) + 1 / 2( v+ 5 ) = 1 / 2

Take the least common multiple of the denominator we have,

⇒ [( v - 2 ) + 2 ] / 2v( v+ 5 ) = 1 / 2

⇒ v  / 2v( v+ 5 ) = 1 / 2

Multiply both the sides of the expression by 2 we get,

⇒ 1 / 2( v+ 5 ) = 1 / 2

⇒ 1 /( v + 5 ) = 1

Cross multiply the expression we get,

⇒ v + 5 = 1

Subtract 5 from the both the sides of the equation we get,

⇒ v = -4

Therefore , the value of v in the given expression is equal to -4.

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Answer:

0 ≤ < 2

Step-by-step explanation:

Answer:1.02 5.27 are correct

Step-by-step explanation:We can solve this quadratic equation in cos(θ) by using the substitution u = cos(θ):

3u^2 + 6u - 4 = 0

Now we can use the quadratic formula to solve for u:

u = (-b ± sqrt(b^2 - 4ac)) / 2a

where a = 3, b = 6, and c = -4. Substituting these values, we get:

u = (-6 ± sqrt(6^2 - 4(3)(-4))) / 2(3)

u = (-6 ± sqrt(84)) / 6

u = (-3 ± sqrt(21)) / 3

Therefore, either:

Note that where the above conditions are given, m∠FCA is also = 158°. This is due to the principle of alternate angles.

A pair of angles that are created by a transversal intersecting two parallel lines, known as alternate angles, have key properties which contribute to their importance in geometry.

They take place on opposite sides of the transversal and have equal magnitudes.

These concepts aid in understanding vertical angles and parallel lines within geometrical systems.

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Answer: ur correct answer is 158

Step-by-step explanation: not doing this to tke points!!

The measures of the three intercepted arcs from least to greatest are;

AC < BC < AB

We have,

We are given that

Angle ACB = 63 degrees

Arc BC is 118 degrees.

Now, from the Triangle angle sum theorem, we know that the sum of all interior angles of any triangle is always equal to 180 degrees.

Thus, the sum of the interior angles of the inscribed triangle given to us id also 180 degrees.

Now, since Angle ACB = 63 degrees

Then Arc AB = 63 * 2 = 126

Arc AC = 360 - (Arc BC + Arc AB)

We are given BC = 118 degrees

Thus;

Arc AC = 360 - (118 + 126)

Arc AC = 116°

Thus, the least arc angle is Arc AC while the greatest is Arc AB

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complete question:

Triangle ABC is inscribed in a circle centered at point O.

The figure shows triangle ABC is inscribed in a circle centered at point O and has an angle ACB is 63 degrees and arc BC is 118 degrees.

Order the measures of the three intercepted arcs from least to greatest.

Answer:

16.5

Step-by-Step Explanation:

16.5 would be the average because it is right between 16 and 17. hope that helps :)

The probability that the student is a sophomore, given that they are in work, is given as follows:

P(sophomore|work) = 30%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The outcomes for this problem are given as follows:

Hence the probability is obtained as follows:

P(sophomore|work) = 3/10 = 0.3 = 30%.

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Based on the standard deviation given and the sample size, the test statistic can be found to be 134.20.

The test statistic can be found by the formula:

= ((Sample size - 1) x Standard deviation of sample²) / Standard deviation of population

Solving gives:

= ((173 - 1) x 10.6²) / 12²

= 19,325.92 / 144

= 134.20

Hence, the test statistic is 134.20.

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The solid that is produced rotating the triangle about the line m is given as follows:

A cylinder with height of 2 units.

The cylinder has a circular base, hence we must start there, then we consider that;

Rotating the line, the features of the cylinder generated are given as follows:

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Answer:

To find the mean of the data, we add up all the heights and then divide by the number of players:

Mean = (58 + 60 + 61 + 59 + 63) / 5

Mean = 301 / 5

Mean = 60.2 inches

Therefore, the mean height of the players on the Tracy basketball team is 60.2 inches.

The answer is 60.2in.

Step-by-step explanation:

Answer:125

Step-by-step explanation:

the drawing (fountains) was 10 inches apart, but real life was 400. 400/10 = 40. this means that the real life distance is 40x the distance of the drawing.

therefore, multiply 3.125 by 40 as well. you get 125.

Answer: 125 ft

Step-by-step explanation:

Let x be the real remove between the two shows.

At that point, we are able set up the extent:

10 inches / 400 feet = 3.125 inches / x

To illuminate for x, able to cross-multiply and disentangle:

10 inches * x = 400 feet * 3.125 inches

10x = 1250

x = 125 feet

Answer:

been wondering that my self

Step-by-step explanation:

that's why they asked

Answer:

5cm

Step-by-step explanation:

To find the radius of the base of the cone, we need to use the formula for the volume of a cone, which is V=31​πr2h

, where V is the volume, r is the radius, and h is the height1

We are given the volume and the height, so we can plug them into the formula and solve for r:

200π=31​πr2(24)

Simplify and isolate r:

r2=8π200π​

r2=25

r=25​

r=5

Therefore, the radius of the base of the cone is 5 cm.

To solve using the elimination method, we need to eliminate one of the variables.

Multiplying the first equation by 3, we get:

9x = 165 - 21y

Now we can write the system as:

9x = 169 - 22y

9x = 165 - 21y

Subtracting the second equation from the first, we get:

0 = 4 - y

Solving for y, we get:

y = 4

Substituting y = 4 in the first equation, we get:

3x = 55 - 7(4)

3x = 27

x = 9

Therefore, the solution to the system is (x, y) = (9, 4).

The system is consistent and independent, since it has a unique solution.

Answer:

a. The given equation is (y - 3)^2 -10 = 71. To determine the number and type of solutions, we need to use the discriminant, which is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in standard form (ax^2 + bx + c = 0). In this case, the equation can be rewritten as (y - 3)^2 = 81, which is in the form of (y - k)^2 = r^2, where k = 3 and r = 9. Therefore, the equation can be written as (y - k)^2 - r^2 = 0, which is a quadratic equation with a = 1, b = -6, and c = -72. The discriminant is then b^2 - 4ac = (-6)^2 - 4(1)(-72) = 300. Since the discriminant is positive, there are two real solutions.

b. To solve the equation (y - 3)^2 -10 = 71, we first add 10 to both sides to get (y - 3)^2 = 81. Then, we take the square root of both sides to get y - 3 = ±9. Adding 3 to both sides, we get y = 3 ± 9, which gives us two solutions: y = 12 and y = -6.

Therefore, the equation (y - 3)^2 -10 = 71 has two real solutions, which are y = 12 and y = -6.

Step-by-step explanation:

Answer:  type: real   number of solutions:2   y=12,-6

Step-by-step explanation:

see images for explanations

Answer:

68

Step-by-step explanation:

the median is the term in the middle (53) the upper quartile is the median of the terms above the median. Since there are an even number of terms, you need to average them (58+78)/2

Answer:

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

To find the slope use the slope formula:

Substitute the coordinates into slope formula to get the slope:

The matching choice is B.

Answer:

The normalization property of the distribution law states that the sum of probabilities of all possible outcomes must equal 1. In this case, we have:

P(X=k) = c/2^k, for k = 0, 1, 2, ...

To find the value of the constant c, we need to use the normalization property:

∑ P(X=k) = ∑ c/2^k = c/2^0 + c/2^1 + c/2^2 + ... = 1

To simplify this expression, we can use the formula for the sum of an infinite geometric series:

∑ a*r^n = a/(1-r), where a is the first term, r is the common ratio, and n goes from 0 to infinity.

In this case, a = c, r = 1/2, and n goes from 0 to infinity. So we have:

∑ P(X=k) = ∑ c/2^k = c/2^0 + c/2^1 + c/2^2 + ... = c/(1-1/2) = 2c

Setting this expression equal to 1, we get:

2c = 1

c = 1/2

Therefore, the value of the constant c is 1/2.

Step-by-step explanation:

Answer:

The answer to your problem is, 3x+6y=9 then -2x-4y=4

Step-by-step explanation:

So first we are going to multiply the first equation by 2  and second equation by 3 and add it.

If there is both x and y terms gets cancelled then we can say there were no solutions at all.

Shown;

A) -2x+4y=4

-3x+6y=6

Which then becomes;

-4x + 8y = 8

-9x + 18y = 18

For, -13x + 26y = 26

B) 3x+y=12-

3x+6y=6

Which we then multiply first equation by 2 and second equation by 3

6x + 2y = 24

-9x + 18x = 18

For, -3x + 20y = 42

C)3x+6y=9

-2x-4y=4

Again multiply first equation by 2 and second equation by 3

6x + 12y = 18

-6x - 12y = 12

Which can equal to:

30

Which both x and y terms becomes 0.

Thus the anwer to your problem is, 3x+6y=9 then -2x-4y=4

Answer:

  6.34°

Step-by-step explanation:

You want to know the bearing of a second plane from the first at 10 a.m. when the first leaves at 9 a.m. and flies 200 mph SW, while the second leaves at 9:30 a.m. and flies 320 mph NW.

The first plane flies for an hour at 200 mph, so is 200 miles from the airport on a bearing of 180° + 45° west of south, or 225°.

The second plane flies for half an hour at 320 mph, so is 160 miles from the airport on a bearing of 360° -45° west of north, or 315°.

The vector in the direction of the second plane from the first is found by subtracting the location of the first plane from the second:

  vector of interest = (160∠315°) -(200∠225°)

A suitable calculator can tell you the difference is ...

  vector of interest = 256∠6.34°

The second plane is on a bearing of 6.34° from the first.

__

Additional comment

You are correct that an angle of 83.66° is involved. That would be the measure of the direction vector counterclockwise from the +x axis. You want the bearing angle, measured clockwise from the +y axis (North), so the angle you're looking for is 90° -83.66° = 6.34°.

If you work enough of these problems, you will find that it doesn't matter how you measure the angles, as long as you do it consistently. The second attachment shows the arithmetic using distance∠bearing. The effect is to swap the x, and y coordinates so that they are (N, E) coordinates, rather than the usual (E, N) coordinates. This doesn't have to be confusing. A diagram usually helps.

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The option or function that is NOT an exponential function is (4/7)ˣ.

The function (4/7)ˣ is no exponential because the base (4/7) is smaller or lesser than 1. All exponential functions must have a base that is less than -1 or greater than 1.

The other functions

1) (3/5)ˣ

2) 4ˣ

are both exponential function.

3/5ˣ is a decay function and it's base is less than 1. 4ˣ on the other hand is a growth function. Of course, it's base is greater than 1.

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Answer: 504 different ways.

Step-by-step explanation:

      This situation uses a permutation. We will use the given formula and simplify to solve. n is equal to the number of contestants and r is 3 since we are selecting and first-, second-, and third-place winner.

Given:

      [tex]\displaystyle P = \frac{n!}{(n-r)!}[/tex]

Substitute values:

      [tex]\displaystyle P = \frac{9!}{(9-3)!}[/tex]

Subtract:

      [tex]\displaystyle P = \frac{9!}{6!}[/tex]

Multiply:

      [tex]\displaystyle P = \frac{9*8*7*6*5*4*3*2*1}{6*5*4*3*2*1}[/tex]

Simplify using the knowledge that something divided by itself is 1:

      P = 9 * 8 * 7

Multiply:

      P = 504

      We can also think of it as 9 different people could win 1st, 8 different people could win 2nd, and 7 different people could win 3rd. 9 * 8 * 7 = 504. However, the steps above show why this works with the formula as well.

The division of the given algebraic expression is:  (x + 4)/2x

The division algorithm for polynomials says, if p(x) and g(x) are the two polynomials, where g(x) ≠ 0, we can write the division of polynomials as: p(x) = q(x) × g(x) + r(x).

Where:

p(x) is the dividend.

q(x) is the quotient.

We want to solve:

[(x + 4)/x²] ÷ 2/x

Thus, we can simplify to:

[(x + 4)/x²]  * x/2

= (x + 4)/2x

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Answer:

To evenly space the letters of "EAGLES" across a 63 inch wide banner with no margins, you would need to divide the remaining space between each letter of the word equally. With five letters total and each letter being 8 inches wide, the total width occupied by the letters is 40 inches. Therefore, the remaining space is 23 inches (63-40=23). To find out how many inches should exist between each pair of letter, you would divide the remaining space by the number of gaps between the letters, which is 4 (one less than the total number of letters in the word).

23 inches ÷ 4 gaps = 5.75 inches between each pair of letters.

Step-by-step explanation:

Answer:

6.7 = π(r^2)(33.5/12)

r = .874

75/(2 × .874) = 42.9 = 42 barrels

6.7 = (3.14)(r^2)(33.5/12)

r = .874

75/(2 × .874) = 42.9 = 42 barrels