If a, b, c are in H.P.; prove that b+ca+ca+b ac' ab bc are also in A.P.​

The synthetic division of the polynomial indicates that other factor of the polynomial is; x² - 3·x + 13

A polynomial consists of terms that have positive integer powers or index of variables joined together by addition and subtraction symbols.

The possible polynomial in the question is; x³ + 5·x² - 11·x + 104

The factor of the polynomial is; (x + 8)

The steps for the synthetic division includes on the left side of the vertical line, placing the opposite sign of the constant term of the factor.

The other steps of synthetic division can then be presented, summarily as follows;

Therefore;

-8 | 1[tex]{}[/tex]   5     -11   104

   |   [tex]{}[/tex]   -8    24  104          

     1   -3    [tex]{}[/tex] 13      0

The factors of the polynomial

The values in the above last row of the long division indicates that the other factor of the polynomial is; x² - 3·x + 13

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Answer:

15

Step-by-step explanation:

Let's call the number we try to find "[tex]x[/tex]"

In the problem, we know that "26 is less than twice a number" is the same as "4". So we can write this as an equation:

[tex]2x - 26 = 4[/tex]

Now we can solve for x by isolating it on one side of the equation.

First, we isolate "2x" by flipping "-26" to the other side, which will now be "26".

[tex]2x = 4 + 26[/tex]

Then we add

[tex]2x = 30[/tex]

finally, we isolate the x by dividing each side with 2

2x ÷ 2 = 30 ÷ 2
x = 15

The length of 'l' is approximately equal to 331.13 units

Volume refers to the amount of space occupied by a three-dimensional object. In other words, it is the measure of how much space an object takes up.

The formula to calculate the volume of a square pyramid is:

Volume = (1/3) x Base Area x Height

where the base area is the area of the square base, and the height is the perpendicular distance from the apex to the base.

Let's substitute the given values in the formula and solve for the height:

48³ = (1/3) x l² x h

Multiplying both sides by 3, we get:

3 x 48³ = l² x h

Dividing both sides by l², we get:

h = (3 x 48³) / l²

Now, we can substitute the value of h in the original formula to get:

48³ = (1/3) x l² x [(3 x 48³) / l²]

Simplifying this equation, we get:

48³ = 48³

This equation is true, which means our solution is correct. Therefore, the length of 'l' is equal to the square root of (3 x 48³), which is approximately equal to 331.13 units (rounded to the nearest hundredth).

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Answer: x=4, sides = 12, 8, 6, 8

Step-by-step explain

Perimeter means the sum of all sides of a shape, in this example there are four sides add the sides and set it equal to 34.

X+8+x+4+2x-2+x²-2x=34

x²+2x+10=34

x²+2x-24

(x-4)(x+6)

x=4, x=-6

x can not be -6 because then the x+4 side would be -2, therefore it is extraneous and x=4 is the correct answer.

Now to find side lengths just plug in 4 for x

4+8=12

4+4=8

2*4-2=6

4*4-2*4=8

Answer:

a) In 2 hours

b) 16 km

Step-by-step explanation:

Let's first find out how far Abel has traveled in 2 hours. We can use the formula:

distance = speed x time

distance = 4 km/h x 2 h = 8 km

So when Jack starts cycling, Abel has already covered 8 km.

Now, let's say it takes Jack t hours to catch up to Abel. During this time, Abel has continued walking for t + 2 hours. So we can write:

distance covered by Abel = distance covered by Jack

4 km/h x (t + 2 h) = 8 km/h x t

Simplifying this equation, we get:

4t + 8 = 8t

4t = 8

t = 2

So Jack will overtake Abel after 2 hours of cycling. To find out how far they will have traveled, we can use the formula:

distance = speed x time

distance = 8 km/h x 2 h = 16 km

Therefore, when Jack overtakes Abel, they will have traveled 16 km in total.

The equation of the ellipse  derived as is

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1.

The equation of foci is

F₁ = (h - √(a^2 - b^2), k) , F₂ = (h + √(a^2 - b^2), k)

An ellipse is  described as a set of points in a plane such that the sum of the distances from each point to two fixed points, called foci, is constant.

We can write the equation of an ellipse as:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

where (h, k) = the center of the ellipse,

a = the semi-major axis,

and b = the semi-minor axis.

The foci of the ellipse are located along the major axis and are equidistant from the center, with a distance of : √(a^2 - b^2).

we know also the formula to find the foci of an ellipse is:

F₁ = (h - √(a^2 - b^2), k)

F₂ = (h + √(a^2 - b^2), k)

The sum of the distances from each point on the ellipse to the foci is constant. The equation can then be written as:

2a = √((x - h + √(a^2 - b^2))^2 + (y - k)^2) + √((x - h - √(a^2 - b^2))^2 + (y - k)^2)

Simplifying, we then can write  the equation of the ellipse as:

(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1

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The area needed for the green region is 40 square feet.

To do so, we can take the area of the whole green area, and then remove the areas of the two holes.

Remember that the area of any rectangle is the product of the two dimensions.

For the green one:

A = 4ft*12ft = 48 ft²

Removing the two areas for the holes we have:

a1 = 3ft*2ft = 6ft²

a2 = 1ft*2ft = 2ft²

Remove that:

green area=  48 ft² - 6ft² - 2t² = 40ft²

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The total money raised is (15/7) * $88,000 = $188,571.43.

a) Let x be the total amount raised.

Charity A received (1/3)(x - A), Charity C received (1/4)(x - C). We have A = (1/3)(B+C) and C = (1/4)(A+B).

Solving these equations, we find B's share to be 7/15 of the total amount.

p) Since Charity B received $88,000, which is 7/15 of the total amount, the total money raised is (15/7) * $88,000 = $188,571.43.

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The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

The y-coordinate of this point is 1, so sin(183π/2) = sin(16,470 degrees) = 1.

It should be noted that go find cos(5π), we first need to convert 5π to degrees. We know that π radians is equal to 180 degrees, so 5π radians is equal to 5π × (180/π) = 900 degrees. The x-coordinate of this point is -1, so cos(5π) = cos(900 degrees) = -1.

Also, to find sin(-9π/2), we first need to convert -9π/2 to degrees. We know that π radians is equal to 180 degrees, so -9π/2 radians is equal to -9π/2 × (180/π) = -810 degrees. The y-coordinate of this point is -1, so sin(-9π/2) = sin(-810 degrees) = -1.

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Answer:

3.36%

Step-by-step explanation:

(10.14-9.81)/(9.81)(100)=3.36% error

By looking at the y-values of the points, we can see that the first four terms are.

2, -2, -6, -10

We know that the x-value of each of the points is the index of the term, so:

x = 1 is the first term

x = 2 is the second term.

And so on.

The y-value will be the actual value of the term.

Here we can see 4 points graphed, these are:

(1, 2), (2, -2), (3, -6), (4, -10)

Then the values of the first four terms of the sequence are:

2, -2, -6, -10

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Answer:

A

Step-by-step explanation:

according triangle inequality,

the sum of any two sides of a triangle is larger than the remaining side.

so a + b > c is correct

Answer:

The walking distance that is saved by cutting across the lot is approximately 18 feet.

Answer:

complex

Step-by-step explanation:

The given sentence "Although Kevin has money, he is not spending it." is a complex sentence.

This is because it contains one independent clause, "he is not spending it," which can stand alone as a complete sentence. The other part of the sentence, "Although Kevin has money," is a dependent clause because it cannot stand alone as a complete sentence.

The dependent clause "Although Kevin has money" introduces a contrast or concession to the independent clause that follows it. Therefore, this sentence is an example of a complex sentence that uses a dependent clause to add meaning and complexity to the independent clause.

A ski lodge inspects 80 skis and discovers four that are faulty. The skis that are likely to be defective are 21.

It is assumed that there are 80 skis, four of which are faulty.

As a result, the total number of outcomes is 80.

Positive outcomes = 4

Let [tex]E_{1}[/tex] represent the event of selecting a faulty. As a result, the likelihood of selecting a defective ski is expressed mathematically as follows.

[tex]E_{1}[/tex] = Favorable case of defective / Total number of skis

= 4 / 80

= 1 / 20

=0.05

Skis will be discovered in 5% of cases.

So, the event of choosing a defective is 4 / 80.

If the total number of skis is 420, then the number of defective skis will be:

4 / 80 × 420

= 1/20 × 420

= 21

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Correct question:

A ski lodge inspects 80 skis and finds 4 to be defective. The lodge has 420 skis. Predict the number of skis that are likely to be defective.

The volume of the solid obtained by rotating the region over the function is given by the integral and V = 44.46 units³

Given data ,

The volume of the solid obtained by rotating the region underneath the graph of the function f(x)=√x² +9 over the interval [0,2] about the y-axis

The formula for Area under definite integral is

∫ₐᵇ f ( x ) = f ( b ) - f ( a )

Since we are rotating the region about the y-axis, the radius is simply the x-coordinate of each point on the curve, or r = x

The height h of each shell is equal to the difference between the y-coordinates of the curve at x and x + Δx, or h = f(x + Δx) - f(x)

Using these expressions for r and h, we can write the volume of each cylindrical shell as:

V(x) = 2πx[f(x + Δx) - f(x)]Δx

V = ∫₀² 2πx[f(x + Δx) - f(x)]dx

As Δx approaches zero, this integral becomes:

V = ∫₀² 2πx√(1 + (f'(x))²) dx

where f'(x) is the derivative of f(x), which is:

f'(x) = x/√(x² + 9)

Substituting this expression for f'(x) into the integral, we get:

V = ∫₀² 2πx√(1 + (x/√(x² + 9))²) dx

This integral can be evaluated using a substitution, u = x² + 9, du/dx = 2x, and dx = du/2x, to get:

V = ∫₉¹³ 2π(x² + 9)^(3/2)/2 dx

V = [4/5 π(x² + 9)^(5/2)]₉¹³

V = 44.46 units³

Hence , the volume of the solid is 44.46 units³

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Let x be the length of each of the equal sides of the isosceles triangle. Then the length of the base is x - 4. The perimeter of the triangle is the sum of the lengths of the three sides, which is:

x + x + (x - 4) = 3x - 4

We know that the perimeter is 32, so we can set 3x - 4 equal to 32 and solve for x:

3x - 4 = 32

Adding 4 to both sides gives:

3x = 36

Dividing by 3 gives:

x = 12

Therefore, one of the equal sides of the triangle is 12 inches long, and the length of the base is 12 - 4 = 8 inches.

So the three sides of the triangle are 12 inches, 12 inches, and 8 inches.

The correct equation to solve the problem is (C) 3x - 4 = 32.

Answer:  

C is the answer

Given :   dilation with a scale factor of 3

To find :  Which statement is true

Step-by-step explanation:

Solution: A dilation is a transformation that produces an image that is the same shape as the original, but is a different size

if Dilation factor  N then image Size is multiplied  by  N

if Scale factor  < 1 then image get reduced

if scale factor > 1  then image get enlarged

scale factor = 1 lead to Same size image

Here is given that Scale factor of 3

=> image will be enlarged to 3 times the original image

Hence correct option is :

C)Each side in the pre-image is multiplied by three to find the corresponding side length in the image

Answer: C

Step-by-step explanation: Edge Unit Test

To find the multiplicative inverse of 2i, we need to find a complex number z such that:

(2i) * z = 1

Multiplying both sides by -i, we get:

z = -i/2

Therefore, the multiplicative inverse of 2i is -i/2.

An equation for L, in terms of t, representing the amount Kadeesha owes her friend after t weeks is L = 8t + 120.

The table of values is as follows;

Number of Weeks               Amount of Money Owed

0                                                    $120

1                                                    $128

2                                                    $136

3                                                    $144

In order to write a linear equation to describe this situation, we would assign variables to the amount of money that Kadeesha borrowed originally and the number of weeks respectively, and then translate the word problem into a linear equation as follows:

Since Kadeesha originally borrowed $120 and agreed to pay back her friend $8 per week, a linear equation which can be used to model the total amount of money that Kadeesha owed is given by;

L = 8t + 120

When t = 0, the value of L is given by;

L = 8(0) + 120

L = $120.

When t = 1, the value of L is given by;

L = 8(1) + 120

L = $128.

When t = 2, the value of L is given by;

L = 8(2) + 120

L = $136.

When t = 3, the value of L is given by;

L = 8(3) + 120

L = $144.

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The sine of angle θ is given as follows:

sin(θ) = -8/17.

For an angle [tex]\theta[/tex] the unit circle is a circle with radius 1 containing the following set of points: [tex](\cos{\theta}, \sin{\theta})[/tex].

The point in this problem has the coordinates given as follows:

(15/17, -8/17).

Hence the trigonometric ratios are given as follows:

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Answer: The answer is 6.25

Step-by-step explanation:

Answer:

6.25

Step-by-step explanation:

Without knowing the exact number of turtles in 2005, we cannot determine the exact number of turtles in 2010. However, we can use the information given to set up an equation.

Let x be the number of sea turtles in 2005.

According to the problem, the number of sea turtles in 2010 was twice the amount in 2005, or 2x.

We can set up an equation:

2x = number of sea turtles in 2010

We don't know the exact value of x, so we cannot solve for the number of turtles in 2010.

a

1.5/4

b

0.5/4

c

0.5/4

because half of 3 is 1.5

Answer:

I'm pretty sure the answer is B(

Answer:

\( A=\frac{d^2}{2} \).

Step-by-step explanation:

Area of Square by Applying Formula for the Diagonal

We can use the relationship between diagonal length d and the side a length of a square to find its area A. So, the formula for area of square using diagonal is \( A=\frac{d^2}{2} \).

There are possible 4,782,969 different telephone numbers.

We will multiply the number of options for each digit to calculate the total number of possible telephone numbers,.

Using the multiplication principle to estimate the number of possible telephone numbers for each digit:  

n1 x n2 x n3 x ... x nk

Here:

the first three digits are fixed: 011

The 4th digit (the first digit of the local phone number) can be any digit from 1 - 9 (i.e. 9 options).

The remaining 6 digits (the second to seventh digit of the local phone number) can be any digit from 0 to 9, except 0 (9 options for each digit).

Therefore, the total number of possible telephone numbers:

9 x 9 x 9 x 9 x 9 x 9 x 9 = 4,782,969,

Therefore, 4,782,969 different telephone numbers are possible.

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Dice game for a particular child’s game are numbered 1-10. Then, the  likelihood of rolling a single one and ending up with an even number is 1/20.

The probability of rolling a single one is 1/10, since there is only one die face with the number one out of ten possible outcomes.

The probability of rolling an even number is 5/10, since half of the die faces are even numbers (2, 4, 6, 8, 10) out of ten possible outcomes.

To find the likelihood of both events happening together, we need to multiply their individual probabilities;

P(rolling a single one and ending up with an even number) = P(rolling a single one) x P(ending up with an even number)

= (1/10) x (5/10)

= 1/20

Therefore, the likelihood of rolling a single one and ending up with an even number is 1/20.

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The value of the points is,

(1/5, 7/5)  or (0.2, 1.4)

The given equation may be simplified as follows:

x² + 14xy + 49y² = 100

(x + 7y)(x + 7y) = 100

(x + 7y)² = 10²

x + 7y = 10

This is a straight line with the equation.

y = -(1/7)x + 10/7

The minimum distance from the origin to this line is provided by a straight line that passes through the origin and which is perpendicular to the straight line.

The slope of the perpendicular line is 7 because the product of the two slopes should be -1.

The perpendicular line is of the form

y = 7x + c.

Because the line passes through (0,0), therefore c = 0.

The line y = 7x intercepts the original line when

y = 7x = -(1/7)x + 10/7

Therefore

7x = -(1/7)x + 10/7

Multiply through by 7.

49x = -x + 10

50x = 10

x = 1/5

y = 7x = 7/5

Hence, The minimum distance is

d = √(x² + y²)

  = √[(1/5)² + (7/5)²]

  = √2

Thus, The point is (1/5, 7/5).

So, Solution are, (1/5, 7/5)  or (0.2, 1.4)

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The first 5 terms of the sequence are 1/3, 2/3, 4/3, 8/3 ans 16/3

From the question, we have the following parameters that can be used in our computation:

The sequence is generated by using gn = 2(gn-1)

Such that g1 = 1/3 and n is a whole number greater than 1.

Using the above as a guide, we have the following:

The sequence means that we multiply the current term by 2 to get the next term

So, we have the following representations

a(2) = 2 * 1/3 = 2/3

a(3) = 2 * 2/3 = 4/3

a(4) = 2 * 4/3 = 8/3

a(5) = 2 * 8/3 = 16/3

Hence, the first 5 terms of the sequence are 1/3, 2/3, 4/3, 8/3 ans 16/3

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The measure (in radians) of central angle θ in the circle below is π/3.

We have,

radius = 3 cm

Arc length = π

We can use the formula relating the arc length (s) to the central angle (θ) and radius (r) of a circle:

s = r Ф

Now,

π = 3 x Ф

Ф = π/3

Thus,

The measure (in radians) of central angle θ in the circle below is π/3.

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