absolute value functions . what is the vertex of this function ?? help please

A 5th degree polynomial have five complex roots. This means that the polynomial can be written as the product of five linear or quadratic polynomials.

The general formula to calculate the roots of a 5th degree polynomial is:

\(x^5 + ax^4 + bx^3 + cx^2 + dx + e = 0\)

The roots of the polynomial can be found by solving the following equation:

\(x^4 + (a - x)x^3 + (b - a x + x^2)x^2 + (c - bx + ax^2 - x^3)x + (d - cx + bx^2 - ax^3 + x^4) = 0\)

To calculate the roots of the polynomial, we need to solve the quartic equation, which can be done by using the quadratic formula. Then, we can use the solutions of the quartic equation to find the roots of the 5th degree polynomial.

For example, if we have the polynomial x^5 + 2x^4 + 3x^3 + 4x^2 + 5x + 6 = 0, then the roots can be calculated as follows:

x1 = -1.7177

x2 = -0.3181

x3 = 0.5813

x4 = 1.2520

x5 = 2.1771

These are the five complex roots of the polynomial.

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The exterior angle adjacent to the largest interior angle in a triangle with a 2:2:4 angle ratio is 90 degrees.

The exterior angle adjacent to the largest interior angle in a triangle is equal to the sum of the other two interior angles.

Let's call the measures of the angles in the triangle x, x, and 2x. Then,

x + x + 2x = 180 degrees,

so 4x = 180 degrees, and

x = 45 degrees.

The largest interior angle is 2x, or 90 degrees. The exterior angle that is adjacent to this angle is equal to 180 degrees minus the interior angle, which is 180 - 90 = 90 degrees.

So the exterior angle adjacent to the largest interior angle in a triangle with a 2:2:4 angle ratio is 90 degrees.

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Answer

1.98

Step-by-step explanation:

1.54 + 0.44 = 1.98

hope it helps.

Answer:

x- intercept=6/2=3

y-intercept=6/3=2

Step-by-step explanation:

Answer:

The answer to this question should be A=18.84 and B=28.26

Answer:

x ≈ 14.4 cm

Step-by-step explanation:

the third angle of the triangle = 180° - 40° - 46° = 94°

Using the Sine rule in the triangle

\(\frac{x}{sin46}\) = \(\frac{20}{sin94}\) ( cross- multiply )

x × sin94° = 20 × sin46° ( divide both sides by sin94° )

x = \(\frac{20sin46}{sin94}\) ≈ 14.4 cm ( to the nearest tenth )

Answer:

14 cm (rounded to nearest whole number)

Step-by-step explanation:

Law of Sine:

\(\displaystyle \large{\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c} = 2R}\)

Given angles are:

Definition of Euclidean Triangle:

Find another angle:

So another angle is 94°.

To find:

Determine:

Therefore:

\(\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} = \dfrac{\sin 94^{\circ}}{20}}\)

Multiply both sides by 20x:

\(\displaystyle \large{\dfrac{\sin 46^{\circ}}{x} \cdot 20x = \dfrac{\sin 94^{\circ}}{20} \cdot 20x}\\\displaystyle \large{20\sin 46^{\circ}= x\sin 94^{\circ}}\)

Divide both sides by \(\displaystyle \large{\sin 94^{\circ}}\):

\(\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = \dfrac{x\sin 94^{\circ}}{\sin 94^{\circ}}}\\\displaystyle \large{\dfrac{20\sin 46^{\circ}}{\sin 94^{\circ}} = x}\)

Evaluate the expression, hence:

\(\displaystyle \large{x = 14.42...}\)

Round to nearest whole number:

\(\displaystyle \large{x = 14}\)

Therefore, the value of x is 14 cm.

The length that the caliper reads can be found to be the value of A. 0. 69 cm.

In order to ascertain the measurement of a caliper's length, it is imperative to recognize both the scales inscribed on it. The larger exterior scale on the caliper is known as the primary scale. The narrower scale situated inside the caliper is known as the vernier scale.

Easily determine the measurement on the primary scale by locating the numeral that corresponds with the zero indication on the vernier scale. This represents the total count of millimeters in whole.

As we can see, the main scale has a value of that is almost at 0. 70 cm but not quite there. We will therefore go to the lower one which is 0.6 cm. Then we look at the venier scale and see that at 0.09 cm, there is an alignment. This means that the length is:

= 0. 6 + 0. 09

= 0. 69 cm

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Options include:
A. 0.69 cm

B. 1.6cm

C. 0.7cm

D. 2.1 cm

As the series diverges, it does not have a finite sum.

To determine whether the series converges or diverges, we can analyze the pattern of the terms.

The series given is:

9 + 4/81 + 729/6561 + ...

Notice that each term in the series is a power of 4, starting from 4⁰ (which is 1) and increasing by powers of 4. Specifically, the nth term is (4ⁿ)²/4ⁿ = 4ⁿ/4ⁿ = 1.

Since each term is 1, the series is essentially an infinite sum of 1's:

1 + 1 + 1 + ...

This series is a geometric series with a common ratio of 1.

For a geometric series to converge, the absolute value of the common ratio must be less than 1.

In this case, the common ratio is 1, which is not less than 1.

Therefore, the series diverges.

Since the series diverges, it does not have a finite sum.

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a) The medical researcher will measure the blood pressure of each of the 100 individuals both before and after the study.

b) There are various reasons for it;Fewer samples: The sample size of this research is small. 100 people are not enough to reach any conclusive result.

a) Controlled experiment

The given research is a controlled experiment in which the medical researcher has assigned 50 people to pursue an exercise program and 50 people to refrain from vigorous activity. The medical researcher will measure the blood pressure of each of the 100 individuals both before and after the study.

b)The conclusion of the researcher that the exercise decreases blood pressure is not better justified than the conclusion that something else was responsible for the difference between the exercise group and the no-exercise group.

There are various reasons for it;Fewer samples: The sample size of this research is small. 100 people are not enough to reach any conclusive result.There could be a number of other reasons such as lifestyle, diet, genetics, age, weight, etc. which are not taken into consideration in this study. Because of these other factors, it is not a good idea to reach any conclusions based on this research.

A randomized controlled trial is one of the most powerful epidemiological research studies, but the sample size should be large enough to account for other variables. Moreover, more variables should be considered and controlled.

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Answer:

Step-by-step explanation:

The dimension of the basis is {[1 0 0 2], [-1 1 0 0]}.

To find a basis for the subspace S = span {[1 -1 0 2], [3 -5 4 8], [0 1 -2 -1]}, we can use the same method as in the example. First, we put the vectors in a matrix and row-reduce it:

[1 -1 0 2]
[3 -5 4 8]
[0 1 -2 -1]

R2 - 3R1 -> R2
R3 -> R3 + 2R1

[1 -1 0 2]
[0 -2 4 2]
[0 1 -2 -1]

-1/2R2 -> R2

[1 -1 0 2]
[0 1 -2 -1]
[0 1 -2 -1]

R3 - R2 -> R3

[1 -1 0 2]
[0 1 -2 -1]
[0 0 0 0]

We can see that the last row is all zeros, so we have only two pivots and one free variable. This means that the dimension of the subspace S is 2. To find a basis, we can write the pivots as linear combinations of the original vectors:

[1 -1 0 2] = [1 0 0 2] + [-1 1 0 0]
[0 1 -2 -1] = [0 1 -2 -1]

Therefore, a basis for S is {[1 0 0 2], [-1 1 0 0]}.

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Answer:

B is the correct answer

Therefore, Pump B will have a longer average cycle time compared to Pump A.

To determine which pump will have a longer average cycle time, we need to calculate the cycle time for each pump based on the given information and compare the results. The cycle time for a single-server system can be calculated using Little's Law: Cycle Time = (1 / Arrival Rate) * (1 / (1 - Utilization))

Given:

Arrival Rate = 0.2 car per minute

Squared-Coefficient of Variation (CV^2) for Arrival Rate = 1

Utilization can be calculated as the product of the mean effective process time and the arrival rate:

Utilization = Mean Effective Process Time * Arrival Rate

For Pump A:

Mean Effective Process Time (A) = 4 minutes

Squared-Coefficient of Variation for Pump A = 0.5

Utilization (A) = 4 * 0.2 = 0.8

For Pump B:

Mean Effective Process Time (B) = 3 minutes

Squared-Coefficient of Variation for Pump B = 5

Utilization (B) = 3 * 0.2 = 0.6

Now, let's calculate the cycle time for each pump:

Cycle Time (A) = (1 / 0.2) * (1 / (1 - 0.8))

= 5 minutes

Cycle Time (B) = (1 / 0.2) * (1 / (1 - 0.6))

= 2.5 minutes

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Option (A) is the correct option and on simplifying the expression by combining like terms we get,

24 - 3p.

Here, we have,

Given, the expression 6(4 - 2p) + 9p

On simplifying, we get

24 - 12p + 9p

24 - 3p

So, on simplifying the given expression, we get

24 - 3p

Hence, Option (A) is the correct option and on simplifying the expression by combining like terms we get,

24 - 3p

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Answer:

Some of the equations can be written as-

1. 2x = 4

2. 3x + 9 = 15

3. x + 5 = 7

4. 5x - 2 = 8

Step-by-step explanation:

Given - Mr. Turney wrote the value statement "x = 2" on the whiteboard.

To find - He asked his students to create an equation or inequality with

              this value as a solution.

Proof -

Given that, The value statement is ''x = 2''

We have to find out that equation in such a way that if we solve the equation , we get the value x = 2

There can be infinite many equation that give the value statement '' x = 2''

Some of the equations can be written as-

1. 2x = 4

2. 3x + 9 = 15

3. x + 5 = 7

4. 5x - 2 = 8

And many more.

Verification -

1. 2x = 4

⇒x = \(\frac{4}{2}\)

⇒x = 2

Verified

2. 3x + 9 = 15

⇒3x = 15 - 9

⇒3x = 6

⇒x = \(\frac{6}{3}\)

⇒x = 2

Verified

3. x + 5 = 7

⇒x = 7 - 5

⇒x = 2

Verified

4. 5x - 2 = 8

⇒5x = 8 + 2

⇒5x = 10

⇒x = \(\frac{10}{5}\)

⇒x = 2

Verified

The magnitude of the resultant force is 135.83 pounds, and its direction angle is 67.7°.

To find the magnitude and direction of the resultant force, we can use vector addition. We first convert the forces into their x and y components. The x-component of the first force is 90 cos(68°) = 34.48 pounds, and the y-component is 90 sin(68°) = 82.73 pounds.

Similarly, the x-component of the second force is 50 cos(61°) = 22.69 pounds, and the y-component is 50 sin(61°) = 41.06 pounds.

To find the x and y components of the resultant force, we add the corresponding components of the two forces. The x-component of the resultant force is 34.48 + 22.69 = 57.17 pounds, and the y-component is 82.73 + 41.06 = 123.79 pounds.

Using the Pythagorean theorem, we can find the magnitude of the resultant force as the square root of the sum of the squares of its x and y components. Thus, the magnitude of the resultant force is sqrt(57.17^2 + 123.79^2) = 135.83 pounds (rounded to the nearest hundredth).

To find the direction angle of the resultant force, we can use trigonometry. The direction angle is the arctan of the y-component divided by the x-component, but we need to adjust for the appropriate quadrant.

Thus, the direction angle is arctan(123.79/57.17) + 180° = 67.7° (rounded to the nearest tenth of a degree), where we add 180° to account for the fact that the resultant force is in the third quadrant.

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Answer:

Step-by-step explanation:

// Solve equation [2] for the variable  x

 [2]    x = y + 1

// Plug this in for variable  x  in equation [1]

  [1]    (y +1) - 2y = 9

  [1]     - y = 8

// Solve equation [1] for the variable  y

  [1]    y = - 8

// By now we know this much :

   x = y+1

   y = -8

// Use the  y  value to solve for  x

   x = (-8)+1 = -7

Solution :

{x,y} = {-7,-8}

Answer:

12 boxes of bookmarks and 8 packs of dust covers.

Step-by-step explanation:

Shelley wants to have at least 250 bookmarks and dust covers for her 250 books.

One box of bookmarks contain 24 bookmarks.

One pack of dust cover contain 36 dust covers.

Shelley buys same number of bookmarks and dust covers.

So, we need to find out LCM(Least Common Multiple) of 24 and 36 to come to a common number that can be bought.

LCM of 24 and 36 is 72.

But, at least 250 bookmarks or dust covers have to be bought. So, we need to find a multiple of 72 which is greater than or equal to 250.

Let us consider the Table of 72:

72, 144, 216, 288, 360 .....

For having enough count of bookmarks and dust covers, at least 288 bookmarks and dust covers have to be bought.

Number of boxes of bookmarks bought = \(\frac{288}{24} \Rightarrow 12\)

Number of packs of dust covers bought = \(\frac{288}{36} \Rightarrow 8\)

12 boxes of bookmarks and 8 packs of dust covers.

Greetings from Brasil...

Here we will use our knowledge of parallel lines....

→ Angles opposite the vertices are equal, so

1 = 6

2 = 5

3 = 8

4 = 7

9 = 14

10 = 13

11 = 16

12 = 15

→ Corresponding angles are equal, so

1 = 9

2 = 10

3 = 11

4 = 12

Given the information above we can write:

1 = 6 = 9 = 14

2 = 5 = 10 = 13

3 = 8 = 11 = 16

4 = 7 = 12 = 15

from the statement of the question it is said that: 3 = 110° and 10 = 95°, so

2 = 5 = 10 = 13 = 95°

3 = 8 = 11 = 16 = 110°

→ from internal/external collateral angles OR supplementary angles:

1 + 2 = 180°

1 + 95° = 180°

1 = 180° - 95°

1 = 85°

3 + 4 = 180°

110° + 4 = 180°

4 = 180° - 110°

4 = 70°

Now we can say that:

Answer:

Step-by-step explanation:

(a) and (b) see diagram

(c) you can see from the graph, the purple line hits the parabola twice which is y=6   or k=6

(d)  Solving simultaneously can mean to set equal

6x - x² = k                        >subtract k from both sides

6x - x² - k = 0                  >put in standard form

- x² + 6x - k = 0               >divide both sides by a -1

x² - 6x + k = 0

(e) The new equation is the same as the original equation just flipped (see image)

(f)  The discriminant is the part of the quadratic equation that is under the root. (not sure if they wanted the discriminant of new equation or orginal.  I chose new)

discriminant formula = b² - 4ac

equation:   x² - 6x + 6 = 0            a = 1      b=-6    c = 6

discriminant = b² - 4ac

discriminant= (-6)² - 4(1)(6)

discriminant = 36-24

discriminant = 12

Because the discriminant is positive, if you put it back in to the quadratic equation, you will get 2 real solutions.

Width: 2.62 meters

Length: 7.62 meters

Let x = the width of the rectangle

Let x + 5 = the length of the rectangle

Area of a rectangle: length * width

Area = x(x + 5)

20 = x^2 + 5x

x^2 + 5x - 20 = 0

Factor: (completing the square)

x^2 + x*2*5/2 + (5/2)^2 - (5/2)^2 - 20 = 0

(x + 5/2)^2 - 26.25 = 0

(x + 5/2)^2 = 26.25

x + 5/2 = 5.1234...

x = 2.6234...

The width is approximately 2.62 meters

The length is approximately (2.62 + 5) --> 7.62 meters

Answer:

3/20

Step-by-step explanation:

stefan sell 1/5 × 3/4 = 3/20 boxes of candy

The best description about the sampling distribution of means for the student's sample is mu x = 45, sigma x = 3.75, shape of distribution unknown.

Sampling distribution is defined as the probability distribution of a statistical measure which is got by the repeated sampling of a specific population.

Given that,

μ = 45 minutes and σ = 15 minutes

Sampling distribution of means for the student's sample is :

μₓ = μ = 45

σₓ = σ / √n, where n is the sample size.

σₓ = 15 / √(16) = 15 / 4 = 3.75

Now since the sample size 16 which is less than 30, shape of the distribution is unknown.

Hence the sampling distribution is, μₓ = 45, σₓ = 3.75, and the shape of distribution is unknown.

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This equation shows that the amount Steven charges (y) is directly proportional to the number of hours he babysits (x) with a constant of proportionality of $5.25 per hour.

To write the equation to represent the relationship between the number of hours Steven babysits and the amount he charges, we need to first find the rate of charging per hour.

Then we can use this rate to write the equation.Let x represent the number of hours Steven babysits and y represent the amount he charges.We can find the rate of charging per hour by dividing the amount he charges by the number of hours he babysits.

Rate of charging per hour when Steven babysits for 4 hours = $21 ÷ 4 = $5.25/hourRate of charging per hour when Steven babysits for 6 hours = $31.50 ÷ 6 = $5.25/hour

We see that Steven charges a flat rate of $5.25 per hour for babysitting.Using this information, we can now write the equation to represent the relationship between x and y:y = 5.25x

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Given the equations:

3y - 6x = 24..............................1

8 + 2x = y..................................2

Substitute (8 + 2x) for y in equation 1.

3(8 + 2x) - 6x = 24

24 + 6x - 6x = 24

24 = 24

Since, we have 24 = 24, we can say the system of equations have infinite solutions.

ANSWER:

A. infinite solutions

Answer:

The answer to your problem is, C. Car C has the best rate at 25 miles per gallon.

Step-by-step explanation:

Well given,

Car C = 450 / 18 = 25

Car B = 361 / 19 = 19

Car A = 340 / 17 = 20

We can see that Car C has the best MPG.

Thus the answer to your problem is, C. Car C has the best rate at 25 miles per gallon.

The sample space in the tree diagram regarding the probability will be HR, HY, HB, TR, TY, TB.

It should be noted that probability simply means the likelihood of the occurence of an event.

In this case, the sample space in the tree diagram will be HR, HY, HB, TR, TY, TB where,

H= head

T = tail

B = blue

r = red

y = yellow

The number of outcomes are in the event "Toss a tails, spin yellow will be 1.

The probability of tossing tails and spinning yellow will be:

= 1/2 × 1/3

= 1/6

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When graphing an exponential function, a T-chart is commonly used to determine the values. The correct answer is option A.

The T-chart employs positive real numbers since this is the most typical form of exponential function.

Exponential functions are utilized to represent processes that increase or decrease exponentially, as well as to model phenomena in many different disciplines, including science, economics, and engineering.

The exponential function can be represented by the following equation:

\(y=a^x\), where a is the base, x is the exponent, and y is the outcome.

When a is a positive number greater than one, the function is called exponential growth, while when a is a fraction between 0 and 1, the function is called exponential decay.

The T-chart is used to determine the values to use in the graph and connect the dots as required. Positive real numbers are used as the values in the T-chart in order to effectively graph the exponential function.

Therefore, the correct answer is option A.

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An angular resolution of 0.56 arcseconds is required to see the Sun and Jupiter as separate objects. This is an extremely small angle and would necessitate the use of a large telescope.

Angular resolution is defined as the minimum angle between two objects that enables a viewer to see them as distinct objects rather than as a single one. A better angular resolution corresponds to a smaller minimum angle. The angular resolution formula is θ = 1.22 λ / D, where λ is the wavelength of light and D is the diameter of the telescope. Thus, the angular resolution formula can be expressed as the smallest angle between two objects that allows a viewer to distinguish between them. In arcseconds, the answer should be given to two significant figures.

To see the Sun and Jupiter as distinct points of light, we need to have a good angular resolution. The angular resolution is calculated as follows:

θ = 1.22 λ / D, where θ is the angular resolution, λ is the wavelength of the light, and D is the diameter of the telescope.

Using this formula, we can find the minimum angular resolution required to see the Sun and Jupiter as separate objects. The Sun and Jupiter are at an average distance of 5.2 astronomical units (AU) from each other. An AU is the distance from the Earth to the Sun, which is about 150 million kilometers. This means that the distance between Jupiter and the Sun is 780 million kilometers.

To determine the angular resolution, we need to know the wavelength of the light and the diameter of the telescope. Let's use visible light (λ = 550 nm) and assume that we are using a telescope with a diameter of 2.5 meters.

θ = 1.22 λ / D = 1.22 × 550 × 10^-9 / 2.5 = 2.7 × 10^-6 rad

To convert radians to arcseconds, multiply by 206,265.θ = 2.7 × 10^-6 × 206,265 = 0.56 arcseconds

The angular resolution required to see the Sun and Jupiter as distinct points of light is 0.56 arcseconds.

This is very small and would require a large telescope to achieve.

In conclusion, we require an angular resolution of 0.56 arcseconds to see the Sun and Jupiter as separate objects. This is an extremely small angle and would necessitate the use of a large telescope.

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Answer:

Ok, sabemos que Carlos ha construido un 3/7 de una casa.

Queremos saber cuanto le falta para terminar de construirla.

Entonces podemos realizar el cálculo:

1 casa - lo que Carlos ya construyo = Lo que le falta.

Entonces:

1 casa - 3/7 de una casa = X

Podemos reescribir:

1 casa = 7/7 de una casas

así tenemos:

7/7 de una casa  - 3/7 de una casa = X

(7/7 - 3/7) de una casa = X

4/7 de una casa  = X

Así podemos concluir que a Carlos le falta construir 4/7 de una casa para acabar.

Answer: A) The decay constant (λ) is a measure of the rate at which a radioactive isotope decays. It is defined as the probability per unit time that an atom of the isotope will decay. The decay constant can be calculated using the formula:

λ = -ln(1 - x) / t

Where x is the fraction of the original activity that has decayed (in this case, 0.40), and t is the time over which the decay occurred (in this case, one week).

B) The half-life (T1/2) is the amount of time it takes for half of the original activity of a radioactive isotope to decay. It can be calculated using the formula:

T1/2 = ln(2) / λ

C) The mean lifetime (τ) is the average amount of time an atom of a radioactive isotope will survive before it decays. It can be calculated using the formula:

τ = 1 / λ

It's important to note that the above formulas are based on the exponential decay model, which assumes that the decay process is random and that the decay constant is constant over time. If the isotope does not decay in this way, these formulas may not give accurate results.

Step-by-step explanation: