what would a number line representing g≠7 look like help asap
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A graph of the equation -5 < x < -3 is shown in the image attached below.

In Mathematics, an inequality simply refers to a mathematical relation that is typically used for comparing two (2) or more numerical data and variables in an algebraic equation based on any of the inequality symbols;

Next, we would rewrite the given compound inequality in pairs as follows;

-5 < x < -3 ≡ x > -5 or x < -3

In this scenario, we would use an online graphing calculator to plot the inequality as shown in the graph attached below.

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Answer:

The value of m that satisfies the equation 6200 = 200^m is approximately 1.94.

Step-by-step explanation:

To find the value of m, we can take the logarithm of both sides of the equation:

log(6200) = log(200^m)

By the laws of logarithms, we can simplify this to:

log(6200) = m log(200)

Now we can solve for m by dividing both sides by log(200):

m = log(6200) / log(200)

Using a calculator, we can evaluate this expression to find:

m ≈ 1.94

Therefore, the value of m that satisfies the equation 6200 = 200^m is approximately 1.94.

Hopefully this helps! I'm sorry if it doesn't. If you need more help, ask me! :]

Answer:

500 Milliliters

Step-by-step explanation:

Because a liter is 100x a milliliter so 0.5x100=500

Answer:

B. 0.25x² + 0.8x - 8 = 0

C. 5x² + 7x = 15

Step-by-step explanation:

Let's try by solving all equations;

(x + 3)² = 36

x² + 3² = 36

x² + 9 = 36

x² = 36 - 9

x² = 27

x = √27

x ≈ 5.2

No quadratic

What about others

0.25x² + 0.8x - 8 = 0

Hmm.... that can only be solved by quadratic

[tex] \sf { \fbox{0.25x² + 0.8x - 8 = 0} \: can \: only \: be \: solved \: by \: quadratic}[/tex]

Also, 5x² + 7x = 15

[tex] \sf { \fbox{5x² + 7x = 15} \: can \: only \: be \: solved \: by \: quadratic}[/tex]

(x + 8)(x + 9) = 0

x² + 9x + 8x + 72 = 0

x² + 17x + 72 = 0

(x + 8)(x + 9) = 0

Not truly quadratic but can be solved with quadratic.

That leaves us with only B and C as the answer

The factors that would most likely reduce the monthly premium a policyholder who pays for automobile insurance is -

Option A: I, II

What is insurance?

Insurance is a legal agreement, evidenced by a policy, under which a policyholder receives financial security or compensation from an insurance provider against losses. In order to make payments to the insured more manageable, the company pools the risks of its clients.

The factors that can affect the monthly premium a policyholder pays for automobile insurance include the type of vehicle, driving record, and other personal factors.

Out of the options given, buying a minivan  and maintaining an accident-free driving record over time would most likely reduce the monthly premium a policyholder pays for automobile insurance.

Minivans are generally considered safer and less expensive to repair than other types of vehicles, and having an accident-free driving record indicates that the policyholder is a lower risk for filing a claim in the future.

Paying a speeding ticket and taking a job with a longer commute by car are unlikely to reduce the monthly premium a policyholder pays for automobile insurance.

Paying a speeding ticket indicates a violation of traffic laws and a higher risk for accidents, and a longer commute by car increases the risk of accidents and damages.

Therefore, the answer is buying a minivan and maintaining an accident-free driving record over time.

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Answer: option a (I, II)

Step-by-step explanation: Person above is correct!

In this particular case, a 2-year old child should receive a dosage of approximately 16.15mL of the drug. This figure is calculated by using the following formula: (adult dosage/150) x (child's weight in kg).

Dosage is a term used to describe the amount of medicine or drug prescribed for a particular individual to take at a given time. It is usually prescribed by a doctor or healthcare provider based on the patient's age, weight, and medical condition. Dosage may vary from person to person depending on their health condition and other factors.

The appropriate dosage of a drug for a 2-year old child is calculated by taking into account the weight, age and health status of the child. Before administering any medications, it is important to consult with a doctor or a pharmacist to ensure that the appropriate dosage is used.

The adult dosage of 257mL is divided by 150 to obtain the base dosage of 1.71mL. This is then multiplied by the child's weight in kilograms (which is usually around 9 kg) to obtain the final dosage of 16.15mL.

It is important to note that the dosage should be rounded to the nearest hundredth. Therefore, the recommended dosage for the 2-year old child in this case would be 16.2mL. It is important to ensure that the dosage is not rounded up, as this could lead to an overdose of the drug, which could have serious consequences.

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The tοtal return οn investment after 20 years is $55,275.00.

By dividing the tοtal number οf shares οutstanding by the number οf shares that a sharehοlder οwns, and multiplying the result by 100, a sharehοlder can determine hοw much οf a firm they οwn.

a. Tο cοmplete the table, we plug in the values οf x and evaluate the fοrmula:

Investment Balance

Years          (x)

3             $14,157.00

5             $18,564.38

10            $32,435.85

20           $67,275.00

b. Tο find the tοtal return οn investment after 20 years, we need tο calculate the difference between the investment balance after 20 years and the initial investment οf $12,000:

Tοtal return = Investment balance after 20 years - Initial investment

Tοtal return = $67,275.00 - $12,000

Tοtal return = $55,275.00

Therefοre, the tοtal return οn investment after 20 years is $55,275.00.

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Answer:

6,400,500 = 660,000(0.66)^t

Step-by-step explanation:

Answer:

1.65 inches

Step-by-step explanation:

3.63/2.2= 1.65

Check

2.2 x 1.65 x 1/2 = 1.815

Repeat for the other half of the triangle

2.2 x 1.65 x 1/2 = 1.815

Add

1.815 + 1.815= 3.63

Answer:

1.65"

Step-by-step explanation:

I did the test

Hope this helps :)

The equation of the graph is y = 0, which means that the graph is a horizontal line at y=0.

An equation is a mathematical statement that asserts that two expressions are equal. It is typically written with an equals sign (=) separating the two expressions. For example, the equation "2x + 1 = 7" asserts that the expression "2x + 1" is equal to the expression "7". Equations can contain variables, which are symbols that represent unknown values. In the example equation, "x" is the variable that we want to solve for. Equations can be used to solve problems and find solutions to unknown values. There are various techniques and methods to solve equations depending on their complexity and nature.

Since the graph is on a semi-log scale, we know that the vertical axis is a logarithmic scale and the horizontal axis is a linear scale. Therefore, the equation of the graph is of the form:

y = a * bˣ

where a and b are constants to be determined.

We are given two points on the graph: (1,0) and (0.5,0). Let's use these points to find the values of a and b.

Using the point (1,0):

0 = a * b¹

0 = a * b

Using the point (0.5,0):

0 = a * b^0.5

0 = √(a)

Therefore, we have:

a = 0

b can be any non-zero number

So the equation of the graph is:

y = 0 * bˣ

which simplifies to:

y = 0

So the equation of the graph is y = 0, which means that the graph is a horizontal line at y=0.

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The decrease in reserve rate from 5% to 2% would increase the amount of money that results from fractional reserve banking on a $25,000 deposit in a bank.

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

If the Federal Reserve decreases the reserve rate from 5% to 2%, it means that banks are required to hold a lower percentage of deposits as reserves and can lend out more money. This can lead to an increase in the amount of money that results from fractional reserve banking.

Assuming a fractional reserve ratio of 10%, which means that banks are required to hold 10% of deposits as reserves, here's how the change in the reserve rate can affect the amount of money that would result from a $25,000 deposit in a bank:

Initially, the bank would hold $2,500 (10% of $25,000) as reserves and can lend out $22,500 ($25,000 - $2,500).

After the decrease in the reserve rate to 2%, the bank would only be required to hold $500 (2% of $25,000) as reserves and can now lend out $24,500 ($25,000 - $500).

If this process continues through multiple rounds of lending and deposits, the total amount of money that can be created through fractional reserve banking can increase significantly.

However, it's important to note that the actual impact of a change in the reserve rate on the money supply depends on a variety of other factors, such as the demand for loans and the willingness of banks to lend out money.

Therefore, the decrease in reserve rate from 5% to 2% would increase the amount of money that results from fractional reserve banking on a $25,000 deposit in a bank.

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The number of intersection points is (a) 1 point or 2 points.

Given that

Line AB and circle P lie on the same plane.

And, we have

AP = 6 mm

BP = 8 mm

The intersection points can be 0, 1, or 2.

Since the radius of circle P is 6mm, any intersection point must also be 6mm away from P.

Next, we have the following test cases

Hence, the answer is (A) 1 point or 2 points.

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Answer:

Step-by-step explanation:

The first question is 3 * 2x which equals 6x + 15. That's your answer

Answer:

[tex]\tt \: Hope \: it \: helps \: you \\[/tex]

Answer:

52 : 39 56 : 42 60 : 45

64 : 48 68 : 51

Step-by-step explanation:

Answer:

About a 7.3 percent Decrease

Step-by-step explanation:

First, calculate the diffrence between the two numbers:

122-113=9

Then, divide this number by the original number, it being 122, and then multiply that number by 100.

the simplified version of this number to the tenths place is 7.38%

We can simplify cosec(t)tant(t) to sec(t). A trigonometric function is a mathematical function that relates the angles of a triangle to the ratios of its sides.

The most common trigonometric functions are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

To simplify the expression cosec(t)tant(t), we need to use the trigonometric identity:

cosec(t) = 1/sin(t)

tant(t) = sin(t)/cos(t)

Substituting these expressions into the original expression, we get:

cosec(t)tant(t) = (1/sin(t))(sin(t)/cos(t))

The sin(t) term in the numerator and denominator cancel out, leaving:

cosec(t)tant(t) = 1/cos(t)

Recalling the definition of secant, sec(t) = 1/cos(t), we can express the simplified expression as:

cosec(t)tant(t) = 1/sec(t)

Therefore, we can simplify cosec(t)tant(t) to sec(t).

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The ratio is [tex]\frac{3}{2}[/tex] in its simplest form.

If the probability of choosing an even card is [tex]\frac{3}{5}[/tex] , then the probability of choosing an odd card is 1- [tex]\frac{3}{5}[/tex] = [tex]\frac{3}{2}[/tex].

Let e be the number of even cards Alisha has, and let o be the number of odd cards. Then, we have:

[tex]\frac{e}{e+0}[/tex]

which simplifies to:

5e=3(e+o)

Expanding the right side gives:

5e=3e+3o

Simplifying further gives:

2e=3o

Dividing both sides by 2 gives:

e= [tex]\frac{3}{2}[/tex]o

[tex]\frac{3}{2}[/tex]

Probability is a branch of mathematics that deals with the study of random events or experiments. It involves the analysis of the likelihood or chance of an event occurring, usually expressed as a number between 0 and 1, where 0 indicates that the event will not occur and 1 indicates that the event is certain to occur.

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Therefore , the solution of the given problem of equation comes out to be  ratio of y to x is not constant in this situation, y does not directly change with x.

The use of the same variable word in mathematical formulas frequently ensures agreement between two assertions. Mathematical equations, also referred to as assertions, are used to demonstrate expression the equality of many academic figures. Instead of dividing 12 into 2 parts in this instance, the normalise technique adds b + 6 to use the sample of y + 6 instead.

Here,

Checking whether there is a fixed ratio between y and x will help us determine whether y directly changes with x.

In general, straight variation is calculated as follows:

=>  y = kx

where k is the variational constant.

Let's split both sides by x to check if the equation 3y = -7x - 18 can be expressed in this way:

=>  3y/x = -7 - 18/x

Now, 3y/x ought to equal some constant k if the relation between y and x is constant:

=>  3y/x = k

When we add this to the solution we previously determined, we get:

=> k = -7 - 18/x

Since the ratio of y to x is not constant in this situation, y does not directly change with x.

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The simplified form of each rational equation is:

Case 1: f(x) = (6 · x³ + 11 · x² - 5 · x - 12) / (3 · x + 4) → f(x) = 2 · x² + x - 3

Case 2: f(x) = (6 · x⁴ + 7 · x³ - 9 · x² + 13 · x - 12) / (3 · x² - x + 3) → f(x) = 2 · x² + 3 · x - 4

Herein we find two rational equations, whose simplified form has to be found. Rational equations are algebraic equations of the form:

R(x) = P(x) / Q(x)

Where:

The procedure to simplify a rational equation is summarized below:

Case 1

f(x) = (6 · x³ + 11 · x² - 5 · x - 12) / (3 · x + 4)

f(x) = [(x - 1) · (3 · x + 4) · (2 · x + 3)] / (3 · x + 4)

f(x) = (x - 1) · (2 · x + 3)

f(x) = 2 · x² - 2 · x + 3 · x - 3

f(x) = 2 · x² + x - 3

Case 2

f(x) = (6 · x⁴ + 7 · x³ - 9 · x² + 13 · x - 12) / (3 · x² - x + 3)

f(x) = [6 · (x + 3 / 4 - √41 / 4) · (x + 3 / 4 + √41 / 4) · (x - 1 / 6 - i √ 35 / 6) · (x - 1 / 6 + i √35 / 6)] / [3 · (x - 1 / 6 - i √ 35 / 6) · (x - 1 / 6 + i √35 / 6)]

f(x) = 2 · (x + 3 / 4 - √41 / 4) · (x + 3 / 4 + √41 / 4)

f(x) = 2 · [x² + (3 / 2) · x + [(3 / 4)² - (√41 / 4)²]]

f(x) = 2 · [x² + (3 / 2) · x - 2]

f(x) = 2 · x² + 3 · x - 4

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(a) Keiko's monthly payment is $536.82. (b) The total amount Keiko will repay is approximately $64,419.19 over the 10-year term. c. (c) The total amount is approximately $14,419.19 in interest.

(a) To find Keiko's monthly payment, we can use the formula for the monthly payment of an amortized loan:

P = (r * A) / (1 - (1+r)^(-n))

where:

P = monthly payment

A = loan amount

r = monthly interest rate (annual interest rate / 12)

n = total number of payments

Plugging in the values we have:

A = $50,000

r = 0.055 / 12

n = 10 * 12 = 120

P = (r * A) / (1 - (1+r)^(-n))

P = (0.055/12 * $50,000) / (1 - (1+0.055/12)^(-120))

P ≈ $536.82

Therefore, Keiko's monthly payment is $536.82.

(b) If Keiko pays the monthly payment each month for the full term of 10 years (120 months), her total amount to repay the loan will be:

Total amount = P * n

Total amount = $536.82 * 120

Total amount ≈ $64,419.19

Therefore, Keiko will repay a total amount of approximately $64,419.19 over the 10-year term.

(c) If Keiko pays the monthly payment each month for the full term of 10 years (120 months), the total amount of interest she will pay can be calculated as:

Total interest = P * n - A

= $536.82 * 120 - $50,000

Total interest ≈ $14,419.19

Therefore, Keiko will pay a total of approximately $14,419.19 in interest over the 10-year term.

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The vertices are (0,3) and (0,-3). The center of the ellipse is (0,0). the equation of the ellipse is (x²)/9 + (y²)/5 = 1.

The center of the ellipse is at the midpoint of the foci, which is (0,0).

The distance between the center and each focus is 2, so we know that c = 2.

The length of the major axis is 6, so we know that a = 3.

The formula for the distance between the center and each vertex is a, so we can find the vertices at (0,3) and (0,-3).

The formula for the minor axis is b² = a² - c², so we have b² = 5. Therefore, b = √(5).

The equation of the ellipse is (x - h)²/a² + (y - k)²/b² = 1, where (h,k) is the center. Plugging in our values:

(x - 0)²/3² + (y - 0)²/5 = 1

Simplifying:

(x²)/9 + (y²)/5 = 1

So the equation of the ellipse is (x²)/9 + (y²)/5 = 1.

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Answer:

Below

Step-by-step explanation:

Volume = L  X  W  X  H    For a CUBE, all of the dimesions L, W, and H are the same ...so this becomes   volume =L  x  L   x  L  =   L^3

Volume = L^3

64 = L^3             take the cube root of both sides of the equation:

L = 4 meters  

Therefore , the solution of the given problem of expressions comes out to be  the expression's precise number is 8/15.

There is a need for calculations like variable multiplication, dividing, joining, and currently removing. If you combined them, you'd get the following: A mathematical formula, some data, and an equation. Values, elements, mathematical operations like equation additions, deductions, errors, and subdivisions, as well as mathematical formulas, make up a statement of truth. It is possible to assess and analyse words and sentences.

Here,

A right triangle with an opposite side of 8 and a hypotenuse of 17 can be completed by adding the missing side using the Pythagorean equation. Make x the neighbouring side. Then:

=> x² + 8² = 17²

=> x² = 17² - 8²

=> x² = 225

=> x = 15

The triangle therefore has edges of 8, 15, and 17. As a result, the neighbouring angle's tangent is 15/8 and the sine of the angle across from the side of length 8 is 8/17. (since tangent is opposite over adjacent). In order to determine the cotangent, we can take the inverse of this tangent:

Coefficient

=> [sin⁻¹ 8/17] = 8/15

As a result, the expression's precise number is 8/15.

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4) Where the hypotenuse is 100 and the adjacent side is x and α = 45°, the x = 70.71

5) Where the opposite side is 100, and  α = 45°, the hypotenuse is 141.42

6) Where the hypotenuse is 88 and β = 45°, x (the opposite side) is 62.23

7) Therefore, the length of each leg of the 45°-45°-90° triangle with a hypotenuse length of 26 is approximately 18.33 units.

8) Therefore, the approximate length of the hypotenuse of the 45°-45°-90° triangle with a leg length of 50 centimeters is 70.71cm.

9) Where the Base is 11 and α = 30°, Hypothenus (x) = 22 and Opposite (y) = 19.05

10) Where the hypotenuse  is 8√3, and β = 45°, opposite (x) = 11.99 and adjacent (y) = 6.93

11) Where the adjacent is 5√3  and α = 30°, opposite (x) = 14.99 and Hypothenus (y) = 17.32

12) where hypotenuse = 30, and the β = 60°, the adjacent (x) = 15 and the opposite (y) = 25.98

13) where opposite is 36.372 and the β = 60°, the hypothenus (x) = 41.99 and the adjacent (y) = 20.99

14) where the hypotenuse is 90.316 and  β = 60°, the opposite (x) = 78.22 and the adjacent (y) is 45.16

15) An equilateral triangle has an altitude length of 27 feet, and the length of a side is 15.588 feet.

16) the length of a side of the equilateral triangle is 22 feet.

The basic rule applied here (from 4-14) is the principle of SOHCAHTOA.


The principle of SOHCAHTOA is a mnemonic device used to remember the three basic trigonometric ratios in a right triangle:

Sine (sin) = opposite/hypotenuse

Cosine (cos) = adjacent/hypotenuse

Tangent (tan) = opposite/adjacent

In SOHCAHTOA, each letter represents one of the ratios:

"S" stands for sine, which relates the opposite side to the hypotenuse.

"O" stands for opposite, which is the side opposite to the angle of interest in the right triangle.

"H" stands for the hypotenuse, which is the longest side of the right triangle and is opposite to the right angle.

"C" stands for cosine, which relates the adjacent side to the hypotenuse.

"A" stands for adjacent, which is the side adjacent to the angle of interest in the right triangle.

"T" stands for the tangent, which relates the opposite side to the adjacent side.

By using SOHCAHTOA, we can quickly and easily find any of the three basic trigonometric ratios, given two sides of a right triangle and an angle.


In 4) This is an example of a right triangle, where one of the angles is 90 degrees. In a right triangle, the side opposite the 90-degree angle is called the hypotenuse, while the other two sides are called the adjacent and opposite sides.

In this case, the hypotenuse is given as 100 units, and the angle between the hypotenuse and adjacent side (x) is given as 45 degrees. To find the length of the adjacent side, we can use the trigonometric function cosine, which relates the adjacent side to the hypotenuse and the angle between them:

cos(α) = adjacent/hypotenuse

Substituting the given values, we get:

cos(45°) = x/100

Simplifying, we get:

x = 100 * cos(45°)

Using the identity cos(45°) = √2/2, we get:

x = 100 * √2/2

Simplifying further, we get:

x = 50√2

To find the approximate value of x in decimal form, we can use a calculator or approximate the value of √2 as 1.414. Then, we get:

x ≈ 50 * 1.414 = 70.71

Therefore, x is approximately equal to 70.71 units.

In 8) Where we needed to find the length of the hypotenuse of a 45°-45°-90° triangle with a leg length of 50 centimeters.

In a 45°-45°-90° triangle, the two legs are congruent, and the length of the hypotenuse is √2 times the length of each leg.

Therefore, if one leg of the triangle has a length of 50 centimeters, then the other leg also has a length of 50 centimeters, and the hypotenuse has a length of:

hypotenuse length = leg length x √2

hypotenuse length = 50 cm x √2

To simplify the expression, we can multiply the numerator and denominator of the fraction by √2:

hypotenuse length = 50 cm x √2 x √2 ÷ √2

hypotenuse length = 50 cm x 2 ÷ √2

hypotenuse length = 100 cm ÷ √2

To rationalize the denominator, we can multiply the numerator and denominator of the fraction by √2:

hypotenuse length = 100 cm ÷ √2 x √2 ÷ √2

hypotenuse length = 100√2 ÷ 2

hypotenuse length = 50√2

Therefore, the length of the hypotenuse of the 45°-45°-90° triangle with a leg length of 50 centimeters is 50√2 centimeters or 70.71cm.

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After answering the presented question, we can conclude that equation Therefore, the solution to the given system of equations is (x, y) = (2, 2).

An equation is a mathematical statement that proves the equality of two expressions connected by the equal symbol '='. 2x - 5 Equals 13, for example. Expressions include 2x-5 and 13. The character '=' joins the two expressions. A mathematical formula with two algebraic expressions on either side of an equal sign (=) is known as an equation. It demonstrates the relationship of equivalence between the left and right formulas. In every formula, LHS = RHS (left side = right side).

[tex]4x + y = 10 ...(1)\\y = 2x - 2 ...(2)\\y = 2x - 2 ...(2)\\y + 2 = 2x ...(2)\\x = (y+2)/2 ...(3)\\4x + y = 10 ...(1)\\4[(y+2)/2] + y = 10 \\[/tex]

[tex]2(y+2) + y = 10 \\3y + 4 = 10\\3y = 6\\y = 2 \\y = 2x - 2 ...(2)\\2 = 2x - 2 \\4 = 2x \\x = 2 \\[/tex]

Therefore, the solution to the given system of equations is (x, y) = (2, 2).

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when running a line, in a right-triangle, from the 90° angle perpendicular to its opposite side, we will end up with three similar triangles, one Small, one Medium and a containing Large one.  Check the picture below.

[tex]\cfrac{6}{x}=\cfrac{x}{8}\implies 48=x^2\implies \sqrt{48}=x\implies \sqrt{4^2\cdot 3}=x\implies 4\sqrt{3}=x[/tex]

If the given zeros are 4 and -2+√5, then the polynomial function of the least degree with integral coefficients that has these zeros can be obtained by using the fact that if a polynomial has a root at x=a, then it has a factor of (x-a).

Therefore, the polynomial function with these zeros can be written as:

(x - 4)(x - (-2+√5))(x - (-2-√5))

Expanding the factors using the difference of squares, we get:

(x - 4)((x + 2) - √5)((x + 2) + √5)

Simplifying the expression further, we get:

(x - 4)((x + 2)^2 - 5)

Expanding the square, we get:

(x - 4)(x^2 + 4x - 1)

Therefore, the polynomial function of the least degree with integral coefficients that has the zeros 4 and -2+√5 is:

f(x) = (x - 4)(x^2 + 4x - 1)

Expanding the factors, we get:

f(x) = x^3 + 4x^2 - x - 16

It takes Carlos 2 minutes longer than Ariel to sing the song.

To find out how much longer it takes Carlos to sing the song, we need to find the difference between the time it takes Ariel and Carlos to sing the song.

Time taken by Ariel = 5.92 minutes

Time taken by Carlos = 7.73 minutes

So, we have

Difference = 7.73 - 5.92 = 1.81 minutes

Rounding to the nearest whole number, we get:

Difference = 2 minutes (rounded to the nearest whole number)

Therefore, the time taken is approximately 2 minutes longer

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The length of the curve y=1/3x³/2-x⁻¹/² from (1, -2/3) to (4, 2/3) is approximately 0.236 units.

what is curve?

In mathematics, a curve refers to a continuous and smooth line or a geometric object that is formed by joining an infinite number of points. Curves can be defined algebraically or geometrically, and they can have different shapes and properties. Some examples of curves include lines, circles, ellipses, parabolas, hyperbolas, and spirals.

Curves are often used in various fields of mathematics, science, and engineering to represent real-world phenomena, such as the trajectory of a moving object, the shape of a surface, or the behavior of a system over time. They are also important in computer graphics and design, where they are used to create visual effects, animations, and models.

In calculus, the study of curves is an essential part of differential and integral calculus. The concepts of limits, derivatives, integrals, and differential equations are used to analyze the properties and behavior of curves, such as their slope, curvature, area, and length.

To find the length of the curve y=1/3x³/2-x¹/² from (1, -2/3) to (4, 2/3), we can use the formula for arc length:

L = ∫[a,b] √(1 + (dy/dx)²) dx

where a and b are the x-coordinates of the starting and ending points of the curve.

First, we need to find the derivative of y:

dy/dx = (d/dx) (1/3 x^³/²- x¹/²) = (1/2) x⁻¹/² - (1/2) x⁻¹/²= x⁻¹/²

Next, we need to find the definite integral of the square root of 1 + (dy/dx)² from 1 to 4:

L = ∫[1,4] √(1 + (x⁽⁻¹/²⁾⁾²) dx

L = ∫[1,4] √(1 + 1/x) dx

To evaluate this integral, we can use the substitution u = 1 + 1/x, which gives du/dx = -1/x²and dx = (1/u) du.

Substituting, we get:

L = ∫[u(1),u(4)] √u (1/u²) du

L = ∫[u(1),u(4)] u⁻¹/² du

L = 2(u(4)¹/²- u(1)¹/²

To find u(1) and u(4), we substitute x=1 and x=4 into the equation for u:

u = 1 + 1/x

u(1) = 2 and u(4) = 1.25

Substituting these values into the expression for L, we get:

L = 2(1.118 - 1)

L = 0.236

Therefore, the length of the curve y=1/3x³/²-x¹/²from (1, -2/3) to (4, 2/3) is approximately 0.236 units.

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