According to the Rational Root Theorem, the following are potential roots of f(x) = 60x2 – 57x – 18. Which is an actual root of f(x)? , 3 6, -1/4,-6/5

Answer:

h^2=a^2+b^2.

h^2=(6x^2y)^2+(9x^2y)^2.

h^2=36x^4y^2+81x^4y^2.

h^2=117x^4y^2.

h=sqrt(117x^4y^2).

=3 √13 x^2y

After applying Pythagoras' theorem the length of the hypotenuse we get is approximately 10.8 x²y unit.

Use the concept of the triangle defined as:

A triangle is a three-sided polygon, which has three vertices and three angles which has a sum of 180 degrees.

And the Pythagoras theorem for a right-angled triangle is defined as:

(Hypotenuse)²= (Perpendicular)² + (Base)²

Given that,

Base = 6x²y

perpendicular = 9x²y

Now apply the Pythagorean theorem,

(Hypotenuse)²= (6x²y)² + (9x²y)²

(Hypotenuse)²= 36x⁴y² + 81x⁴y²

(Hypotenuse)²= 117x⁴y²

Take square root on both sides we get,

Hypotenuse = √117 x²y

Hypotenuse ≈ 10.8 x²y

Hence,

The length of the hypotenuse is approximately 10.8 x²y unit.

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I think its both because An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. Consecutive means one after the other. The constant value is called the common difference. Another way to think about an arithmetic sequence is that each term in the sequence is equal to the previous term plus the common difference and also A geometric sequence is a sequence in which the ratio of any two consecutive terms is constant. This constant value is called the common ratio. Another way to think about a geometric sequence is that each term is equal to the previous term times the common ratio. So when u look at it check out if the 16, 8, 4, and 2 see if they fit these terms

The ratios are constant (0.5), so the sequence is geometric.

Since the sequence is not arithmetic but is geometric, the correct answer is: B. Geometric

To determine whether the sequence 16, 8, 4, 2 is arithmetic, geometric, both, or neither, let's analyze the differences between consecutive terms and the ratios between consecutive terms.

Arithmetic Sequence:

An arithmetic sequence is a sequence in which the difference between consecutive terms is constant.

Calculating the differences:

8 - 16 = -8

4 - 8 = -4

2 - 4 = -2

The differences are not constant, so the sequence is not arithmetic.

Geometric Sequence:

A geometric sequence is a sequence in which the ratio between consecutive terms is constant.

Calculating the ratios:

8 / 16 = 0.5

4 / 8 = 0.5

2 / 4 = 0.5

The ratios are constant (0.5), so the sequence is geometric.

Since the sequence is not arithmetic but is geometric, the correct answer is:

B. Geometric

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Answer:

C){(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)}

Step-by-step explanation:

The one-to-one function has inverse where the inverse is also function.

That is, there should be unique output for each input values.

Look at the options, Option C) only has unique output for each input values.

Therefore, the answer C){(-1, 3), (0, 4), (1, 14), (5, 6), (7, 2)}

Hope this will helpful.

Thank you.

Final answer:

The prediction method for the number of broken light bulbs in an order is based on a proportional relationship. Using the ratio from a smaller sample, it's calculated that an order of 125 light bulbs would result in 20 broken bulbs, assuming the breakage rate stays constant.

Explanation:

Given that 4 out of 25 light bulbs are broken in the initial order, we can predict the number of broken light bulbs in a larger order of 125 bulbs. The ratio of broken bulbs to total bulbs in the initial order is 4 broken bulbs for every 25 bulbs.

To find the predicted number of broken bulbs in a larger order, you multiply the total number of bulbs in the larger order by the ratio of broken bulbs in the smaller order. The calculation for the larger order of 125 bulbs would be:

(4 broken bulbs / 25 total bulbs) × 125 total bulbs in the larger order = 20 broken bulbs

Therefore, if 25 light bulbs yield 4 broken ones, an order of 125 light bulbs is predicted to have 20 broken bulbs, assuming the rate of breakage remains consistent.

Answer:

So the answer is D

Step-by-step explanation:

The value of x is option (D) 2

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y.

Given,

f(x) = [tex]2^{x}[/tex]

we have to find the value of x when f(x) =4

f(x) = [tex]2^{x}[/tex] =4

[tex]2^{x}=4\\ 2^{x}=2^{2}[/tex]

Therefore value of x is 2

Hence, the value of x is option (D) 2

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In the 2005 regular season, the Chicago White Sox won 99 games and the Detroit Tigers won 71 games.

In the 2005 regular season, let's denote the number of games won by the Chicago White Sox as 'x' and the number of games won by the Detroit Tigers as 'y'. We know that the Chicago White Sox won 28 more games than the Detroit Tigers, so we can write x = y + 28.

Together, they won a total of 170 games, so we can write x + y = 170.

Now we can solve the system of equations:

Therefore, the Chicago White Sox won 99 games and the Detroit Tigers won 71 games.

The Chicago White Sox won 99 games and the Detroit Tigers won 71 games in the 2005 regular season.

To find out how many games each team won, we need to set up a system of equations using the given information. Let x represent the number of games won by the Chicago White Sox and y represent the number of games won by the Detroit Tigers.

We are given two pieces of information:

1. The White Sox won 28 more games than the Tigers, so we have the equation x = y + 28.

2. Together, both teams won a total of 170 games, so we have the equation x + y = 170.

We can use these equations to solve for the values of x and y. Substituting the first equation into the second equation, we get (y + 28) + y = 170. Combining like terms, we have 2y + 28 = 170. Subtracting 28 from both sides, we get 2y = 142. Dividing both sides by 2, we find that y = 71.

Now we can substitute the value of y back into the first equation to find x. x = 71 + 28, so x = 99. Therefore, the Chicago White Sox won 99 games and the Detroit Tigers won 71 games.

To determine the probability of a patron waiting less than 18 minutes or more than 25 minutes at a restaurant, we calculate the Z-scores for each time, look up the corresponding probabilities in a standard normal distribution table, and add the probabilities together.

To find the probability that a patron will wait less than 18 minutes or more than 25 minutes at a restaurant with an average waiting time of 23.5 minutes and a standard deviation of 3.6 minutes, assuming a normal distribution, we need to calculate two separate probabilities and then add them together.

First, we calculate the Z-score for 18 minutes, which is (18 - 23.5) / 3.6. Then, we find the corresponding probability from the standard normal distribution table. This gives us the probability of waiting less than 18 minutes.

Secondly, we calculate the Z-score for 25 minutes, which is (25 - 23.5) / 3.6. The corresponding probability gives us the probability of waiting more than 25 minutes. However, we want the probability of waiting longer, so we subtract this probability from 1 to find the probability of waiting more than 25 minutes.

Adding both probabilities gives us the total probability of a patron waiting either less than 18 minutes or more than 25 minutes.

Answer with explanation:

Question asked by Mr.Maddox to find the sum of fractions with the help of number line :

    [tex]\rightarrow 3 \frac{2}{3}+1 \frac{3}{4}+\frac{2}{3}\\\\ \text{Using Associative Property}\\\\\rightarrow a+(b+c)=(b+c)+a\\\\\rightarrow [\frac{2}{3}+ 1\frac{3}{4}]+3 \frac{2}{3}\\\\\rightarrow [\frac{2}{3}+\frac{7}{4}]+\frac{11}{3}\\\\\rightarrow \frac{21+8}{12}+ \frac{11}{3}\\\\\rightarrow\frac{29+44}{12}\\\\ \rightarrow\frac{73}{12}\\\\\rightarrow 6\frac{1}{12}[/tex]

Answer:

add the parallel sides and divide by 2

then multiply it by the perpendicular side so  

Step-by-step explanation:

A=side one+side two x2

                2

1.) 9(4) + 8(5)

36 + 40 = 74

2.) 2(3) + 8(3)

6 + 24 = 30

3.) 4(5) + 7(5)

20 + 35 = 55

4.) 10(4) + 18(5)

40 + 90 = 130

5.) 2 + 8(1/4)

2 + 2 = 4

6.) 9(1/3) + 8(1/4)

3 + 2 = 5

7.) 3(8) + 7(4)

24 + 28 = 52

8.) 12(1/4) + 16(5)

4 + 80 = 84

Answer:

the answer is add 2 feet on each side!

Step-by-step explanation:

The equivalent fractions with the same denominator are 6/ 15 and 5 /15.

The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.

Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.

Given that the two fractions are 2/5 and 1/3. The two equivalent expressions with the common base will be written as:-

2/5 = ( 2 x 3 ) / ( 5 x 3 )

2//5 = 6 / 15

1/3 = ( 1 x 5 ) / ( 3 x 5 )

1 / 3 = 5 / 15

Therefore, the two fractions are 6/ 15 and 5 /15.

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Final answer:

The general solution to the given second-order differential equation, 3y'' + 2y' + y = 0, is found using the characteristic equation method, resulting in complex roots. The solution is expressed in terms of sine and cosine functions multiplied by an exponential decay factor.

Explanation:

To find the general solution of the given second-order differential equation, 3y'' + 2y' + y = 0, we first convert it into its characteristic equation. This is done by substituting y = ert into the differential equation, where r is the root of the characteristic equation and t is an independent variable. This approach transforms the given differential equation into a quadratic equation.

The characteristic equation for this differential equation is 3r2 + 2r + 1 = 0. Solving this quadratic equation using the formula r = [-b ± sqrt(b2 - 4ac)] / 2a, where a=3, b=2, and c=1, gives the roots of the characteristic equation. In this case, the discriminant (b2 - 4ac) is less than zero, indicating complex roots.

The roots can be found to be r = -1/3 ± i(sqrt(2)/3). Therefore, the general solution to the differential equation is y(t) = e-t/3[C1cos(sqrt(2)t/3) + C2sin(sqrt(2)t/3)], where C1 and C2 are constants determined by initial conditions.

The general solution is [tex]\( y(t) = e^{-\frac{t}{3}} (C_1 \cos\left(\frac{\sqrt{2} t}{3}\right) + C_2 \sin\left(\frac{\sqrt{2} t}{3}\right)) \)[/tex].

To solve the second-order linear homogeneous differential equation [tex]\( 3y'' + 2y' + y = 0 \)[/tex], we follow these steps:

1. Write the characteristic equation associated with the differential equation.

2. Solve the characteristic equation for its roots.

3. Write the general solution based on the roots of the characteristic equation.

Step 1: Write the Characteristic Equation

The given differential equation is:

[tex]\[ 3y'' + 2y' + y = 0 \][/tex]

We assume a solution of the form [tex]\( y = e^{rt} \)[/tex]. Substituting [tex]\( y = e^{rt} \)[/tex] into the differential equation, we get:

[tex]\[ 3(r^2 e^{rt}) + 2(r e^{rt}) + e^{rt} = 0 \][/tex]

Dividing through by [tex]\( e^{rt} \)[/tex] (which is never zero), we obtain the characteristic equation:

[tex]\[ 3r^2 + 2r + 1 = 0 \][/tex]

Step 2: Solve the Characteristic Equation

The characteristic equation is a quadratic equation:

[tex]\[ 3r^2 + 2r + 1 = 0 \][/tex]

To find the roots of this quadratic equation, we use the quadratic formula:

[tex]\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

where [tex]\( a = 3 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = 1 \)[/tex].

Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the quadratic formula:

[tex]\[ r = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 3 \cdot 1}}{2 \cdot 3} \][/tex]

[tex]\[ r = \frac{-2 \pm \sqrt{4 - 12}}{6} \][/tex]

[tex]\[ r = \frac{-2 \pm \sqrt{-8}}{6} \][/tex]

[tex]\[ r = \frac{-2 \pm 2i\sqrt{2}}{6} \][/tex]

[tex]\[ r = \frac{-1 \pm i\sqrt{2}}{3} \][/tex]

Thus, the roots of the characteristic equation are:

[tex]\[ r_1 = \frac{-1 + i\sqrt{2}}{3} \][/tex]

[tex]\[ r_2 = \frac{-1 - i\sqrt{2}}{3} \][/tex]

Step 3: Write the General Solution

Since the roots are complex conjugates [tex]\( r_1 = \alpha + i\beta \)[/tex] and [tex]\( r_2 = \alpha - i\beta \)[/tex] with [tex]\( \alpha = -\frac{1}{3} \)[/tex] and [tex]\( \beta = \frac{\sqrt{2}}{3} \)[/tex], the general solution to the differential equation is of the form:

[tex]\[ y(t) = e^{\alpha t} (C_1 \cos(\beta t) + C_2 \sin(\beta t)) \][/tex]

Substitute [tex]\( \alpha \)[/tex] and [tex]\( \beta \)[/tex]:

[tex]\[ y(t) = e^{-\frac{t}{3}} \left( C_1 \cos\left( \frac{\sqrt{2} t}{3} \right) + C_2 \sin\left( \frac{\sqrt{2} t}{3} \right) \right) \][/tex]

Thus, the general solution of the differential equation [tex]\( 3y'' + 2y' + y = 0 \)[/tex] is:

[tex]\[ y(t) = e^{-\frac{t}{3}} \left( C_1 \cos\left( \frac{\sqrt{2} t}{3} \right) + C_2 \sin\left( \frac{\sqrt{2} t}{3} \right) \right) \][/tex]

where [tex]\( C_1 \)[/tex] and [tex]\( C_2 \)[/tex] are arbitrary constants.

Answer:

The fifth term of the sequence is:

49

Step-by-step explanation:

We are given the general term([tex]n^{th}[/tex] term) of the sequence as:

an=5.2n-1

We have to find the fifth term

i.e. we have to find the value of an for n=5

a5=5×2×5-1

   =50-1

  =49

Hence, the fifth term of the sequence is:

49

Final answer:

To calculate the percentage discount of a book, subtract the sale price from the original price, divide by the original price, and multiply by 100. The detailed calculation shows that the book had a 30% discount during the sale.

Explanation:

The question asks how to calculate the percentage discount of a book that has been reduced from its normal price to a sale price. To find the percentage discount, we subtract the sale price from the original price, and then divide this difference by the original price. Finally, we multiply by 100 to get the percentage.

Step-by-Step Solution:

The percentage discount offered on the book during the sale was 30%.

The given points do not form a rhombus.

A rhombus is a quadrilateral that has four equal sides.

Some of the properties we need to know are:

- The opposite sides are parallel to each other.

- The opposite angles are equal.

- The adjacent angles add up to 180 degrees.

We have,

To determine if the given points form a rhombus, we need to check if the sides are congruent (have equal length) and if the opposite angles are congruent (have equal measure).

First, we can find the lengths of all four sides of the quadrilateral using the distance formula:

AB = √((3 - 0)² + (4 - 0)²) = 5

BC = √((8 - 3)² + (4 - 4)²) = 5

CD = √((5 - 8)² + (0 - 4)²) = 5

DA = √((0 - 5)² + (0 - 4)²) = 5

Since all four sides have the same length of 5 units, the quadrilateral satisfies the property of having congruent sides.

Next, we need to check if the opposite angles are congruent.

We can do this by finding the slopes of the two diagonals and checking if they are perpendicular. If the slopes are perpendicular, then the opposite angles are congruent.

The slope of diagonal AC can be found as:

m(AC) = (4-0)/(8-0) = 1/2

The slope of diagonal BD can be found as:

m(BD) = (0-4)/(5-3) = -2/2 = -1

Since the product of the slopes is:

m(AC) x m(BD) = (1/2) x (-1) = -1/2

which is not equal to -1, the diagonals are not perpendicular and the opposite angles are not congruent.

Therefore,

The given points do not form a rhombus.

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The derivative of the function f'(x)= arccsc 8x is f'(x)= -1/(|8x|√((8x)² - 1)). We achieved this result by using the rules for derivatives of arcsine, arcsecant and arccosecant along with the chain rule for differentiation of composite functions.

To find the derivative of the function f'(x)= arccsc 8x, we will first need to understand that the derivative of the arcsine of x, also known as the inverse sine of x, is 1/(√(1 - x²)). Similarly, the derivative of arcsecant of x, known as the inverse secant of x, is 1/(|x|√(x² - 1)).

However, here we have arccosecant of x, known as the inverse cosecant of x. With this, we find that the derivative of arccosecant of x is -1/(|x|√(x² - 1)). Now in context of our function, where we replace x with 8x, we get f'(x)= -1/(|8x|√((8x)² - 1)).

We used the rules mentioned as well as the mutation rule ( df (u) = [du f(u)] dx ). This mutation rule is also known as the chain rule in differentiation, which allows us to differentiate composite functions. The function present here is a composite function where we have 8x in place of x in the arccosecant function.

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