8. A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. (i) What is the car's angular speed in radians per hour? (ii) What is the car’s linear speed in miles per hour?

The integral of  [tex]\int\limits^e_1 {x^{1/2}lnx } \, dx[/tex]  is (2/3)[tex]e^{3/2}[/tex] - (4/9).

To find the integral of   [tex]\int\limits^e_1 {x^{1/2}lnx } \, dx[/tex]  , we can use integration by parts. Let u = ln x and dv = [tex]x^{1/2}[/tex] dx. Then du/dx = 1/x and v = (2/3) [tex]x^{3/2}[/tex] .

Using the formula for integration by parts, we have:

∫[tex]x^{1/2}[/tex]lnx dx = uv - ∫v du/dx dx

= ln x * (2/3) [tex]x^{3/2}[/tex]  - ∫(2/3) [tex]x^{3/2}[/tex]  * (1/x) dx

= ln x * (2/3) [tex]x^{3/2}[/tex]  - (2/3) ∫ [tex]x^{1/2}[/tex]  dx

= ln x * (2/3) [tex]x^{3/2}[/tex]  - (4/9) [tex]x^{3/2}[/tex]  + C

where C is the constant of integration.

To evaluate the definite integral from 1 to e, we substitute e for x in the expression above and subtract the result when x = 1:

 [tex]\int\limits^e_1 {x^{1/2}lnx } \, dx[/tex]   = [(2/3) [tex]e^{3/2}[/tex] ln e - (4/9) [tex]e^{3/2}[/tex] ] - [(2/3)[tex]1^{3/2}[/tex]ln 1 - (4/9)[tex]1^{3/2}[/tex]]

= (2/3) [tex]e^{3/2}[/tex] - (4/9)

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An estimate for the mean time spent watching television is approximately 24.375 hours.

Midpoint of 0 < h ≤ 10 = (0 + 10) / 2 = 5

Midpoint of 10 < h ≤ 20 = (10 + 20) / 2 = 15

Midpoint of 20 < h ≤ 30 = (20 + 30) / 2 = 25

Midpoint of 30 < h ≤ 40 = (30 + 40) / 2 = 35

Sum of observations = (5 × 5) + (15 × 15) + (25 × 50) + (35 × 10) = 125 + 225 + 1250 + 350 = 1950

Total number of observations = 5 + 15 + 50 + 10 = 80

Mean = Sum of observations / Total number of observations = 1950 / 80 = 24.375

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The  equation accurately represents this statement is "Negative 3 less than 4.9 times a number, x, is the same as 12.8"

The statement is "Negative 3 less than 4.9 times a number, x, is the same as 12.8." We can break it down into two parts:

"4.9 times a number, x": This means we need to multiply a number, x, by 4.9. So the first part of the equation is 4.9x.

"Negative 3 less than 4.9 times a number, x, is the same as 12.8":

This means we need to subtract 3 from the result of multiplying x by 4.9. So the second part of the equation is -3.

Putting it all together, we get:

4.9x - 3 = 12.8

This equation accurately represents the given statement.

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The value of theta in the given trigonometry function is 45⁰.

The value of theta in the given trigonometry function is calculated as follows;

9 sin²θ tanθ  -  9 sin²θ = 0

Solve the equation as follows;

9 sin²θ tanθ  =  9 sin²θ

divide both sides by  9 sin²θ;

(9 sin²θ tanθ)/9 sin²θ = 9 sin²θ /9 sin²θ

tanθ = 1

Now, solve for θ as follows;

θ = tan⁻¹ (1)

θ = 45⁰

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Therefore, the solution to the equation -2 - 5/4x = 13 is x = -12.

A mathematical statement proving the equality of two expressions is known as an equation. Variables, numbers, and mathematical operations like addition, subtraction, multiplication, and division are frequently included. In addition to representing a connection between variables, an equation may also be used to find an unknown number. As they enable us to define and evaluate connections between things and make predictions about how they will behave under various circumstances, equations are a crucial tool in many domains, including mathematics, physics, engineering, and many more.

To solve for x in the equation -2 - 5/4x = 13, you can follow these steps:

Add 2 to both sides of the equation to isolate the fraction on the left side:

-2 - 5/4x + 2 = 13 + 2

-5/4x = 15

Multiply both sides of the equation by -4/5 to get x by itself:

(-4/5) * (-5/4x) = (-4/5) * 15

x = -12

Therefore, the solution to the equation -2 - 5/4x = 13 is x = -12.

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18. C

(m / 2) - 6 = (m / 4) + 2

---Multiply everything by the LCM of the denominators

---LCM = 4

2m - 24 = m + 8

m - 24 = 8

m = 32

19. A

k / 12 = 25 / 100

---We can simplify 25/100

---We want to simplify enough to where the denominator of 25/100 is a multiple or factor of 12

k / 12 = 1 / 4

---4 x 3 = 12, 1 x 3 = 3

k = 3

20. A

9 / 5 = 3x / 100

---Cross multiply and solve algebraically

(5 * 3x) = (9 * 100)

15x = 900

x = 60

Hope this helps!

The coordinates of Y' are (6, 4) and the coordinates of Z' are (5, -1).

To find the coordinates of Y' and Z', we need to apply the same translation that maps X to X'.

We know that the vector that connects X to X' is (3 - (-1), 1 - 0) = (4, 1).

So, to translate Y to Y', we add the vector (4, 1) to the coordinates of Y:

Y' = Y + (4, 1)

= (2, 3) + (4, 1)

= (6, 4)

Similarly, to translate Z to Z', we add the vector (4, 1) to the coordinates of Z:

Z' = Z + (4, 1)

= (1, -2) + (4, 1)

= (5, -1)

Therefore, the coordinates of Y' are (6, 4) and the coordinates of Z' are (5, -1).

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Answer:

Step-by-step explanation:

We are given a figure, consisting of a right triangle with a circle cut out of it.

To Calculate : area of the shaded region below,

First Let's find the Area of the right triangle:

Area(triangle) = 1/2 × Base × Height

= 1/2 × 56 × 56

= 1/2 × 3136

= 3136/2

= 1568 m².

Now, We are diameter of the circle = 20 m

Radius of the circle = 20/2 = 10 m

Acc. to question we have to use 3.14 as an

approximation for pi.

Area (circle) = 3.14 × 10 × 10

Area (circle) = 3.14 × 100

Area (circle) = 314 m²

Area of shaded region = Area of triangle - Area of circle =

= 1568 m² - 314 m²

= 1.254 square meters

Therefore, The approximate area of the shaded region is 1.254 square meters.

Option (a) is the required answer

Answer:

A ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

Step-by-step explanation:

Let's use variables to represent the cost of each dinner. Let's say that the cost of a ribeye steak dinner is "r" and the cost of a grilled salmon dinner is "s".

From the first day's sales, we know:

From the second day's sales, we know:

We now have two equations with two variables, which we can solve using elimination or substitution.

Let's use elimination. If we multiply the first equation by 15 and the second equation by 11, we can eliminate the "r" variable:

Subtracting the second equation from the first, we get:

Dividing both sides by 133, we get:

So a grilled salmon dinner costs $14.04.

We can substitute this value back into one of the original equations to solve for "r". Let's use the first equation:

So a ribeye steak dinner costs $32.23.

Therefore, a ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

The addition of the vectors [tex]\vec a + \vec b[/tex] is (3, -5) and it has been represented by using the head to tail method in the graph below.

In Mathematics and Science, a vector can be defined as an element of a vector space, which represents an object that is composed of both magnitude and direction.

In Mathematics and Science, a vector typically comprises two (2) points. First, is the starting point which is commonly referred to as the "tail" and the second (ending) point that is commonly referred to the "head."

Generally speaking, the head to tail method of adding two (2) vectors involve drawing the first vector ([tex]\vec a[/tex]) on a cartesian coordinate and then placing the tail of the second vector ([tex]\vec b[/tex]) at the head of the first vector. Lastly, the resultant vector is then drawn from the tail of the first vector ([tex]\vec a[/tex]) to the head of the second vector ([tex]\vec b[/tex]).

By solving the given vectors algebraically, we have the following:

[tex]\vec a + \vec b[/tex] = (4, -3) + (-1, -2)

[tex]\vec a + \vec b[/tex] = [(4 + (-1)), (-3 + (-2))

[tex]\vec a + \vec b[/tex] = [(4 - 1), (-3 - 2)]

[tex]\vec a + \vec b[/tex] = (3, -5).

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Answer:

yes

Step-by-step explanation:

the factor theorem says if (x + a) is a factor of a polynomial f(x) then f(- a) = 0

given (x+ 5) then if f(- 5) = 0 , (x + 5) is a factor

4[tex](-5)^{4}[/tex] + 16(- 5)³ - 6(- 5)² + 73(- 5) + 15

= 4(625)+ 16(- 125) - 6(25) - 365 + 15

= 2500 - 2000  - 150 - 350

= 500 - 500

= 0

then (x + 5) is a factor of the given polynomial

A. The slopes are the same but the y - intercepts are different.

In a graph, the slope is the steepness of a line, which is a measure of how quickly the y-coordinate changes with respect to the x-coordinate. It is represented by the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.

The slope is often denoted by the letter m, and is calculated as:

m = (y₂ - y₁) / (x₂ - x₁)

where (x₁, y₁) and (x₂, y₂) are any two points on the line.

In a graph, the y-intercept is the point where a line or curve intersects the y-axis. It is the value of the dependent variable (usually denoted by y) when the independent variable (usually denoted by x) is equal to zero.

The y-intercept is often denoted by the letter b, and is expressed as a coordinate point (0, b), where 0 represents the x-coordinate and b represents the y-coordinate. For example, in the equation of a line y = mx + b, the y-intercept is the value of b.

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The given polynomial is a prime polynomial.

The given polynomial is

x² - x + 35

Since we know that discriminant of quadratic polynomial

p(x)= ax² + bx + c  is b² - 4ac

Therefore, discriminant of the given polynomial be,

⇒ (-1)² - 4x1x35

⇒ - 139

Since discriminant is negative number so it has no real roots.

And an irreducible or primal polynomial is a polynomial with integer coefficients that cannot be factored into polynomials of lower degree and also with integer coefficients.

Hence it is not factored completely.

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Answer:

3.36%

Step-by-step explanation:

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

The correct option regarding which bus has the least spread among the travel times is given as follows:

Bus 14, with an IQR of 6.

To obtain measures of spread, we can use the dot plot to visualize the frequency of each observation in the data-set, and then calculate the interquartile range (IQR).

For a group of 15 students, we can find the quartiles as follows:

The first quartile (Q1) is the median of the first half of the data-set, which consists of the first seven students.

Looking at the dot plot, the fourth dot represents the median of the first half. For Bus 14, Q1 = 12. For Bus 18, Q1 = 9.

The third quartile (Q3) is the median of the second half of the data-set, which consists of the last seven students. Looking at the dot plot, the eleventh dot represents the median of the second half. For Bus 14, Q3 = 18.

For Bus 18, Q3 = 16.

We can then calculate the IQR as the difference between Q3 and Q1:

For Bus 14, IQR = Q3 - Q1 = 18 - 12 = 6.

For Bus 18, IQR = Q3 - Q1 = 16 - 9 = 7.

Based on the IQR values, we can conclude that Bus 14 is more consistent than Bus 18, as it has a lower IQR.

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The total cost of the fence needed for the Perimeter of the work site is $8,800. So the answer is option B.

Because the question involves many units of measurement, we must convert them all to the same unit in order to determine the answer.

1/4 mile is equivalent to 1320 feet (1 mile = 5280 feet).

The length of the fence required to surround the work site equals the perimeter of the work site. To calculate the length of the fence in yards, divide 1320 feet by 3 (since a yard is 3 feet long).

1320 feet ÷ 3 = 440 yards

So the length of the fence needed is 440 yards.

The fence costs $20.00 per yard, thus to get the total cost, multiply the length of the fence by the cost per yard:

440 yards x $20.00/yard = $8,800.00

Therefore, the total cost of the fence needed for the perimeter of the work site is $8,800.00.

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The expression "the cube root of the product of x and y, less twelve" can be mathematically represented as ³√(xy) - 12.

To understand this expression, let's break it down step by step:

Product of x and y:

We multiply the values of x and y together to find their product, denoted as xy.

Cube root of the product:

We take the cube root of the product obtained in the previous step.

The cube root of a number x is the value that, when raised to the power of 3, equals x. In this case, we take the cube root of xy, represented as ³√(xy).

Subtract twelve:  

Finally, we subtract twelve from the result obtained in the previous step, ³√(xy). This gives us the expression ³√(xy) - 12.

The expression ³√(xy) - 12 represents a mathematical operation involving the cube root of the product of x and y, followed by subtracting twelve from the result.

This can be useful in various mathematical contexts, such as solving equations or modeling real-world problems.

It's important to note that the expression is dependent on the values of x and y.

By substituting specific values for x and y, we can evaluate the expression to obtain a numerical result.

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The number of kg of maize was left with Ali is 73350 Kg.

Given that, Ali had 3469 bags of maize each weighing 90 kg.

Ali sold 2654 of them.

Number of bags left = 3469-2654

= 815

Number of Kg's of maize left = 815×90

= 73350 Kg

Therefore, the number of kg of maize was left with Ali is 73350 Kg.

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2a. To calculate the average number of violations committed, we add up all the violations and divide by the number of offenders:

Average number of violations = (15+20+10+25+5+22+8+12+18+7+23+6+24+19+11) / 15

Average number of violations = 187 / 15

Average number of violations ≈ 12.47

Therefore, the average number of violations committed by the offenders in the sample is approximately 12.47.

2b. To calculate the sample mean for the given data, we add up the five samples and divide by the number of samples:

Sample mean = (22 + 8 + 19 + 7 + 24) / 5

Sample mean = 80 / 5

Sample mean = 16

Therefore, the sample mean for the five samples is 16.

2c. To estimate the mean interval with 95% confidence level, we can use the t-distribution and the formula:

Sample mean ± t-value (α/2, n-1) x (standard deviation / square root of n)

We are not given the standard deviation of the population, so we need to estimate it using the sample standard deviation:

s = sqrt [ Σ(xi - x)^2 / (n - 1) ]

where xi is the i-th sample, x is the sample mean, and n is the number of samples.

Using the five samples given in part (b), we can calculate the sample standard deviation:

s = sqrt [ ((22-16)^2 + (8-16)^2 + (19-16)^2 + (7-16)^2 + (24-16)^2) / (5-1) ]

s = sqrt [ (36 + 64 + 9 + 81 + 64) / 4 ]

s ≈ 9.17

Using a t-distribution table with α/2 = 0.025 and degrees of freedom = n-1 = 4, we find that the t-value is 2.776.

Plugging in the values, we get:

Sample mean ± t-value (α/2, n-1) x (standard deviation / square root of n)

16 ± 2.776 x (9.17 / sqrt(5))

16 ± 9.55

Therefore, with 95% confidence, we estimate that the mean number of violations committed by the population of lawbreakers is between 6.45 and 25.55.

   To estimate the number of violations when the road area is 6 meters, we need to use the regression equation Y = a + bX. However, we are not given the values of a and b.

Without knowing the values of a and b, we cannot estimate the number of violations when the road area is 6 meters or use the regression equation to make any predictions.Answer:

Step-by-step explanation:

For a population of 300 lawbreakers:

2. a. To calculate the average number of violations committed, find the mean of the given data set.

Mean = (15 + 20 + 10 + 25 + 5 + 22 + 8 + 12 + 18 + 7 + 23 + 6 + 24 + 19 + 11) / 15

Mean = 180 / 15

Mean = 12

Therefore, the average number of violations committed is 12.

b. To calculate the sample mean, we need to find the mean of the given sample data set.

Sample mean = (22 + 8 + 19 + 7 + 24) / 5

Sample mean = 80 / 5

Sample mean = 16

Therefore, the sample mean is 16.

c. To estimate the mean interval with 95% confidence level, we can use the t-distribution with n-1 degrees of freedom, where n is the sample size. The formula for the confidence interval is:

Confidence interval = sample mean ± (t-value x standard error)

where t-value is the value obtained from the t-distribution table for a 95% confidence level and n-1 degrees of freedom, and standard error is the standard deviation of the sample data divided by the square root of the sample size.

First, find the standard deviation of the sample data set.

Standard deviation = √[(Σ(x - μ)²) / (n - 1)]

where Σ is the sum of the values, x is each value in the sample data set, μ is the sample mean, and n is the sample size.

μ = 16 (from part b)

n = 5

x values = 22, 8, 19, 7, 24

Standard deviation = √[((22-16)² + (8-16)² + (19-16)² + (7-16)² + (24-16)²) / (5 - 1)]

Standard deviation = √[(36 + 64 + 9 + 81 + 64) / 4]

Standard deviation = √(254 / 4)

Standard deviation = √63.5

Standard deviation ≈ 7.97

Next, find the t-value for a 95% confidence level and 4 degrees of freedom. From the t-distribution table, the t-value is 2.776.

Confidence interval = 16 ± (2.776 x (7.97 / √5))

Confidence interval = 16 ± (2.776 x 3.57)

Confidence interval = 16 ± 9.91

Therefore, the mean interval with 95% confidence level is (16 - 9.91, 16 + 9.91), or approximately (6.09, 25.91).

3. To estimate the number of violations for a road area of 6 meters, we need to use the regression equation Y = a + bX, where Y is the number of violations and X is the road area in meters.

First find the values of a and b from the given data.

Using the formula:

b = [(nΣXY) - (ΣX)(ΣY)] / [(nΣX²) - (ΣX)²]

a = (ΣY - bΣX) / n

where n is the number of data points, Σ is the sum of the values, and X and Y are the variables.

n = 4

ΣX = 18

ΣY = 15

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Image transcribed:

QUESTIONS

2. There are 300 lawbreaker as a population

X = the number of law violations by the 1st lawbreaker in 1 year. Researched as many as 15 offenders as a random sample. It turns out that the number of violations committed by them is :

15  20  10  25  5

22  8  12  18  7

23  6  24  19  11

Questions:

a. Calculate the average number of violations committed

b. If 5 samples are taken, namely: 22 8 19 7 24

Calculate the sample mean

c. Estimate the mean interval with 95% confidence level

3. The number of traffic violations in city A every day (several times) is:

Road area (X) in meter | 4   3.5   5   5.5

Violation (Y)                 | 3   3   2   7

Approximately how many violations if the road area is 6 meters? If the regression equation is:

Y = a+bX

Answer:

x=9km

Step-by-step explanation:

Pythagorean Theorem: [tex]a^{2}[/tex] + [tex]b^{2}[/tex] = [tex]c^{2}[/tex]

The hypotenuse is [tex]c^{2}[/tex], so the equation will be

[tex]12^{2}[/tex] + [tex]b^{2}[/tex] = [tex]15^{2}[/tex]

Evaluating the equation gives us

144 + [tex]b^\\{2}[/tex] = 225

Subtract 144 from each side to have [tex]b^{2}[/tex] alone

[tex]b^{2}[/tex] = 81
square root on each side to get rid of the exponent

[tex]\sqrt{b^2[/tex] = [tex]\sqrt{81}\\[/tex]

[tex]\sqrt{b^2[/tex]'s square root is b and          [tex]\sqrt{81[/tex]'s is 9,-9, giving us b=±9

As distance cannot be negative, x = 9km

Answer:

Step-by-step explanation:

11000*14539.95

7.1483737364203627356954981215625 × 10^24

is the answer

Answer:

False.

Step-by-step explanation:

A integer in math is defined as a whole number. Therefore, the presence of fractions (which, in this case, is 1/2) and decimals, would mean that the term is in fact, not a integer.

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Answer:

2(4x-3)

Step-by-step explanation:

Find common numbers between 8 and 6, which is 2. 2x4= 8 2x3=6

x is only on 8 so you can't put x on the outside of the bracket. so you put it with the 4 so it comes out correct

The estimate of the cost for a 20-ft cord is given as follows:

$74.66.

To find the regression equation, which is also called called line of best fit or least squares regression equation, we need to insert the points (x,y) in the calculator.

The points in the context of the problem are given as follows:

(3, 12.75), (5, 16), (6, 25.99), (50, 185).

Inserting these points into a calculator, the regression equation is given as follows:

y = 3.6794x + 1.06462.

Hence the estimate of the cost for a 20-ft cord is given as follows:

y = 3.6794(20) + 1.06462

y = $74.66.

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Check the picture below.

Answer:

No

Step-by-step explanation:

In order for a triangle to be a right angle, the sum of the squares of the two legs (a and b) must be equal the square of the longest side, known as the hypotenuse (c):

[tex]a^2+b^2=c^2[/tex]

The square root of 155 is the longest side as it is approximately 12.45.  Thus, if the triangle is a right triangle, the square root of 155 must be the hypotenuse.

We can now check to see whether the sum of the squares of 7 and 9 equal the square of the square root of 155:

[tex]7^2+9^2=(\sqrt{155})^2\\ 49+81=155\\130=155[/tex]

The equation is not true since 130 is less than and not equal to 155.  Therefore, a triangle with side lengths 7, 9, and the square root of 155 cannot be a right triangle.

The probability that a respondent has both portable earbuds and a smart speaker is 0.16

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

This means that

P(earbuds) = 54.5%

P(smart speaker) = 30%

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

P = P(earbuds) * P(smart speaker)

Substitute the known values in the above equation, so, we have the following representation

P = 54.5% * 30%

Evaluate

P = 0.16

Hence, the probability is 0.16

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There are 2000 milligrams in 2 grams of a substance, and it constitutes 40% of a 5-gram sample.

Convert grams to milligrams:

2 grams = 2 * 1000 milligrams

2 grams = 2000 milligrams

Calculate the percentage in a 5-gram sample:

Percentage = (2000 milligrams / 5000 milligrams) * 100

Percentage = (2 / 5) * 100

Percentage = 0.4 * 100

Percentage = 40%

Thus, there are 2000 milligrams in 2 grams of a substance, and it constitutes 40% of a 5-gram sample.

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​2 grams = ​ milligrams

How many milligrams are there in 2 grams of a substance, and what is the percentage of this amount in a 5-gram sample?

Answer: D

Step-by-step explanation: 60x9=540 and 540x80=43,200