evaluate the integral. (remember to use absolute values where appropriate. use c for the constant of integration.) ∫ (3x^2 + 2x – 3) / (x^3 – x) dx
_____________

Step-by-step explanation:

The length and the width of the rectangle are LEGS of a RIGHT triangle

    the 91 yards is the hypotenuse .....so you can use the Pythagorean Theorem for right triangles

    hyp^2  =  leg1^2   + leg2^2

     91^2   = 35^2 +   width^2

      91^2 - 35^2  = width^2

          7056 = width^2  

                 then width = sqrt (7056) = 84 yards

Answer:

$100 for air conditioning and $40 for water heater

Step-by-step explanation:

$200÷2= 100

200÷5= 40

they spent a total of $215.76 including tax and tip.

What is the arithmetic operation?

The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.

Therefore, the total cost per person without tax and tip would be:

$28 + $30 = $58

Since there were three people, the total cost without tax and tip would be:

$58 × 3 = $174

The sales tax would be 4% of the total cost:

$174 × 0.04 = $6.96

The tip left was about 5 times the amount of tax, so:

Tip = $6.96 × 5 = $34.80

Therefore, the total cost including tax and tip would be:

$174 + $6.96 + $34.80 = $215.76

So, they spent a total of $215.76 including tax and tip.

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Answer:

[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12

Hope this helps!

Step-by-step explanation:

[tex]t^{4}[/tex] - 1.5[tex]t^{2}[/tex] + t - 12 + 9.5[tex]t^{2}[/tex] + 7t : Combine like terms

[tex]t^{4}[/tex] + ( 9.5 - 1.5 )[tex]t^{2}[/tex] + ( 7t + t ) - 12

[tex]t^{4}[/tex] + 8[tex]t^{2}[/tex] + 8t - 12

Answer: To find the value of the car after 8 years, we can use the given information to create a linear equation in the form:

y - y1 = m(x - x1)

Where:

y is the value of the car after 8 years (what we want to find)

y1 is the value of the car after a given time (either 3 or 5 years)

x is the given time (either 3 or 5 years)

x1 is the starting time (when the car was purchased)

m is the rate of decrease (slope of the line)

We can use the two given data points to find the slope of the line:

m = (y2 - y1) / (x2 - x1)

m = (11,000 - 15,000) / (5 - 3)

m = -2,000 per year

Now, we can use one of the data points to solve for the y-intercept of the line:

y - y1 = m(x - x1)

y - 15,000 = (-2,000)(3 - x1)

y - 15,000 = (-2,000)(3 - x1)

y - 15,000 = (-6,000 + 2,000x1)

y = 2,000x1 - 6,000 + 15,000

y = 2,000x1 + 9,000

So the linear equation that represents the value of the car after x years is:

y - 15,000 = (-2,000)(x - 3)

Simplifying this equation gives:

y - 15,000 = -2,000x + 6,000

Now, we can use x = 8 to find the value of y:

y - 15,000 = -2,000(8) + 6,000

y - 15,000 = -2,000

y = $13,000

Therefore, the value of the car after 8 years will be $13,000.

So the completed expression is:

y - 15,000 = -2,000(x - 3)

The value of the car after 8 years will be $13,000.

Step-by-step explanation:

A histogram of the average age for every possible combination of cases from the population is referred to as a scatter plot

A scatter plot matrix is a plot that presents all feasible scatter plots for a given collection of variables in a matrix structure. It is helpful for investigating the connections between related factors in a collection. A scatter plot matrix is not a histogram that shows the average age for every mix of instances drawn from the community.

Instead, it is referred to as a sampling distribution or a distribution of sample means. This kind of histogram displays the range of means for all conceivable community samples of a specific number. It is a crucial idea in statistics because it aids in understanding a population's characteristics based on a subset of data.

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A rational inequality for which -2 < x < 1 is the solution is (3x - 1)/(x + 2) > 2.

An inequality that incorporates one or more rational expressions—that is, expressions that can be expressed as the quotient of two polynomial expressions—is referred to as a rational inequality. The same techniques used to resolve other kinds of inequalities, such as graphing, testing intervals, or algebraic manipulation, can also be used to resolve rational inequalities. The denominator of a rational expression cannot, however, equal zero, hence extra care must be made to identify and rule out any values of the variable that would do so, as these values would render the expression undefinable.

A rational inequality for which -2 < x < 1 is the solution is:

(3x - 1)/(x + 2) > 2

In the equation when,

For x -2, the inequality holds because both the denominator and the numerator are negative.

When -2 x 1, the inequality is true since both the denominator and the numerator are positive.

When x > 1, the denominator and numerator are both positive, but the expression's value is less than 2, rendering the inequality untrue.

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The length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.

The Pythagoras rule states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 

Pythagoras rule we can evaluate for b as follows:

65² = 63² + b²

b = √(65² - 63²) {make b the subject}

b = √(4225 - 3969)

b = √256

b = 16

Therefore, the length of the missing leg for the right triangle is 16 yd using the Pythagoras rule.

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Debbie sat for 9 tests with a total score of 756, based on her average of 84 marks for all tests.

The average refers to the quotient of the total value divided by the number of data items.

The average is also known as the mean.

The total number of marks obtained in x tests = 504

The additional number of tests taken = 3

The total number of marks obtained in x + 3 tests = 756

The number of marks obtained in 3 tests = 252 (756 - 504)

The average marks obtained in the 3 tests = 84 (252 ÷ 3)

The average marks obtained in x tests = 84

x = 6 (504 ÷ 84)

Therefore, the total number of tests taken = x + 3 = 6 + 3 = 9.

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The probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.

Based on the table presented, Steve simulated the situation by rolling a six-sided die 5 times. Each time an odd number appears, it represents a tip. This table shows the results of 100 trials: number of times an odd number appears 0 1 2 3 4 5 number of trials 2 15 33 31 16 3.

The customer leaves a tip 50% of the time, we can calculate the probability by multiplying the probability of a single tip (0.5) by itself 5 times.

Probability = (0.5) * (0.5) * (0.5) * (0.5) * (0.5)

Probability = 0.03125

From the table presented, it can be concluded that Steve got all of the five customers tipping him in 3 trials. Therefore, the probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.

The probability that an event will occur is the ratio of the number of ways that the event can occur to the number of total possible outcomes. In this scenario, the probability of a customer giving Steve a tip is 0.5.

Since the customer either gives a tip or doesn't.

In other words, there are two possible outcomes (gives a tip or doesn't give a tip) and only one outcome results in a tip (gives a tip).

Therefore, the probability of getting a tip from one customer is 1/2 or 0.5.

Since there are five customers, the probability of all five customers giving Steve a tip is calculated by multiplying the probability of each customer giving a tip.

This is because the occurrence of each event is independent of the others.

Therefore, the probability of all five customers giving Steve a tip is:

0.5 x 0.5 x 0.5 x 0.5 x 0.5 = 0.03125 or 3.125%.

From the table presented, it can be concluded that Steve got all of the five customers tipping him in 3 trials.

Therefore, the probability that all of the customers will give him a tip is 3/100 or 0.03 or 3%.

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The total number of muffins the bakeries make each day is 36,552.

As per the given information, A muffin company owns 6 bakeries. Each

bakery makes 6,092 muffins per day.

To find out how many muffins in total the bakeries make each day, you

should multiply the number of bakeries by the number of muffins each

bakery makes per day.

Identify the number of bakeries (6) and the number of muffins per bakery

(6,092).

Multiply the number of bakeries by the number of muffins per bakery (6 × 6,092).

Therefore, each day, bakers produce a total of 36,552 muffins.

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The rate of change (slope) for a linear relationship can be found by taking the difference in the y-values divided by the difference in the corresponding x-values.

In this case, the change in y from x=1 to x=9 is 32-8=24, and the change in x is 9-1=8. Therefore, the rate of change is:

rate of change = (change in y) / (change in x) = 24/8 = 3

So the answer is A. 3.

It will take approximately 13.6 years for the account value to reach $9000.

What is an account value?

We can use the formula for compound interest to solve this problem:

A = P*[tex](1 + r/n)^{nt}[/tex]

where A is the final amount, P is the principal or initial amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

We want to find t, so we can rearrange the formula as:

t = (1/n) × log(A/P) / log(1 + r/n)

Substituting the given values:

P = $3200

r = 8% = 0.08

A = $9000

n = 1 (interest is compounded annually)

Using a calculator:

t = (1/1) × log(9000/3200) / log(1 + 0.08/1)

t ≈ 13.6

Rounding to the nearest tenth of a year, it will take approximately 13.6 years for the account value to reach $9000.

What is an interest ?

Interest is the cost of borrowing money, or the compensation paid by a borrower to a lender for the use of money. When you borrow money, you are usually required to pay back the amount borrowed (the principal) plus an additional amount of money, which is calculated as a percentage of the principal and represents the interest.

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Answer:

Need to be more specific. What units?

Step-by-step explanation:

The sample size was 1000 Americans, and 20% of the survey respondents reported owning 2 pets. To calculate the number of Americans who own 2 pets, we can use the formula (20/100) x 1000 = 0.20 x 1000 = 200. However, we cannot directly conclude that 20% of all Americans own 2 pets.

The 1000 Americans that were polled in this case make up the sample. This indicates that a sample rather than the complete American population was used to get the data on pet ownership.

A statistic, which is a quantitative description of the sample, is the finding that 20% of survey respondents reported having two pets. It is crucial to remember that while this statistic provides useful information about the sample, it cannot be used to draw conclusions about the full population without additional research or inference.

Based on the poll, we can apply the following calculation to get the proportion of Americans who own two pets:

Number of Americans who own 2 pets = (Percent of survey respondents who own 2 pets / 100) x Sample size

Plugging in the given values, we get:

Number of Americans who own 2 pets = (20 / 100) x 1000

= 0.20 x 1000

= 200

As a result, we may assume that 200 Americans possess 2 pets based on the poll results. However, as this would entail assuming the representativeness of the sample and the generalizability of the results, we cannot immediately conclude that 20% of all Americans own 2 pets.

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a. The z-table, the probability that a sample mean will be within 0.02 of the population mean is:

P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999

b. The probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.

a. We can use the formula for the standard error of the mean:

SE = σ/√n

where σ is the population standard deviation, n is the sample size, and SE is the standard error.

For a sample size of n = 25:

SE = 0.0588/√25 = 0.01176

Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:

P(-0.02/0.01176 < Z < 0.02/0.01176) = P(-1.70 < Z < 1.70) = 0.9045

For a sample size of n = 50:

SE = 0.0588/√50 = 0.00832

Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:

P(-0.02/0.00832 < Z < 0.02/0.00832) = P(-2.41 < Z < 2.41) = 0.9872

For a sample size of n = 100:

SE = 0.0588/√100 = 0.00588

Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:

P(-0.02/0.00588 < Z < 0.02/0.00588) = P(-3.40 < Z < 3.40) = 0.9984

For a sample size of n = 200:

SE = 0.0588/√200 = 0.00416

Using the z-table, the probability that a sample mean will be within 0.02 of the population mean is:

P(-0.02/0.00416 < Z < 0.02/0.00416) = P(-4.81 < Z < 4.81) = 0.9999

b. The advantage of a larger sample size is that it leads to a smaller standard error of the mean, which means that the distribution of sample means is narrower and more closely centered around the population mean. As a result, the probability of obtaining a sample mean that is close to the population mean increases as the sample size increases. This means that larger sample sizes provide more accurate estimates of the population mean.

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The value of angle EFG is 151°.

What is angle?

An angle is a measure of the amount of rotation or turn between two lines, rays, or planes that meet at a common point called the vertex. The unit of measurement for angles is degrees or radians.

An angle is formed when two rays share a common endpoint, known as the vertex. The rays are often referred to as the arms or legs of the angle. The angle is measured in degrees, with a full circle measuring 360 degrees.

Angles can be classified according to their size. An acute angle measures less than 90 degrees, a right angle measures exactly 90 degrees, an obtuse angle measures more than 90 degrees but less than 180 degrees, and a straight angle measures exactly 180 degrees.

Angles can also be complementary or supplementary. Two angles are complementary if their sum is equal to 90 degrees, and they are supplementary if their sum is equal to 180 degrees.

Angles are used in many fields, including mathematics, physics, engineering, and architecture. They are used to measure distances, calculate trajectories, and design buildings and structures.

Here given,

[tex]\angle EFH = {44}^{o} [/tex] and

[tex] \angle \: HFG = {107}^{o} [/tex]

Here

[tex]\angle EFG = \angle EFH + \angle HFG \\ \angle EFG = {44}^{o} + {107}^{o} \\ = {151}^{o} [/tex]

So, required value of angle EFG is 151°.

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The total number of unique ways in which the type of monsters can be collected by the collector as per the conditions are 6.

The monster collector wants to catch 22 little monsters.

The combination in one case is given by the formula,

C(n, a) = ⁿCₐ = n!/(n-a)!(a)!

Now the question says he intends to catch at least two of each type.

This is represented mathematically because the has twenty small sample container assemblies. Collector can only access non-unique types. They are small monsters with attributes of water, fire and grass.

C(3, 2) = 3!/(2!(3-2)!)

= 3!/2! x1!

= 3

He also intends to catch at least three fire monsters.

(3-1)!)

= 3!/1!x2!

= 3

The number of different combinations is therefore 6.

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Answer:

16/25

Step-by-step explanation:

64/100 which simplifies to 16/25 if the asked for the simplified answer

The sum of the angles in triangle MNP is 180, and subtracting 135 from both sides gives 45 degrees. With the angle sum property of triangle, ∠MNP = 121°.

An angle is the result of the intersection of two lines.

An "angle" is the length of the "opening" between these two beams.

Angles are commonly measured in degrees and radians, a measurement of circularity or rotation.

In geometry, an angle can be created by joining the extremities of two rays. These rays are intended to represent the angle's sides or limbs.

The two primary components of an angle are the limbs and the vertex.

The joint vertex is the common terminal of the two beams.

According to the given question,

45° + 90° + ∠MNP = 180°

135° + ∠MNP = 180°

∠MNP = 45°

Hence, With the angle sum property of triangle, ∠MNP = 121°.

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An equation that represents the transformed function include the following: A. y = |1/4x|.

In Mathematics and Geometry, a transformation is the movement of a point from its initial position to a new location. This ultimately implies that, when a geometric figure or object is transformed, all of its points would also be transformed;

In Geometry, dilation can be defined as a type of transformation which typically changes the size of a geometric object, but not its shape.

Since the vertex of the two absolute value functions is located at points (0, 0), we would have to dilate the coordinates of the pre-image by using a scale factor of 1/4 centered at the origin as follows:

y = |1/4x|

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The change of base formula is:  log_b(x) = log_a(x) / log_a(b)

The term "change of base" refers to the process of converting a logarithm from one base to another. A logarithm is an exponent that indicates the power to which a base must be raised to obtain a given number. The most common logarithmic bases are 10, e (Euler's number), and 2, but logarithms can be taken to any base.

The change of base formula is a mathematical formula that allows you to convert logarithms from one base to another. It is useful when you need to calculate logarithms that cannot be easily evaluated on a calculator or when you need to work with logarithms in different bases.

The change of base formula is:

log_b(x) = log_a(x) / log_a(b)

Where:

log_b(x) is the logarithm of x to the base b

log_a(x) is the logarithm of x to the base a

log_a(b) is the logarithm of b to the base a

To use the change of base formula, you first need to identify the base of the logarithm that you want to convert and the base that you want to convert it to. Then, you can plug the values into the formula and simplify the expression.

For example, let's say you want to convert log_2(8) to base 10. Using the change of base formula, you can write:

log_2(8) = log_10(8) / log_10(2)

To evaluate this expression, you can use a calculator to find that:

log_10(8) = 0.9031

log_10(2) = 0.3010

Substituting these values into the formula gives:

log_2(8) = 0.9031 / 0.3010

Simplifying this expression gives:

log_2(8) = 3

Therefore, log_2(8) is equal to 3 when evaluated in base 10.

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The perimeter to the nearest tenth, we get: Perimeter ≈ 18.4 units.

What is perimeter?

We can start by using the Pythagorean theorem to find the length of AC:

AC² = AB² + BC²

AC² = 2² + 1.5²

AC² = 4.25

AC = √(4.25)

AC ≈ 2.06 units

Next, we can use the fact that AC is common to both triangles to find the length of CD:

ACD is a right triangle, so we can use the Pythagorean theorem:

CD² = AD² - AC²

CD² = 6.5² - 2.06²

CD² ≈ 39.86

CD ≈ 6.31 units

Now we can find the perimeter of the figure by adding the lengths of all three sides:

Perimeter = AB + BC + CD + DA + AC

Perimeter = 2 + 1.5 + 6.31 + 6.5 + 2.06

Perimeter ≈ 18.37 units

Therefore, none of the given options is the correct answer. However, if we round the perimeter to the nearest tenth, we get:

Perimeter ≈ 18.4 units

And the closest option to this value is option 3) 16 units, which is not very close. Therefore, we can conclude that none of the given options is a correct approximation of the perimeter of the figure.

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Answer:

Point K

Step-by-step explanation:

Which Point represents the ordered pair

( -4[tex]\frac{1}{2}[/tex], -1)

Left -4[tex]\frac{1}{2}[/tex] we have points J and K

Down -1 we have point K

So, the answer is point K

Answer:

12.74

Step-by-step explanation:

To find the diameter of a circle with a given circumference, you can use the formula:

Circumference = π x diameter

where π (pi) is a mathematical constant approximately equal to 3.14159.

In this case, the circumference is given as 40. So we can rearrange the formula to solve for the diameter:

diameter = Circumference / π

Substituting the given values, we get:

diameter = 40 / π

To get the answer to the nearest hundredth, we can use a calculator or estimate π to a few decimal places. Let's use π ≈ 3.14.

diameter = 40 / 3.14

diameter ≈ 12.74

Therefore, the diameter of the circle to the nearest hundredth is approximately 12.74.

Step-by-step explanation:

Circumference = pi  * d

circumference / pi = d  =  40 / pi = 12.73 units

The regular polygon with an exterior angle measuring 120 will have 3 equal sides. Thus, an equilateral triangle have three equal sidea and angles is a regular polygon.

Regular polygon: A regular polygon is both "equiangular" and "equilateral," meaning it has all sides that are the same length and interior angles that are the same size.

The following algorithm determines an ordinary polygon's exterior angle:

Exterior Angle = [tex]360 Degrees / Sides[/tex]

We can therefore make up an equation if the exterior angle measurement is [tex]120[/tex] degrees:

[tex]120 = 360/n[/tex] the number of sides being n

[tex]n (sides) = 360/120 = 3[/tex]

Three sides [tex]=[/tex] triangles are polygons with three sides.

An equilateral triangle is a regular three-sided polygon.

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