Find an angle in each quadrant with a common reference angle with 210°, from 0°≤θ<360°

The expression that is not equivalent to the model shown is given as follows:

-4(3 + 2). -> Option C.

Equivalent expressions are mathematical expressions that have the same value, even though they may look different. In other words, two expressions are equivalent if they produce the same output for any input value.

The expression for this problem is given by three times the subtraction of four, plus three times the addition of 2, hence:

3(-4) + 3(2) = -12 + 6 = 3(-4 + 2) = 3(-2) = -6.

Hence the expression that is not equivalent is the expression given in option C, for which the result is given as follows:

-4(3 + 2) = -4 x 5 = -20.

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Answer:

x = 7

Step-by-step explanation:

The two angles are equal so the opposite sides are equal.

5x-2 =33

Add two to each side.

5x-2+2 = 33+2

5x=35

Divide by 5

5x/5 =35/5

x = 7

Answer:

To shift the graph of f(x) = |x| to have a domain of [-3, 6], we need to move the left endpoint from -6 to -3 and the right endpoint from 3 to 6.

A translation to the right by 3 units will move the left endpoint of the graph of f(x) to -3, but it will also shift the right endpoint to 6 + 3 = 9, which is outside the desired domain.

A translation to the left by 3 units will move the right endpoint of the graph of f(x) to 3 - 3 = 0, which is outside the desired domain.

A translation upward or downward will not change the domain of the graph, so options B and D can be eliminated.

Therefore, the correct answer is C g(x) = x - 3. This translation will move the left endpoint to -3 and the right endpoint to 6, which is exactly the desired domain.

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

Having three straight sides and three angles where they intersect, a triangle is a closed, two-dimensional shape. It is one of the fundamental geometric shapes and has a number of characteristics that can be used to study and resolve issues that pertain to it. The triangle inequality theory states that the sum of a triangle's interior angles is always 180 degrees, and that the longest side is always the side across from the largest angle. Triangles can be used to solve a wide range of mathematical issues in a variety of disciplines and can be categorised based on the length of their sides and the measurement of their angles.

given

We can use the following congruence theories or postulates based on the data in the diagram:

A. ASA

B. AAS

C. LL (corresponding angles hypothesis)

F. SAS

Therefore, A, B, C, and F are the proper responses as the congruence theories or postulates based on the data.

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Answer:

Boy = 40%

Girl = 66.6%

Step-by-step explanation:

1) Work out probability that student is a boy who wears cap

8 boys out of 20 wears a cap so to find the probability we have to do 8 divided by 20

2) Work out probability for girls who doesn't wear spectacles

To find the probability of girls who doesn't wear spectacles we have to do 8 divided by 12

Hope this helps, have a lovely day! :)

Here's the completed table and the graph:

x  h(x)

-90°  1.0

-75°  -0.5

-60°  -1.0

-45° -0.0

-30° 1.0

-15° 0.5

0° 1.0

15° 0.5

30° -0.0

45° -1.0

60° -0.5

75° 1.0

90° 1.0

In mathematics, a function is a relation between a set of inputs and a set of possible outputs, with the property that each input is related to exactly one output. Functions are often represented as a set of ordered pairs, where the first element of each pair is an input and the second element is the corresponding output. Functions are a fundamental concept in many areas of mathematics and have many real-world applications, including in science, engineering, and economics.

Here,

To calculate the values of h(c) in the table, we plug in the given values of x into the function h(c) = cos 2x and evaluate. For example, to find h(c) when x = -75°:

h(c) = cos 2x

h(c) = cos 2(-75°) (substitute -75° for x)

h(c) = cos (-150°) (simplify using the double angle identity)

h(c) = -0.5 (evaluate using the unit circle or a calculator)

We repeat this process for each value of x to fill out the table.

To graph the function h(x) = cos 2x, we plot each point from the table on the given system of axes. The x-axis represents the angle x in degrees, and the y-axis represents the value of h(x) = cos 2x. We then connect the points with a smooth curve to obtain the graph.

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Answer: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103

Step-by-step explanation:

54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.

First, we order the numbers by their sign: 54, 1, 1 2/3, 0.5, 0.03, 0, -1/4, -1, -4, -103.

Then we order the positive numbers in decreasing order: 54, 1 2/3, 1, 0.5, 0.03, 0.

Finally, we order the negative numbers in increasing order: -103, -4, -1, -1/4.

Putting it all together, we have: 54, 1 2/3, 1, 0.5, 0.03, 0, -1/4, -1, -4, -103.

Answer: The boat is approximately 357.4 feet from the base of the cliff.

Step-by-step explanation:

Let x be the horizontal distance from the base of the cliff to the boat. Using the tangent function, we can write:

tan(38) = 275 / x

Solving for x, we have:

x = 275 / tan(38)

Using a calculator, we get:

x ≈ 357.4 feet

Therefore, the boat is approximately 357.4 feet from the base of the cliff.

Answer:

352m

Step-by-step explanation:

h = 275m

a = b (alternative angles)

.: b = 38°

Let the base from the boat to the cliff be d

Tan 38° = opposite ÷ adjacent

Tan 38° ° = 275 ÷d

d = 275 ÷ Tan 38 °

d = 352m

.: The boat is 352m away from the foot of the cliff

Answer:

k = -5

Step-by-step explanation:

According to the Remainder Theorem, when we divide a polynomial f(x) by (x − c), the remainder is f(c).

Therefore, if we divide polynomial f(x) = x³ + kx + 9 by (x - 2) and the remainder is 7 then:

To find the value of k, simply substitute x = 2 into the function, equate it to 7 and solve for k.

[tex]\begin{aligned}f(2)=(2)^3 + k(2) + 9 &= 7\\8+2k+9&=7\\2k+17&=7\\2k&=-10\\k&=-5\end{aligned}[/tex]

Therefore, the value of k is -5.

The cardinality of set A, n(A) = 29

The cardinality of a set is the total number of elements in the set

Given the Venn diagram here shows the cardinality of each set. To find the cardinality of set A, n(A), we proceed as follows.

Since the cardinality of a set is the total number of elements in the set, then cardinality of set A , n(A) = 9 + 8 + 3 + 9

= 29

So, n(A) = 29

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The required probability that a household in Maryland with annual income of ,

$90,000 or more is equal to 0.3377.

$50,000 or less is equal to 0.2218.

Annual household income in Maryland follows a normal distribution ,

Median =  $75,847

Standard deviation = $33,800

Probability of household in Maryland has an annual income of $90,000 or more.

Let X be the random variable representing the annual household income in Maryland.

Then,

find P(X ≥ $90,000).

Standardize the variable X using the formula,

Z = (X - μ) / σ

where μ is the mean (or median, in this case)

And σ is the standard deviation.

Substituting the given values, we get,

Z = (90,000 - 75,847) / 33,800

⇒ Z = 0.4187

Using a standard normal distribution table

greater than 0.4187  as 0.3377.

P(X ≥ $90,000)

= P(Z ≥ 0.4187)

= 0.3377

Probability that a household in Maryland has an annual income of $90,000 or more is 0.3377(rounded to four decimal places).

Probability that a household in Maryland has an annual income of $50,000 or less.

P(X ≤ $50,000).

Standardizing X, we get,

Z = (50,000 - 75,847) / 33,800

⇒ Z = -0.7674

Using a standard normal distribution table

Probability that a standard normal variable is less than -0.7674 as 0.2218. This implies,

P(X ≤ $50,000)

= P(Z ≤ -0.7674)

= 0.2218

Probability that a household in Maryland has an annual income of $50,000 or less is 0.2218.

Therefore, the probability with annual income of $90,000 or more and  $50,000 or less is equal to 0.3377 and 0.2218 respectively.

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Answer:

19/20 , 1 1/4 , 1 4/5

Step-by-step explanation:

1.8 = 1 4/5 (fraction)

1.8 converts to 18/10. This can be simplified twice, firstly by making it 9/5 since both 18 and 10 are divisible by two, but can be simplified further to 1 4/5

1.25 = 1 1/4 (fraction)

1.25 converts to 125/100. This can be simplified to 5/4 or 1 1/4

0.95 = 19/20 (fraction)

0.95 converts to 95/100. This can be simplified to 19/20

Ascending Order (smallest to largest)

smallest - 19/20

middle - 1 1/4

largest - 1 4/5

I believe this is the right answer, but haven't done fractions in a while so may want to double check to make sure

Sure, let's solve this step-by-step:

First, we need to solve for x in the equation x + 1/2 = 5.

We can do this by subtracting 1/2 from both sides, giving us x = 4 1/2.

Now, we can substitute x = 4 1/2 into the equation 2*x^2 - 3x + 6 - 3/x +2/x^2.

We can simplify the equation by multiplying both sides by x^2, giving us:

2*x^2 - 3x + 6 - 3/x +2 = 10*x^2 - 3x + 6.

Now, we can combine all of the terms with x:

10*x^2 - 6x + 6 = 0.

Finally, we can solve the equation using the quadratic formula:

x = 3/5 or x = 2.

Therefore, the answer to the equation is 10*(3/5)^2 - 6(3/5) + 6 = 4.8, or 10*2^2 - 6(2) + 6 = 16.

Answer: x = -3, y = -4, or (-3, -4)

Step-by-step explanation:

To solve the system of equations, we can substitute the first equation into the second equation, replacing y with 3x + 5:

x + (3x + 5) = -7

Simplifying:

4x + 5 = -7

Subtracting 5 from both sides:

4x = -12

Dividing by 4:

x = -3

Now that we know x = -3, we can substitute that value into either of the original equations to find y:

y = 3(-3) + 5 = -4

Therefore, the solution to the system of equations is x = -3, y = -4, or (-3, -4).

The value of the x is 5√3 after we successfully do the application of the 30°-60°-90° Triangle theorem.

The 30°-60°-90° Triangle Theorem states that in such a triangle, the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is the product of the length of the hypotenuse and the square root of 3 divided by 2.

Using this theorem, we can write:

y = hypotenuse

Opposite of 30° angle = 5 = hypotenuse/2

Opposite of 60° angle = x = hypotenuse × (√(3)/2)

Solving for the hypotenuse in terms of y from the first equation, we get:

hypotenuse = 5×2 = 10

Substituting this value into the third equation, we get:

x = 10 × (√(3)/2) = 5 × √(3)

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Answer:

The absolute extrema is minimum at (-1, 2/9)

Step-by-step explanation:

Absolute extrema is a logical point that shows whether a the curve function is maximum or minimum.

Forexample a curve in the image attached. A, B and C are points of absolute maxima or absolute maximum. and P and Q are points of absolute minima or minimum.

Remember A, B, C, P, Q are critical points or stationary points.

How do we find absolute extrema?

The find the sign of the second derivative of the function.

From the question;

[tex]{ \sf{g(x) = \sqrt[3]{x} }} \\ \\ { \sf{g(x) = {x}^{ \frac{1}{3} } }} \\ [/tex]

Find the first derivative of g(x)

[tex]{ \sf{g {}^{l}(x) = \frac{1}{3} {x}^{ - \frac{2}{3} } }} \\ [/tex]

Find the second derivative;

[tex]{ \sf{g {}^{ll} (x) = ( \frac{1}{3} \times - \frac{2}{3}) {x}^{( - \frac{2}{3} - 1) } }} \\ \\ { \sf{g {ll}^{(x)} = - \frac{2}{9} {x}^{ - \frac{5}{3} } }}[/tex]

Then substitute for x as -1 from [-1, 1]

[tex]{ \sf{g {}^{ll}(x) = - \frac{2}{9} ( - 1) {}^{ - \frac{5}{3} } }} \\ \\ = \frac{ - 2}{9} \times - 1 \\ \\ = \frac{2}{9} [/tex]

Since the sign of the result is positive, the absolute extrema is minimum

The value of local maximum and local minimum for the function f(x) = x^2/(x -1 ) is equal to f(0) = 0 at x = 0 and f(2) = 4 at x = 4 respectively.

Local maximum and minimum values of the function

f(x) = x^2 / (x - 1),

Use both the first and second derivative tests.

First, let's find the critical points of the function,

By setting its derivative equal to zero and solving for x,

f'(x) = [2x(x - 1) - x^2] / (x - 1)^2

⇒ [2x(x - 1) - x^2] / (x - 1)^2 = 0

Simplifying this expression, we get,

x(x - 2) = 0

This gives us two critical points,

x = 0 and x = 2.

These critical points correspond to local maxima, local minima, or neither.

Use the second derivative test,

f''(x) = [2(x - 1)^2 - 2x(x - 1) + 2x^2] / (x - 1)^3

At x = 0, we have,

f''(0) = 2 / (-1)^3

       = -2

Since the second derivative is negative at x = 0, this critical point corresponds to a local maximum.

f(0) = 0^2/ (0 -1 )

      = 0

At x = 2, we have,

f''(2) = 2 / 1^3

       = 2

Since the second derivative is positive at x = 2, this critical point corresponds to a local minimum.

f(2) = 2^2/ (2 - 1)

     = 4

Therefore, at x = 0, the local maximum value is f(0) = 0, and at x = 2, the local minimum value is f(2) = 4.

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Answer:

We can use the formula for the area of a rectangle to solve this problem. Let's assume that the length of the top of the bookcase is L and the width is b. Then, we can write:

L × b = 300

Solving for b, we get:

b = 300 / L

Since we don't know the length L, we cannot find the exact value of b. However, we can use the given information to make an estimate. Let's say that the length of the bookcase is 60 inches. Then, we have:

b = 300 / 60 = 5

So, if the length of the bookcase is 60 inches, the width needs to be at least 5 inches to accommodate Bria's soap carving collection. However, if the length is different, the required width will also be different.

Answer:

Step-by-step explanation:

1)  b      (acute is less than 90)

2)  a     (obtuse: more than 90, less than 180)

3)  c

4) c

Answer:

1. NOM, JOK, KOL

2. MOL, NOK, MOJ

3. NOJ, JOL

4. NOL, MOK

a.(1) When both r and s are odd integers, the quadratic polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers.

(2) When both r and s are even integers, the polynomial obtained by multiplying out (x - r)(x - s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers.

(3) When one of r and s is even and the other is odd, the polynomial obtained by multiplying out (x - r)(x - s) will have a coefficient of 1 for x^2 term, the coefficient of x term will be an odd integer, while the constant term will be an even integer.

b. x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

a. When both r and s are odd integers, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and both the coefficient of x term and constant term will be odd integers. This is because the sum of two odd integers and the product of two odd integers is also an odd integer.

When both r and s are even integers, the product (x − r)(x − s) will also have a coefficient of 1 for x^2 term, but the coefficient of x term and constant term will be even integers. This is because the sum of two even integers and the product of two even integers is also an even integer.

When one of r and s is even and the other is odd, the product (x − r)(x − s) will have a coefficient of 1 for x^2 term, and the coefficient of x term will be an odd integer, while the constant term will be an even integer. This is because the sum of an odd and even integer is an odd integer, and the product of an odd and even integer is an even integer.

b. If x^2 - 1253x + 255 can be written as a product of two polynomials with integer coefficients, then we can write it as (x - r)(x - s) where r and s are integers. From part (a), we know that both r and s cannot be odd integers since the coefficient of x term would be odd, but 1253 is an odd integer. Similarly, both r and s cannot be even integers since the constant term would be even, but 255 is an odd integer. Therefore, one of r and s must be odd and the other must be even. However, the difference between an odd integer and an even integer is always odd, so the coefficient of x term in the product (x - r)(x - s) would be odd, which is not equal to the coefficient of x term in x^2 - 1253x + 255. Hence, x^2 - 1253x + 255 cannot be written as a product of two polynomials with integer coefficients.

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The function f(x,y) has only minimum value at (1,0) is -1 and maximum value does not exist.

The given function is f(x,y)=x²+y²-2x

First find the partial derivative with respect to x and y

f'(x)=2x-2

f'(y)=2y

f'(x)=0=f'(y)

2x-2=0

x=1

and y=0

Now we will cheak maxima and minima at (1,0)

f''(x,y)=2 and f"(x,y)=2 and f"(x,y)=0( derivative of first order of x with respect to y)

We know that

rt-s²≥0 and r positive then f is minimum and  r negative maximum

r=2 , t=2 and s=0

rt-s²≥0 and r is positive so f(x,y) is minimum at (1,0)

f(1,0)=1-2=-1

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Therefore , the solution of the given problem of standard deviation comes out to be the group standard deviation in order to use (b) z.

Statistics uses variance as a way to quantify difference. The image of the result is used to compute the average deviation between the collected data and the mean. Contrary to many other valid measures of variability, it includes those pieces of data on their own by comparing each number to the mean. Variations may be caused by willful mistakes, irrational expectations, or shifting economic or business conditions.

Here,

We use the z-distribution if the total standard deviation is known; otherwise, we use the t-distribution.

Additionally, for small sample sizes, the t-distribution is used, whereas for big sample sizes, the z-distribution is used.

The fact that z requires that you know the population standard deviation, and that this is frequently not known in practice, is one reason to use a distribution rather than the traditional .

Normal curve to find critical values when computing a level C confidence interval for a population mean.

You must be aware of the group standard deviation in order to use (b) z.

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Therefore, the distance between the 90 degree angle and the hypotenuse is approximately 0.829 units.

A triangle is a two-dimensional geometric shape that is formed by three straight line segments that connect to form three angles. It is one of the most basic shapes in geometry and has a wide range of applications in mathematics, science, engineering, and everyday life. Triangles can be classified by the length of their sides (equilateral, isosceles, or scalene) and by the size of their angles (acute, right, or obtuse). The study of triangles is an important part of geometry, and their properties and relationships are used in many areas of mathematics and science.

Here,

1. To find HF, we can use the angle bisector theorem, which states that if a line bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the adjacent sides. Let's denote the length of HF as x. Then, by the angle bisector theorem, we have:

JF/FH = JG/HG

Substituting the given values, we get:

15/x = 18/21

Simplifying and solving for x, we get:

x = 15 * 21 / 18

x = 17.5

Therefore, HF is 17.5 cm.

2. Let's denote the length of the hypotenuse as h and the length of the leg opposite the 18-unit perpendicular as a. We can then use the Pythagorean theorem to write:

h² = a²  + 18²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = a² + 18²

We are also told that the leg adjacent to the angle opposite the 4-unit segment is divided into segments of length 4 and (a - 4), so we can write:

a = 4 + (a - 4)

Simplifying this equation, we get:

a = a

Now we can substitute this expression for a into the previous equation and solve for x:

(x + 6)² = (4 + (a - 4))² + 18²

Expanding and simplifying, we get:

x² + 12x - 36 = 0

Using the quadratic formula, we get:

x = (-12 ± √(12² - 4(1)(-36))) / (2(1))

x = (-12 ± √(288)) / 2

x = -6 ± 6√(2)

Since the length of a segment cannot be negative, we take the positive root:

x = -6 + 6sqrt(2)

x ≈ 1.46

Therefore, the value of x is approximately 1.46 units.

3. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the 9-unit perpendicular as b. We can then use the Pythagorean theorem to write:

h² = b² + 9²

We are told that the hypotenuse is divided into segments of length x and 6 units, so we can write:

h = x + 6

Substituting this expression into the first equation, we get:

(x + 6)² = b² + 9²

Expanding and simplifying, we get:

x² + 12x - b² = 27

We also know that the length of the leg opposite the 9-unit perpendicular is:

a = √(h² - 9²)

= √((x + 6)² - 9²)

= √(x² + 12x + 27)

Now we can use the fact that the tangent of the angle opposite the 9-unit perpendicular is equal to the ratio of the lengths of the opposite and adjacent sides:

tan(θ) = a / b

Substituting the expressions for a and b, we get:

tan(θ) = √(x² + 12x + 27) / (x + 6)

We also know that the tangent of the angle theta is equal to the ratio of the length of the opposite side to the length of the adjacent side:

tan(θ) = 9 / b

Substituting the expression for b, we get:

tan(θ) = 9 / √(h² - 9²)

Substituting the expression for h, we get:

tan(θ) = 9 / √((x + 6)² - 9²)

Since the tangent function is the same for equal angles, we can set these two expressions for the tangent equal to each other:

√(x² + 12x + 27) / (x + 6) = 9 / √((x + 6)² - 9²)

Squaring both sides, we get:

(x² + 12x + 27) / (x + 6)² = 81 / ((x + 6)² - 81)

Cross-multiplying and simplifying, we get:

x⁴ + 36x³ + 297x² - 1458x - 2916 = 0

Using a numerical method such as the Newton-Raphson method or the bisection method, we can find the approximate solution to this equation:

x ≈ 9.449

Therefore, the value of x is approximately 9.449 units.

4. Let's denote the length of the hypotenuse as h and the length of the leg adjacent to the angle opposite the distance we want to find as b. We can use the Pythagorean theorem to write:

h² = b² + d²

We are told that the hypotenuse is divided into segments of length 9 and 4 units, so we can write:

h = 9 + 4 = 13

Substituting this expression into the first equation, we get:

13² = b² + d²

Simplifying and solving for d, we get:

d = √(13² - b²)

Now, we need to find the value of b. We know that the hypotenuse is divided into segments of length 9 and 4 units, so we can use similar triangles to write:

b / 4 = 9 / 13

Simplifying and solving for b, we get:

b = 36 / 13

Substituting this expression for b into the equation we found earlier for d, we get:

d = √(13² - (36/13)²)

Simplifying and finding a common denominator, we get:

d =√ (169*13 - 36²) / 13²

Simplifying further, we get:

d = √(169169 - 3636) / 169

Calculating this expression, we get:

d ≈ 0.829

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4.5y

subtract 2.9y from 7.4y, and you get 4.5y

Between 3.26 on the low end and 3.30 on the high end is where UGPAS's middle 50% lies.

Provided that the parabola's vertex is at the origin and that it is symmetric about the y-axis. So, depending on whether the parabola expands upward or downward, the equation can take the form x2 = 4ay or x2 = -4ay.

Because we are interested in the top 5%, the region to the right of the z-score is 0.05. n,... As a result, we can apply the following z-score formula:

z = (x - μ) / σ

x = z * σ + μ

Substituting the values we have, we get:

x = 1.645 * 0.15 + 3.25 = 3.66

Therefore, the z-scores corresponding to the 25th and 75th percentiles are:

z1 = -0.675

z2 = 0.675

Using the same formula as before, we can find the UGPAs corresponding to these z-scares:

x1 = -0.675 * 0.15 + 3.25 = 3.26

x2 = 0.675 * 0.15 + 3.25 = 3.30

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The integral of the function expressed as sum of partial frictions is -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C.

First, factor out 2x from the denominator to obtain:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[(x + 3)/(2x)(x² - 4)] dx

Next, we use partial fractions to express the integrand as a sum of simpler fractions. To do this, we need to factor the denominator of the integrand:

2x(x² - 4) = 2x(x + 2)(x - 2)

Therefore, we can write:

(x + 3)/(2x)(x² - 4) = A/(2x) + B/(x + 2) + C/(x - 2)

Multiplying both sides by the denominator, we get:

x + 3 = A(x + 2)(x - 2) + B(2x)(x - 2) + C(2x)(x + 2)

Now, we need to find the values of A, B, and C. We can do this by equating coefficients of like terms:

x = A(x² - 4) + B(2x² - 4x) + C(2x² + 4x)

x = (A + 2B + 2C)x² + (-4A - 4B + 4C)x - 4A

Equating coefficients of x², x, and the constant term, respectively, we get:

A + 2B + 2C = 0

-4A - 4B + 4C = 1

-4A = 3

Solving for A, B, and C, we find:

A = -3/4

B = 7/16

C = -1/16

Therefore, the partial fraction decomposition is:

(x + 3)/(2x)(x² - 4) = -3/(4(2x)) + 7/(16(x + 2)) - 1/(16(x - 2))

The integral becomes:

∫[(x + 3)/(2x³ - 8x)] dx = ∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

Integrating each term separately gives:

∫[-3/(8x) + 7/(8(x + 2)) - 1/(8(x - 2))] dx

= -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

where;

Therefore, the final answer is:

∫[(x + 3)/(2x³ - 8x)] dx = -3/8 ln|x| + 7/8 ln|x + 2| - 1/8 ln|x - 2| + C

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From the discriminant of the give quadratic equation, the temperature of the machine will part after 50 minutes of operation.

The given equation that models the temperature of the machine is;

T = -0.005x² + 0.45x + 125

Let check if there's a value that exists for T = 135

Putting T = 135 in the given equation,

135 = -0.005x² + 0.45x + 125

We can simplify this to;

0.005x² - 0.45x + 10 = 0

From the general form of quadratic equation which is ax² + bx + c = 0, where a = 0.005, b = -0.45, and c = 10.

The discriminant of this quadratic equation is given by:

D = b² - 4ac

= (-0.45)² - 4(0.005)(10)

= 0.2025 - 0.2

= 0.0025

The discriminant of the equation is positive which indicates we have two roots. Therefore, the temperature of the machine part will cross 135°F at some point during the operation.

We can also find the roots of the quadratic equation using the formula:

[tex]x = (-b \± \sqrt(D)) / 2a[/tex]

Substituting the values of a, b, and D, we get:

[tex]x = (0.45 \± \sqrt(0.0025)) / 2(0.005)\\= (0.45 \± 0.05) / 0.01[/tex]

Taking the positive value, we get:

x = 50

Therefore, the temperature of the machine part will cross 135°F after 50 minutes of operation.

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Answer:

0.6267

Step-by-step explanation:

See the picture.

Hope its clear.

The rank of the three circuits consisting of concentric circular arcs according to the magnitudes of the magnetic fields they produce at C, from least to greatest is (3), (2), (1).

We know that, the radial segments don't produce magnetic field at C, so consider arcs.

Assume that the current is counter clockwise and the magnetic field to be positive pointing out of the page.

Understand that, magnetic field at the center from an arc φ of radius R is [tex]\frac{{{\mu _0}i\phi }}{{4\pi R}}[/tex]

Therefore, for (1) :

[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} + \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = \frac{1}{3}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered}[/tex]

For (2) :

[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\pi }}{{4\pi r}}\\ \Rightarrow B = - \frac{1}{6}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered} \\[/tex]

For (3) :

[tex]\begin{gathered}\begin{array}{l}B = \frac{{{\mu _0}i\pi }}{{4\pi \left( {3r} \right)}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi r}} - \frac{{{\mu _0}i\left( {\frac{\pi }{2}} \right)}}{{4\pi \left( {2r} \right)}}\\ \Rightarrow B = - \frac{5}{{48}}\frac{{{\mu _0}i}}{r}\end{array}\end{gathered}[/tex]

Therefore, the magnitude of the magnetic fields at C after arranging them in the order of least to greatest are (3), (2), (1).

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