What is the measure of <×?​

In linear equation, Her profit is rupees 4000.

No. of purses she must sell to have neither profit nor loss is 600 nos.

She made loss of rupees 2135.

What is a linear equation in mathematics?

A linear equation is an algebraic equation of the form y=mx+b. m is the slope and b is the y-intercept. The above is sometimes called a "linear equation in two variables" where y and x are variables. 

The housewife earns,

Profit on 1 purse = 25 rupees

Loss on 1 vanity bag = 20 rupees

So,

Profit on 1000 purses = 25*1000 rupees

                                   = 25000 rupees

Loss on 1050 purses = 20*1050 rupees

                                  = 21000 rupees

Here, Profit > Loss

So,

Total profit = 25000-21000 rupees

                 = 4000 rupees

i) Her profit is 4000 rupees.

If no. of vanity bags sold = 750 nos.

She made loss of = 750*20 rupees

                            = 15000 rupees

ii) No. of purses she must sell to have neither profit nor loss

= 15000/25 nos.

= 600 nos.

Profit on selling 325 purses = 325*25 rupees

                                              = 8125 rupees

Loss on selling 513 vanity bags = 513*20 rupees

                                              =10260 rupees

Here, Profit < Loss

So,

iii) She made loss of = 10260-8125 rupees

                            = 2135 rupees

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The complete question is -

A housewife along with a group of ladies sold bags of different sizes. She earns a profit of 25 on a purse and a loss of 20 on a vanity bag sold. i. She received an order of 1050 vanity bags and 1000 purses. What is her profit or loss? ii. How many purses must she sell to have neither profit nor loss, if the number of vanity bags sold is 750? iii. How much profit/loss did she make in selling 325 purses and 513 vanity bags?

[tex]\cfrac{5^7}{5^{-13}}\times\left( \cfrac{4^3}{7^{-2}} \right)^{-2}\implies 5^7\cdot 5^{13}\times \left( \cfrac{7^{-2}}{4^3} \right)^{+2}\implies 5^7\cdot 5^{13}\times\left( \cfrac{7^{-4}}{4^6} \right) \\\\\\ 5^{7+13}\times\left( \cfrac{1}{4^6\cdot 7^4} \right)\implies \cfrac{5^{20}}{4^6\cdot 7^4}\implies \cfrac{95367431640625}{9834496}[/tex]

If the the a is greater than 1, compared to the parent function the C. Stretched vertically.

The equation y = ax^2 + c represents a quadratic function where "a" is the coefficient of the x^2 term and "c" is a constant term. The parent function of this quadratic function is y = x^2.

If the equation of a quadratic function is given in the form y = ax^2 + c and "a" is greater than 1, then the graph of the function will be stretched vertically and the vertex will be shifted up or down depending on the value of "c".

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Answer: 119

Step-by-step explanation:

We know that we must find the least common multiple of 15 and 27 in order to solve the problem because when we are finding remainders that are the same, there must be some relationship between the integer and the two dividing numbers.

Thus, we have the least common multiple 135 and its multiples which will all have a 0 remainder when divided by 15 and by 27.

We can take each of the numbers (7) and the 15 consecutive numbers after each of them, because of modulo becoming the same after 15. If we take the total of all these numbers, which will have the same remainder after division by 15 and 27, we are left with:

15 * 8 - 1 = 119

The x⁶y³ term in the expansion will be:  35 x⁶y³.

The x⁸y² term in the expansion will be: 21x⁸y².

The binomial expansion of (x + y)ⁿ is given by the binomial theorem, which states:

(x + y)ⁿ = C(n, 0) * xⁿ * y⁰ + C(n, 1) * xⁿ⁻¹ * y¹ + C(n, 2) * xⁿ⁻² * y² + ... + C(n, k) * xⁿ⁻ᵏ * yᵏ + ... + C(n, n) * x⁰ * yⁿ

where;

Given that the term 210x⁴y⁶ appears in the expansion, we can infer that it corresponds to C(n, k) * x⁴ * y⁶, where k is the number of times y appears in the term, and (n - k) is the number of times x appears in the term.

Comparing this with the given term, we can deduce the values of n, k, and x in the following way:

C(n, k) = 210

x⁴ = x⁴

y⁶ = y³ * y³

Comparing the exponents on x and y, we can set up the following equations:

n - k = 4 (1)

k = 3 (2)

Solving equation (2) for k, we get:

k = 3

Substituting this value of k into equation (1), we can solve for n:

n - 3 = 4

n = 7

So, the value of n is 7.

Now, we can use the binomial coefficient formula to calculate C(n, k):

C(n, k) = C(7, 3) = 7! / (3! * (7 - 3)!) = 35

Finally, substituting the values of n, k, and C(n, k) into the general term of the expansion, we can find the specific terms:

The x⁶y³ term in the expansion will be:

C(7, 3) * x⁶ * y³ = 35 * x⁶ * y³

The x⁸y² term in the expansion will be:

C(7, 2) * x⁸ * y² = 21 * x⁸ * y²

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The original cost of a pair of gloves is 3.9

Answer: Using the formula i = prt, we have:

i = (104292.12 - 80040) = 24252.12

p = 80040

t = 3

Substituting these values, we get:

24252.12 = 80040 * r * 3

Solving for r, we get:

r = 0.101 or 10.1%

Therefore, the interest rate is 10.1%.

Step-by-step explanation:

After calculation we get

11) d. m/JML = 90 degrees, a. m/GMH = 45 degrees, b. mLH = 90 degrees, c. m/LKH = 45 degrees

12) a. AGKH is an isosceles triangle, b. Another triangle formed by two radii is an equilateral triangle.

13) a. m/GKH = 33.4 degrees, c. m/KHJ = 33.4 degrees, e. mJL66.8 degrees, b. m/KGH = 66.8 degrees, d. m/JKL = 113.2 degrees

A quadrilateral is a geometric shape with four straight sides and four angles. Examples of quadrilaterals include squares, rectangles, parallelograms, trapezoids, and kites.

According to the given information:

a. Since m/LGH and m/GHJ are both 45°, GH is a bisector of ∠JGL. Therefore, m/GMH = 90° - 45° = 45°.

b. Since GH is a bisector of ∠JGL, m/LGH = m/HGL = 45°. Also, since LH is a straight line, m/LGH + mLH + m/HGL = 180°. Thus, mLH = 90° - 45° = 45°.

c. Since GH is a bisector of ∠LJK, m/GHK = m/JHK = 45°. Also, since LK is a straight line, m/LKH + m/JHK + m/LJK = 180°. Thus, m/LKH = m/LJK = (180° - 2*45°)/2 = 45°.

d. Since GH is a bisector of ∠JGL and JML is a straight line, m/JML = m/JGH + m/HGL = 45° + 45° = 90°.

a. AGKH is a quadrilateral with two sides that are radii of the circle. Since all radii of a circle are equal, AGKH is a kite. Furthermore, since the two radii AG and KH are perpendicular to each other, AGKH is also a rectangle.

b. Another triangle formed by two radii is AKJ, where AK and AJ are radii of the circle and KJ is a chord.

a. Since GH is a diameter of the circle and GKH is a right triangle with ∠GHK = 90°, m/GKH = 180° - 90° = 90°.

b. Since GH is a diameter of the circle and KJ is a chord, m/KGH = m/KJ = 1/2 * m/KHJ = 1/2 * (180° - 113.2°) = 33.4°.

c. Since GH is a diameter of the circle and KHJ is a right triangle with ∠KHJ = 90°, m/KHJ = 180° - m/GKH = 180° - 90° = 90°.

d. Since GH is a diameter of the circle and JKL is a right triangle with ∠JKL = 90°, m/JKL = 180° - m/KJ - m/JKH = 180° - 33.4° - 45° = 101.6°.

e. Since JL is a chord of the circle and ∠JGL is an inscribed angle that intercepts it, m/JGL = 1/2 * m/JL = 1/2 * (180° - m/JKL) = 1/2 * (180° - 101.6°) = 39.2°. Also, since GH is a diameter of the circle and GJL is a right triangle with ∠GJL = 90°, m/GJL = 90° - m/GHL = 90° - 45° = 45°. Therefore, m/JLH = m/JGL - m/GLH = 39.2° - 45° = -5.8°. Note that the negative value indicates that ∠JLH is a reflex angle.

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Answer:

(x - 5)² + (y + 6)² = 16

Step-by-step explanation:

assuming you require the equation of the circle

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r is the radius

here (h, k ) = (5, - 6 ) and r = 4 , then

(x - 5)² + (y - (- 6) )² = 4² , that is

(x - 5)² + (y + 6)² = 16

The values are;

1. x = 6 ,y = 6√2

2. x = 9√2, y = 18

3. x = y = 9

4. x = 12, y = 12√2

5. x = y = 4√2

6. x = y =( 3√2)/2

Trigonometric Ratios are defined as the values of all the trigonometric functions based on the value of the ratio of sides in a right-angled triangle.

There are some special angles , in which 45 is part of them.

sin 45 = 1/√2

cos 45 = 1/√2

tan 45 = 1

1. x = 6 ( isosceles triangle)

y = 6 × √2 = 6√2

2. x = 9√2 ( isosceles triangle)

y = 9√2 × √2 = 9×2 = 18

3. x = 9√2/√2 = 9

x = y = 9 ( isosceles triangle)

4. x = 12 ( isosceles triangle)

y = 12×√2 = 12√2

5. x = 8/√2 = 8√2/2 = 4√2

x = y = 4√2( isosceles triangle)

6. x = 3/√2 = (3√2)/2

x = y =( 3√2)/2 ( isosceles triangle)

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Answer: Option A:

To calculate the total amount Owen will spend on Option A, we need to calculate the monthly payment and then multiply it by the number of months:

First, we need to calculate the total amount of the loan. Owen is making a down payment of $3200, so he will be borrowing $28,820 (the price of the car minus the down payment).

Next, we can use the formula for calculating the monthly payment for a loan:

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

where P is the monthly payment, r is the monthly interest rate, A is the total amount of the loan, and n is the number of months.

For Option A, the monthly interest rate is 1.3% / 12 = 0.01083, the total amount of the loan is $28,820, and the number of months is 36. Plugging these values into the formula, we get:

P = (0.01083 * 28,820) / (1 - (1 + 0.01083)^(-36)) = $860.45

Therefore, the total amount Owen will spend on Option A is:

36 * $860.45 = $30,975.98

Option B:

For Option B, we need to take into account the $1500 cash back that Owen will receive as part of the down payment. This means that the total amount of the loan will be $32,020 - $3200 - $1500 = $27,320.

To calculate the monthly payment, we can use the same formula as before:

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

For Option B, the monthly interest rate is 5.2% / 12 = 0.04333, the total amount of the loan is $27,320, and the number of months is 36. Plugging these values into the formula, we get:

P = (0.04333 * 27,320) / (1 - (1 + 0.04333)^(-36)) = $825.53

Therefore, the total amount Owen will spend on Option B is:

36 * $825.53 + $1500 = $30,316.08

Therefore, Option A will cost Owen a total of $30,975.98, and Option B will cost him a total of $30,316.08. Therefore, Option B is the cheaper option for Owen.

Step-by-step explanation:

0.685 is a decimal that equals 68.5%.

The range of the 98% confidence interval for the percentage of faulty components that differ between machines A and B is about between -0.2448 and 0.1305. (rounded to 4 decimal places).

Given:

Sample proportion from machine A ([tex]p_1[/tex]) = 5/35 = 0.14285714285714285

Sample proportion from machine B ([tex]p_2[/tex]) = 7/35 = 0.2

Sample size from machine A ([tex]n_1[/tex]) = 35

Sample size from machine B ([tex]n_2[/tex]) = 35

Standard error (SE) = [tex]\sqrt{ [(p1 * (1 - p1) / n1) + (p2 * (1 - p2) / n2)}[/tex]

= [tex]\sqrt{[0.14285714285714285 * (1 - 0.14285714285714285) / 35] + [0.2 * (1 - 0.2) / 35] }[/tex]

= 0.07058061453775912 (rounded to 11 decimal places)

Margin of error (ME) = Critical value * Standard error

= 2.660 * 0.07058061453775912 (using z-score for a 98% confidence level)

= 0.18765117789861733 (rounded to 11 decimal places)

Confidence interval (CI) = Sample statistic ± Margin of error

= [tex](p_1 - p_2) \pm ME[/tex]

= (0.14285714285714285 - 0.2) ± 0.18765117789861733

= -0.05714285714285715 ± 0.18765117789861733

The 98% confidence interval for the difference in proportions of defective parts between machine A and machine B is approximately -0.2448 to 0.1305 (rounded to 4 decimal places).

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The range of the function in the graph is  6≤e≤12. So correct option is A.

In mathematics, range is a term used to refer to the set of all possible output values of a function. It is the set of values that the function can take as its input varies over its entire domain. In other words, the range of a function is the set of all output values that can be obtained by evaluating the function for all possible input values.

For example, consider the function f(x) = x². The domain of this function is all real numbers, but the range is only non-negative real numbers, since x² is always non-negative for any real number x.

The range of a function can be determined by analyzing its graph, which is a visual representation of the function. The range corresponds to the set of all y-values that appear on the graph. For instance, the range of the function f(x) = sin(x) is the closed interval [-1, 1], since the sine function oscillates between -1 and 1 as its input varies over all real numbers.

Sometimes, it is useful to restrict the domain of a function in order to obtain a specific range. This process is called domain restriction or range selection. For example, the inverse function of f(x) = x² can be obtained by restricting the domain of f to non-negative real numbers, which ensures that the inverse function is also a function. The resulting function is f^-1(x) = √x, whose domain is non-negative real numbers and range is the same as the domain of f.

The range of the function in the graph is  6≤e≤12. So correct option is A.

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So, the area of triangle DEF is 312.5 square feet, using the scale factor of 5/2.

the context of mathematics and geometry, dilation is a transformation that changes the size of an object. It is a type of transformation that scales an object by a certain factor, without changing its shape or orientation.

In other words, dilation involves multiplying the coordinates of a geometric figure by a fixed constant, which results in an enlarged or reduced version of the original figure. The constant is known as the dilation factor or the scale factor, and it can be any real number greater than zero.

For example, if we dilate a circle by a scale factor of 2, every point on the circle will be moved twice as far away from the center, resulting in a new circle with a diameter twice as large as the original.

Let's start by using the formula for the perimeter of a triangle:

[tex]Perimeter of triangle ABC = AB + BC + AC = 65.5 feet[/tex]

We can also use Heron's formula to find the area of triangle ABC:

[tex]Area of triangle ABC = \sqrt(s(s-AB)(s-BC)(s-AC))[/tex]

where s is the semi perimeter of the triangle:

[tex]s = (AB + BC + AC) / 2[/tex]

We can use these equations to solve for the side lengths of triangle ABC:

[tex]AB + BC + AC = 65.5[/tex]

[tex]s = (AB + BC + AC) / 2[/tex]

[tex]25 = \sqrt(s(s-AB)(s-BC)(s-AC))[/tex]

Solving for AB, BC, and AC gives us:

AB = 15

BC = 20

AC = 30.5

Now, let's dilate triangle ABC by a factor of 5/2 to create triangle DEF. This means that each side of triangle ABC will be multiplied by 5/2 to get the corresponding side length of triangle DEF.

DE = AB * (5/2) = 37.5

EF = BC * (5/2) = 50

DF = AC * (5/2) = 76.25

Now we can use Heron's formula again to find the area of triangle DEF:

s = (DE + EF + DF) / 2 = 81.875

Area of triangle DEF = sqrt(s(s-DE) (s-EF) (s-DF)) = 312.5 square feet

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Statistics is a branch of mathematics that involves the collection, analysis, interpretation, presentation, and organization of data. It is used in a wide range of fields such as science, engineering, social sciences, business, economics, and more.

In statistics, data is collected through various methods such as surveys, experiments, and observations. This data is then analyzed using statistical methods to extract meaningful insights, identify patterns and relationships, and make informed decisions.

Some common statistical techniques include descriptive statistics, inferential statistics, hypothesis testing, regression analysis, and probability theory. These techniques are used to help researchers and analysts to understand and draw conclusions about data, and to test whether their conclusions are statistically significant.

Statistics has many practical applications, such as market research, medical research, quality control, risk assessment, and many others. It plays a critical role in modern society, helping individuals and organizations make informed decisions based on data-driven insights.

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Sin 7π/4 is the value of sine trigonometric function for an angle equal to 7π/4 radians. The value of sin 7π/4 is -(1/√2) or -0.7071 (approx).

315° is comparable to 7 / 4 radians. In general, we multiply the angle measurement in radians by 180/ to translate an angle measurement given in radians to degrees. Therefore, we multiply 7 / 4 by 180 / to convert to radians. We discover that 7/4 radians equals 315 degrees.

We can first convert the angle to degrees as follows:

7π/4 radians = (7/4) × 180 degrees/π ≈ 315 degrees

The trigonometric ratios for 315 degrees (or 7/4 radians) can therefore be calculated using the reference angle of 45 degrees (which is /4 radians), as shown below.

sin(7π/4) = -sin(π/4) = -1/√2

cos(7π/4) = -cos(π/4) = -1/√2

tan(7π/4) = tan(π/4) = 1

csc(7π/4) = csc(-π/4) = -√2/2

sec(7π/4) = sec(-π/4) = -√2/2

cot(7π/4) = cot(-π/4) = 1

Therefore, the trigonometric ratios for the angle 7π/4 radians are:

sin(7π/4) = -1/√2

cos(7π/4) = -1/√2

tan(7π/4) = 1

csc(7π/4) = -√2/2

sec(7π/4) = -√2/2

cot(7π/4) = 1

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if we change the value of 523 to 424 in the list of numbers, then the mean decreases by approximately 3.22 and the median stays the same.

How to calculate the mean?

To calculate the mean, we add up all the numbers in the list and divide by the total number of values. Before the change, the sum of the numbers is:

164 + 225 + 227 + 250 + 261 + 268 + 277 + 379 + 523 = 2494

And there are 9 numbers in the list. So the mean is:

2494 / 9 ≈ 277.11

If we change the value of 523 to 424, then the sum becomes:

164 + 225 + 227 + 250 + 261 + 268 + 277 + 379 + 424 = 2465

And there are still 9 numbers in the list. So the new mean is:

2465 / 9 ≈ 273.89

So the mean decreases by approximately 3.22.

To calculate the median, we find the middle value of the list. If the list has an odd number of values, then the median is the middle value. If the list has an even number of values, then the median is the average of the two middle values. In this case, the list has an odd number of values, so the median is:

261

If we change the value of 523 to 424, then the list becomes:

164, 225, 227, 250, 261, 268, 277, 379, 424

And the median is still:

261

So the median stays the same.

In summary, if we change the value of 523 to 424 in the list of numbers, then the mean decreases by approximately 3.22 and the median stays the same.

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Answer:

x = √17

y = 10.1

Step-by-step explanation:

x² + 8² =  9²

x² = 81 - 64 = 17

x = √17

sin∅ = √17/9

∅ = 27.27°

9/y = cos(27.27)

y = 9/cos(27.27) = 10.13

y = 10.1

The truth table is given above for (A ⋀ B) → C.

What is the logical statement?

A logical statement, also known as a proposition or a statement of fact, is a declarative sentence that is either true or false, but not both. It is a statement that can be evaluated based on the available information or evidence to determine its truth value. In other words, a logical statement is a statement that can be either true or false, but not both.

To create a truth table for the logical statement (A ⋀ B) → C, we need to consider all possible combinations of truth values for propositions A, B, and C.

There are 2 possible truth values (true or false) for each proposition, so there are 2³ = 8 possible combinations.

We can organize these combinations into a table as follows:

| A | B | C | (A ⋀ B) | (A ⋀ B) → C |

|---|---|---|---------|-------------|

| T | T | T |    T    |      T      |

| T | T | F |    T    |      F      |

| T | F | T |    F    |      T      |

| T | F | F |    F    |      T      |

| F | T | T |    F    |      T      |

| F | T | F |    F    |      T      |

| F | F | T |    F    |      T      |

| F | F | F |    F    |      T      |

In this table, the column labeled (A ⋀ B) represents the truth value of the conjunction of A and B (i.e., A AND B), and the column labeled (A ⋀ B) → C represents the truth value of the conditional statement (A ⋀ B) → C.

The symbol "T" represents "true" and the symbol "F" represents "false".

Hence, The truth table is given above for (A ⋀ B) → C.

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For all x, P(x) (using universal quantifier ∀) and  It is not the case that there exists an x such that ~P(x) (using negation ¬ and existential quantifier ∃)

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

Let S be the set of students in your class and P(x) be the predicate "x has a cellular phone". Then, we can represent the statement "everyone in your class has a cellular phone" as:

1. For all x in S, P(x) (using universal quantifier ∀)

2. It is not the case that there exists an x in S such that ~P(x) (using negation ¬ and existential quantifier ∃)

If we want to represent the same statement for all people, we can use the same predicate P(x) and consider the domain of all people. Then, the statement "everyone has a cellular phone" can be represented as:

For all x, P(x) (using universal quantifier ∀)

It is not the case that there exists an x such that ~P(x) (using negation ¬ and existential quantifier ∃)Let S be the set of students in your class and P(x) be the predicate "x has a cellular phone". Then, we can represent the statement "everyone in your class has a cellular phone" as:

For all x in S, P(x) (using universal quantifier ∀)

It is not the case that there exists an x in S such that ~P(x) (using negation ¬ and existential quantifier ∃)

If we want to represent the same statement for all people, we can use the same predicate P(x) and consider the domain of all people. Then, the statement "everyone has a cellular phone" can be represented as:

1. For all x, P(x) (using universal quantifier ∀)

2. It is not the case that there exists an x such that ~P(x) (using negation ¬ and existential quantifier ∃)

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a. i. The function f(x)/g(x) = (2x - 1)/√(3x - 5)

ii.  The domain is x > 5/3

b. i. The function f(x) - g(x) = (2x - 1) - √(3x - 5)

ii.  The domain is x > 5/3

A function is a mathematical relation ship between two variables.

Since we have the functions f and g defined as follows

f(x) = 2x-1

g(x) = √3x-5

a. i To find f/g we note that

(f/g)(x) = f(x)/g(x)

So, substituting the values of the variables into the equation, we have that

f(x)/g(x) = (2x - 1)/√(3x - 5)

ii. The domain of f(x)/g(x) = (2x - 1)/√(3x - 5) is the value for which the denominator g(x) > 0.

So,g(x) > 0

⇒ √(3x - 5) > 0

⇒ 3x - 5 > 0²

⇒ 3x > 5

⇒ x > 5/3

So, the domain is x > 5/3

b. i. to find f - g, we note that

f - g = f(x) - g(x)

So, substituting the values of the variables into the equation, we have that

f(x) - g(x) = (2x - 1) - √(3x - 5)

ii. The domain of f(x) - g(x) is the value of x at which g(x) > 0

So. g(x) > 0

⇒  √(3x - 5) > 0

⇒ [√(3x - 5)]² > 0

⇒  3x - 5 > 0

⇒ 3x > 5

⇒ x > 5/3

So, the domain is x > 5/3

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Answer:

B is the answer

Step-by-step explanation:

The expression that is not equal to the others is B [[2/−5]] The other expressions are A −[[2/5]], C [[−2/5]], and D [[2/5]].

Answer:

39 divided by 8 is equal to 4 with a remainder of 7.

The quotient is the number of times the divisor goes into the dividend. In this case, 8 goes into 39 4 times with a remainder of 7.

The remainder is the number that is left over after the divisor has been divided into the dividend. In this case, 7 is left over after 8 has been divided into 39.

Here is the long division of 39 by 8:

```

39 / 8

4

32

7

```

Step-by-step explanation:

The quotient of 39 divided by 8 is 4, and the remainder is 7.

We have,

When performing long division, we divide the dividend (39) by the divisor (8) to find the quotient and remainder.

  4

--------

8 | 39

  - 32

    ---

     7

Here's how the long division process works for 39 divided by 8:

-We start by dividing the first digit of the dividend (3) by the divisor (8). Since 3 is less than 8, we can't divide it evenly, so we move to the next digit (9).

- We now have 39 as the remaining portion of the dividend. We divide 39 by 8. The largest multiple of 8 that fits into 39 is 4. We place the quotient, which is 4, above the line.

- We multiply the quotient (4) by the divisor (8), which gives us 32. We subtract 32 from 39, which leaves us with a remainder of 7.

- Since there are no more digits to bring down from the dividend, and the remainder (7) is less than the divisor (8), we stop the division process.

Therefore,

The quotient of 39 divided by 8 is 4, and the remainder is 7.

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This sentence relating to the quotient can be expressed mathematically as:

(30 / x) - 2

This sentence can be expressed mathematically as:

(30 / x) - 2

where x represents the unknown number.

The word "quotient" indicates that we are dividing 30 by the unknown number x. The phrase "is decreased by 2" means that we need to subtract 2 from the quotient.

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The statement that (sin 2x) / (1 + cos 2x) = tan x can be proven.

To prove that (sin 2x) / (1 + cos 2x) = tan x, we will use trigonometric identities.

(sin 2x) / (1 + cos 2x)

(2sin x × cos x) / (1 + (cos²x - sin²x))

(2sin x × cos x) / (cos²x + 2sin x × cos x + sin²x)

We can rewrite the denominator using the Pythagorean identity sin²x + cos²x = 1:

(2sin x × cos x) / (1 + 2sin x × cos x)

(2sin x × cos x) × (1 - 2sin x × cos x) / (1 - (2sin x × cos x)²)

((2sin x × cos x) - (4sin²x × cos²x)) / (1 - 4sin²x × cos²x)

(2sin x - 4sin²x) / (1/cos²x - 4sin²x)

Since tan x = sin x / cos x, we can rewrite the expression:

(2tan x - 4tan²x) / (sec²x - 4tan²x)

(2tan x - 4tan²x) / (1 + tan²x - 4tan²x)

(2tan x - 4tan²x) / (1 - 3tan²x)

2tan x × (1 - 2tan²x) / (1 - 3tan²x)

tan x

So, we have proved that (sin 2x) / (1 + cos 2x) = tan x.

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Answer:

x=90°

Step-by-step explanation:

As in the angle, there is a box giving us info that x has to be 90°.

But to be sure that there is no mistake we have to do the following:

So we are looking at the surroundings of angle x (which is on a straight line) we see that it is a right angle and look at the angle on the same line is a right angle too.

The equation right angle=90° helps us see that because there are two right angles on a 180° line (90°+90°+180°).

Therefore the answer is:

x=90°

Answer: Spheres aren’t three-dimensional—they are two-dimensional. This is evident from the fact that in order to specify a point on a sphere, you only need two pieces of information, such as latitude and longitude.

If you include the interior of the sphere, this is instead called a closed ball, and that is three-dimensional. You can specify a point in the closed ball in all sorts of different ways; one of the most convenient would be latitude, longitude, and distance from the center. However, other than convenience, there is no reason to prefer one coordinate system over any other.

(This fact has nothing to do with spheres or closed balls—that is just a statement that is generally true. People who insist that “the three dimensions” are length, width, and height don’t know what they are talking about.)

Step-by-step explanation:

The domain restrictions on the function [tex]\left f(x\right)=-\frac{3x}{81\:-\:x^2}\:\cdot \frac{81\:-\:x^2}{2x^2-6x}\:\div \frac{x^2+2x-6x}{3x^2-30x+63}[/tex] are the x values -9, 0, 3, 4 and 9

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

[tex]\left f(x\right)=-\frac{3x}{81\:-\:x^2}\:\cdot \frac{81\:-\:x^2}{2x^2-6x}\:\div \frac{x^2+2x-6x}{3x^2-30x+63}[/tex]

Next, we set the denominator to 0 and solve for x

So, we have

81 - x²: x = ±9

2x² - 6x: x = 0 and x = 3

x² + 2x - 6x: x = 0 and x = 4

Hence, the domain restrictions are the x values -9, 0, 3, 4 and 9

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