Find the distance between the points (9,8) and (–2,–4).

We're given the equation,

[tex]\longrightarrow\sin\theta+\csc\theta+2=0[/tex]

Take [tex]\sin\theta=x,[/tex] then [tex]\csc\theta=\frac{1}{x}.[/tex] Then the equation becomes,

[tex]\longrightarrow x+\dfrac{1}{x}+2=0[/tex]

Multiply both sides by [tex]x,[/tex] assuming [tex]x\neq 0.[/tex] Then we get,

[tex]\longrightarrow x^2+2x+1=0[/tex]

[tex]\longrightarrow (x+1)^2=0[/tex]

[tex]\Longrightarrow x=-1[/tex]

We're asked to find the value of,

[tex]\longrightarrow y=\sin^{2019}\theta+\csc^{2020}\theta[/tex]

As we've taken [tex]\sin\theta=x,[/tex] this expression becomes,

[tex]\longrightarrow y=x^{2019}+\dfrac{1}{x^{2020}}[/tex]

Now putting [tex]x=-1,[/tex]

[tex]\longrightarrow y=(-1)^{2019}+\dfrac{1}{(-1)^{2020}}[/tex]

[tex]\longrightarrow y=-1+\dfrac{1}{1}[/tex]

[tex]\longrightarrow y=-1+1[/tex]

[tex]\longrightarrow\underline{\underline{y=0}}[/tex]

Hence 0 is the answer.

Value of  [tex]1^{3} + 10^{3} + 11^{3} +12^{3} +......+30^{3}[/tex] is "216225".therefore the correct option is C.

Natural Numbers are the counting numbers that start from 1 and go on till infinity. To find the sum of cube of first n natural numbers means to add the cube of a specific number of natural numbers starting from 1 and get the answer.

There are a lot of patterns in mathematics. One analogous pattern of numbers is the sum of cube of n natural numbers.

Given:  Equation  = [tex]1^{3} + 10^{3} + 11^{3} +12^{3} +......+30^{3}[/tex]

            Here  N=30  , We have given

solution:

                                              [tex]S=\frac{{n^{2} (n+1)^{2}}}{4} \\=\frac{30^{2} (30+1)^{2}}{4}\\=\frac{900 (31)^{2}}{4}\\=\frac{900 (961)}{4}\\=\frac{864900}{4}\\=216225[/tex]

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Answer:

B - 50

Step-by-step explanation:

the exterior angle is 105, and as it's on a straight line, the interior angle is 180-105=75

75+55=130

angles in a triangle are 180

180-130=50

Answer:50 degrees

Step-by-step explanation: The 105degrees angle forms a supplementary angle with the missing degrees in the triangle

so the missing value which is not requested is 180degrees-105 degrees=75 degrees

No we have two angles inside the triangle. We can now work out the missing angle for x

We know that a triangle adds up to 180 degrees

So x=180-55-75=50 degrees as the answer

An employee earns $14 per hour and a trainer charges $20 for every 30 minutes of training plus a $5 tip are a constant rate per unit interval. Therefore, options A and E are the correct answers.

Given that:

an employee earns $14 per hour.

constant rate of change :

When something has a constant rate of change, one quantity changes in relation to the other. For example, for every half hour the pigeon flies, he can cover a distance of 25 miles. We can write this constant rate as a ratio. For ratios, it's always a good idea to state both units in whole terms.

A). Let the number of hours be x and the total money earned be y

So, y = 14x

B). Let the money earn by an employee each year be a and total earning in a year be b.

So, b=a+3% of a

⇒ b = a + 0.03a

⇒ b = 1.03a

C) Let the number of months be m and the total charge be t

So, t=50+30m

D) Let cost of gym equipment be p, rate of change =8% and time period = 6 months

So,

E) Let the number of hour be x.

Here, 30 minutes = 1/2 hours

Total charge = 20 + 5x/2

An employee earns $14 per hour and a trainer charges $20 for every 30 minutes of training plus a $5 tip are a constant rate per unit interval. Therefore, options A and E are the correct answers.

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Full question :

For which situation does one quantity change at a constant rate per unit interval relative to another quantity? responses.

Select all that apply

A. An employee earns $14 per hour.

B. An employee earns a 3% raise each year.

C. A gym charges a $50 down payment plus $30 per month.

D. A piece of gym equipment decreases 8% in value every 6 months.

E. A trainer charges $20 for every 30 minutes of training plus a $5 tip.

The integer set that makes the inequality 3p - 10 > 7p + 6 true is given as follows:

{- infinity, ..., -6, -5}.

The inequality in this problem is defined as follows:

3p - 10 > 7p + 6

To solve the inequality, we treat it similarly to an equality, isolating the variable of interest. The difference is that the solution is a set with infinity values, and not a single value.

Hence the solution of the inequality is obtained as follows:

3p - 10 > 7p + 6

-4p > 16

4p < -16 (multiplying by -1 due to the negative coefficient of x, the sign also changes)

p < -16/4

p < -4.

Hence the integer set of the solution of the inequality is given as follows:

{- infinity, ..., -6, -5}.

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Answer: see below

Step-by-step explanation:

The table is based on the rule y=-2x+5

let's use the last numbers as an example. x=10 and y= -15

since we know the values of x and y, we can substitute them into the rule equation

-15=-2(10)+5, see how I replaced y with 15 and x with 10?

Now we solve and see if they match.

-15=-20+5

-15=-15

since they match, they are correct.

Basically, replace x in the rule with the x values in the table, and if the y value is the same as the y value in the table, it is correct.

The median for the next batch of observations is 5.

The median is the value in the middle of a data set, which indicates that 50% of the data points have a value less than or equal to the median, and 50% have a value greater than or equal to the median. The median is the number in the middle of a set of numbers. The median represents the 50th percentile of the set of numbers. In other words, the median is the value in the center of a set of numbers where half are less than the median and half are more than the median.

Here,

given set of observation,

=2,3,4,5,6,7,8

n=7

median=(7+1)/2

=4th term

=5

The median for following set of observation is 5.

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To solve this equation, use the distributive property to rewrite it as 4x - 2y = 12. Then, solve the equation using either addition or subtraction.

To solve the equation 4x 2y = -12, the first step is to use the distributive property to rewrite the equation as 4x - 2y = 12. This allows us to solve the equation using either addition or subtraction. If we choose to use addition, we can add 2y to both sides of the equation to get 4x + 2y = 12. We can then subtract 2y from both sides to get 4x = 12 - 2y. Dividing both sides of the equation by 4 gives us the solution x = (12 - 2y)/4. If we choose to use subtraction, we can subtract 2y from both sides of the equation to get 4x - 2y = 12. We can then add 2y to both sides to get 4x = 12 + 2y. Dividing both sides of the equation by 4 gives us the solution x = (12 + 2y)/4. In either case, the solution is x = (12 ± 2y)/4.

Addition:

4x + 2y = 12

4x = 12 - 2y

x = (12 - 2y)/4

Subtraction:

4x - 2y = 12

4x = 12 + 2y

x = (12 + 2y)/4

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The original amount is 10 and the new quantity is 14 percent increase  is 40%

An quantity or part that is contained in each hundred is known as a percentage. The symbol "%" signifies that it is a fraction with 100 as the denominator.

Here,

The increase in percentage is equal to (New quantity - Original quantity)/Original quantity 100

(14-10)/10* 100.

4/10 × 100

40%

A 40% increase from the original amount to the new quantity may be calculated as a result.

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The rational roots of the given polynomial are +/- 1, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12, +/- 1/2, +/- 1/4, +/- 1/5, +/- 1/10, and +/- 1/20.

To find the rational roots of the polynomial, we can use the Rational Root Theorem. This theorem states that if a polynomial of the form a_n*x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 has a rational root, then that root must be of the form p/q, where p is a divisor of a_0 and q is a divisor of a_n.

In this case, the polynomial is

f(x) = 20x^4 - 8x^3 - x^2 + 12x,

so a_0 = 12 and a_4 = 20. The divisors of 12 are 1, 2, 3, 4, 6, and 12. The divisors of 20 are 1, 2, 4, 5, 10, and 20. Therefore, the possible rational roots of the polynomial are

p/q = +/- 1, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12, +/- 1/2, +/- 1/4, +/- 1/5, +/- 1/10, and +/- 1/20.

We can then test each of these values to see if they are actually roots of the polynomial. If a value is a root, then substituting it into the polynomial will result in a value of 0.

Therefore, the rational roots of the given polynomial are +/- 1, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12, +/- 1/2, +/- 1/4, +/- 1/5, +/- 1/10, and +/- 1/20.

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This theorem states that if a polynomial of the form a_n*x^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 has a rational root, then that root must be of the form p/q, where p is a divisor of a_0 and q is a divisor of a_n.

f(x) = 20x^4 - 8x^3 - x^2 + 12x,

p/q = +/- 1, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12, +/- 1/2, +/- 1/4, +/- 1/5, +/- 1/10, and +/- 1/20.

Therefore, the rational roots of the given polynomial are +/- 1, +/- 2, +/- 3, +/- 4, +/- 6, +/- 12, +/- 1/2, +/- 1/4, +/- 1/5, +/- 1/10, and +/- 1/20.

Step-by-step explanation:

∠1 + 49° + 49° = 180°

∠1 = 180-98

Answer:

The value of 9^(x) is 3.

Step-by-step explanation:

To find the value of 9^(x), we can use the equation:

(a^x) = (b^y)

Where a and b are different bases, and x and y are different exponents.

We can rewrite this equation as:

x = y * log(a) / log(b)

Plugging in the values given in the question, we get:

x = log(8) / log(4)

Solving for x, we get:

x = 3

Therefore, the value of 9^(x) is 3.

The length of the side of the smaller box is 6 and the length of the side of the larger box is 20. The equation for the difference in volume of the boxes is, y3 - x3 = 7784 .

Let us consider the length of side of small box be x inch & length of large box be y inch.

Length of smallest box is 30% of the length of the side of the largest box 

x=30%*y

   = 30 * y / 100

x = 3y / 10

Volume of the smallest box is x3.

volume of the largest box is y3.

Volume of largest box is 7784 cubic inches more than the volume of smallest box. Therefore,

y3 =  x3 + 7784

     = (3y/ 10 )3 + 7784

     = 27y3 / 1000 + 7784

y3 - 27y3 / 1000 = 7784

973 y3  / 1000 = 7784

973 y3 = 7784 * 1000

y3 = 7784 * 1000 / 973

y3 = 8000

   y = 20

Now find the smaller side.

x = 3y / 10

putting the value of y  this we get,

x = 3 * 20 / 10

   x= 6    

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The general formula for the variance varies according to the distribution, For the discrete and continuous probability distribution, The expected values formula varies. The variance is given by [tex]E(X^{2})-(E(X))^{2}[/tex] .

A probability distribution is a mathematical function used in probability theory and statistics that estimates the likelihood that various possible outcomes of an experiment will occur. In terms of its sample space and event probability, it is a mathematical description of a random phenomena.

The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed throughout a wider range, a low standard deviation suggests that the values tend to be close to the established mean.

If distribution is discrete  , E(X) = x*P(x).

If distribution is continuous , E(x) = x *f(x).

Variance =  [tex]E(X^{2})-(E(X))^{2}[/tex]

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Answer:

range

Step-by-step explanation:

the x values or inputs are called the domain

the y values or outputs are called the range

360° is all the way around a circle. The length of the intercepted arc is equal to the circumference of the circle.

Arc Length of a Circle:

Length. A practical way to determine the length of an arc in a circle is to plot two lines from the arc's endpoints to the center of the circle, measure the angle where the two lines meet the center, then solve for L by cross-multiplying the statement: measure of angle in degrees/360° = L/circumference.

Does arc length have to be in radians?

Arc length is a measurement of distance, so it cannot be in radians. The central angle, however, does not have to be in radians. It can be in any unit for angles you like, from degrees to arcsecs. Using radians, however, is much easier for calculations regarding arc length, as finding it is as easy as multiplying the angle by the radius.

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The example of a rational function  is f ( x) = x 2 − 1 2 x + 21.  , we can represent this function in terms of ratio of p and q.

Other than than a real world example can be that , a cookie being divided among two children , the representation will be , 1 / 2.

A graph of a rational function has asymptotes which are of three types and that is what makes them different from the rest of the functions. Rational functions are the functions which can be represented in the ratio of two polynomials p and q. The solution of this equation is know as the roots of the equation.

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The age of the son and father obtained after solving the equation will be equal to 6 years and 34 years.

Mathematical expressions with two algebraic symbols on either side of the equal (=) sign are called equations.

This relationship is illustrated by the left and right expressions being equal. The left-hand side equals the right-hand side is a basic, straightforward equation.

As per the given information in the question,

Let the age of the father is f and the age of the son be s.

As per the given statement in the question,

The sum of the ages of a father and his son is 40 years.

f + s = 40   (i)    and,

f + s + 2(f -s) = 96    

3f - s = 96       (ii)

Add equations (i) and (ii)

4f = 136

f = 136/4

f = 34 years

Put f = 34 in equation (i)

f + s = 40

34 + s = 40

s = 40 - 34

s = 6 years.

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Answer:120

Step-by-step explanation:

1,440 divided by 12 equals 120 so 120 shelves are needed

Answer:

15 storage towers

Step-by-step explanation:

first you need to find out how many video games can fit into one storage tower.

8 times 12 is 96,meaning 96 per tower.

Then you would one to get the total number of video games and divide it by 96 to find out how many towers are needed.

If done correctly you will end up with 15.

The two distinct real number zeros of f(x) left parenthesis, x, right parenthesis are -2 + sqrt(7) and -2 - sqrt(7).

The discriminant of a polynomial is a quantity that determines the number of distinct real roots of a polynomial equation. For a polynomial of degree n, the discriminant is calculated as the square of the coefficient of the nth degree term multiplied by the product of the coefficients of the remaining terms, minus the square of the product of the coefficients of the lower degree terms. In the case of [tex]f(x)=3x^2+12x+5[/tex], the discriminant is given by[tex]D=12^2 - 4*3*5 = 144 - 60 = 84.[/tex]

Since the discriminant is greater than 0, it follows that the polynomial equation has two distinct real number zeros. To find the zeros of the equation, we must solve the equation by setting it equal to zero and factorising it. Doing so, we find that the zeros are given by [tex]x = (-12 +/- sqrt(84))/6 = -2 +/- sqrt(7)[/tex]. Hence, the two distinct real number zeros of f(x) are -2 + sqrt(7) and -2 - sqrt(7).

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Answer:

80π feet

Step-by-step explanation:

The question is asking us for the circumference of a circle with diameter 80 ft.

The formula for the circumference of a circle is πd. Therefore, the circumference is π * 80 = 80π ft!

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Answer:

The circumference of a circle is equal to the diameter times pi, which is approximately 3.14. Therefore, the distance of one complete loop on a circular path with a diameter of 80 feet is approximately 80 * 3.14 = 251.2 feet.

Step-by-step explanation:

The solution of the given system of equations 3x + 6y = 2 ,  15x -30y = 10 is equal to x = 2/3 and y = 0.

As given in the question,

Given system of equations are :

3x + 6y = 2      ____(1)

15x -30y = 10   _____(2)

Multiply equation (1) by 5 and add it to equation (2) to eliminate y and get the value of x we have,

15x + 30y = 10

15x - 30y = 10

30x      = 20

⇒x = 20 / 30

⇒ x = 2 / 3

Substitute value of x = 2/3 in equation (1) to get the value of y,

3(2/3) + 6y = 2

⇒ 2 + 6y = 2

⇒ 6y = 0

⇒ y = 0

Therefore, the solution of the given system of equations is equal to x = 2/3 and y =0.

The above question is incomplete, the complete question is:

What is the solution of the system of equations 3x + 6y = 2 ,  15x -30y = 10?

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There are nine numbers in this set, so the median is the fifth number. The fifth number in this set is 14, so the median is 14.

The median of a set of numbers is the value that is exactly in the middle of the set when it is ordered from least to greatest. In this case, the set of numbers is 13, 16, 12, 14, 19, 12, 14, 13, 14. When we order this set from least to greatest, we get 12, 12, 13, 13, 14, 14, 14, 16, 19. There are nine numbers in this set, so the median is the fifth number. The fifth number in this set is 14, so the median is 14.

To find the median, we first need to arrange the numbers in order from least to greatest. If there is an odd number of numbers in the set, the median is the middle number. If there is an even number of numbers in the set, the median is the average of the two middle numbers. In this case, there are nine numbers in the set, so the median is the middle number.

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The general case when G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ) is that the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

G⊕H is cyclic

That is G⊕H = <(g, h)>

Let us choose an arbitrary element g₁ ∈ G, then as G⊕H is cyclic

(g₁, h₁) ∈ G⊕H = <( g, h)>

This means that there exists an integer x such that

(g₁, h₁) = (g, h)ˣ

(g₁, h₁) = (gˣ, hˣ)

that is g₁ = gˣ

implies g₁ ∈ <g>

that is G ⊆ <g> as g₁ is an arbitrary element belonging to G

But we know <g> ⊆ G

Hence G = <g>

Thus G is a cyclic group generated by g.

Similarly we can prove that H group is cyclic generated by h, that is H = <h>

So the general case is that G₁⊕G₂⊕G₃...⊕Gₙ is cyclic generated by (g₁, g₂, g₃,...,gₙ), then the subgroups G₁, G₂, G₃,...., Gₙ are cyclic generated by  g₁, g₂, g₃, ..., gₙ respectively.

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Answer:

(a) similar

Step-by-step explanation:

20/10 = 1/2

8/4 =1/2

22/11 =1/2

The sides are proportional in the same ratio hence they are similar triangles

Answer:

a

Step-by-step explanation:

it is not D because if it was a replica it would be exactly the same. If it was a conjourent than it would be if all three corresponding sides are equal and all the three corresponding angles are equal in measure. so it leaves similar

The value of the given expression ( 402 ) ^ (1/2) using binomial theorem is equal to option a. 20.05.

As given in the question,

Given expression is equal to :

( 402 ) ^(1/2)

Standard form of binomial theorem is:

( a + b )ⁿ = ⁿC₀aⁿb⁰ +  ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b² +..............+  ⁿCₙa⁰bⁿ

Rewrite the 402 = 400 + 2 and represent it in binomial theorem form:

( 400 + 2 ) ^(1/2)

Here take out 400 as common factor :

= (400)^(1/2) ( 1 + 2/400)^(1/2)

= 20 ( 1 + 1/200) ^(1/2)

Using one of the result of binomial theorem to get the approximate value if other values are extremely small.

( 1 + x )ⁿ = 1 + nx

Here n = 1/2 and x = 1/ 200

= 20 [ 1 + (1/2) (1/200)]

= 20 ( 1 + 1 /400)

= 20 ( 401/400)

= 401 / 20

= 20.05

Therefore, the value of (402)^(1/2) using binomial theorem is equal to option a. 20.05.

The above question is incomplete , the compete question is:

The value of (402)^(1/2) according to the binomial theorem is

a.  20.05        b. 20.01         c. 20.00          d.  20.2​

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Answer:

A whole number is a number without fractions or intergers

Step-by-step explanation:

An example of this wou;d be 0, 16, etc too find one you have to look at the qualities and see if it fits

The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3.

The vertical asymptotes for the function f(x) = (x^2)(x-6)(x+3)^-1 are x=6 and x=-3. This is because the denominator can be factored into two linear expressions: x-6 and x+3. When either of these expressions equals 0, the fraction becomes undefined and a vertical asymptote is created. To find the asymptotes, set each linear expression in the denominator equal to 0 and solve for x. For x-6, x=6; for x+3, x=-3. Therefore, the vertical asymptotes of the function are x=6 and x=-3. When x is close to either of these values, the fraction will become very large, making the graph approach the vertical asymptotes without ever touching them.

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