Performance task trigonometric identities

If there are 30 skaters competing at the olympic games, then the required number of ways = 24360

The term permutation refers to a mathematical calculation of the number of ways a particular set can be arranged. Put simply, a permutation is a word that describes the number of ways things can be ordered or arranged. With permutations, the order of the arrangement matters.

Given that,

There are 30 skaters competing in the competition.

To find,

The number of ways in which they can win gold, silver, and bronze medal.

So,

We can write,

In distribution of medals(gold, silver, and bronze) we require 3 winners in a particular order, so for this we use Permutations.

According to permutation , number of ways of selecting r things out of ,

n = \(\frac{n!}{(n-r)!}\)

Then, The number of ways in which 18 skaters can win 3 medals,

= \(\frac{30!}{(30-3)!}\)

= \(\frac{30!}{27!}\)

= \(\frac{30*29*28*27!}{27!}\)

= \(30*29*28\)

= 24360

Therefore,

If there are 30 skaters competing at the olympic games, then the required number of ways = 24360

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Answer:

The equation of a line with slope 0 and y-intercept -6 in slope-intercept form is \(y = -6\).

Step-by-step explanation:

From Analytical Geometry we know that equation of the line in slope-intercept form is represented by:

\(y = m\cdot x + b\) (Eq. 1)

Where:

\(x\) - Independent variable, dimensionless.

\(y\) - Dependent variable, dimensionless.

\(m\) - Slope, dimensionless.

\(b\) - y-Intercept, dimensionless.

If we know that \(m = 0\) and \(b = -6\), then the equation of the line is:

\(y = 0\cdot x +(-6)\)

\(y = -6\)

The equation of a line with slope 0 and y-intercept -6 in slope-intercept form is \(y = -6\).

Answer:

ummmmmmmmmmmmmmmmmmmmmme'[4wszmdg

Aw;3;ootgkokgm bkttmo[

q324wky

['

Step-by-step explanation:

Answer:

-240

Step-by-step explanation:

Answer:

-240

Step-by-step explanation:

-50-90-100

-140-100

-240

Adding negatives is the same as subtracting.

Answer: 71.25 meters

Step-by-step explanation:

1. 498.75/7

2. 71.25

Mia made 7 trips to visit her grandmother. she walked 498.75 meters in all, mia walks on each trip is 71.25 m

Mia made 7 trips to visit her grandmother

she walked 498.75 meters in all

In order to calculate the distance she walks on each trip, we calculate it by dividing the total distance by the number of trips

The total distance is 498.75

The number of trips is 7

walk on each trip is 498.75/ 7 = 71.25

Therefore, mia made 7 trips to visit her grandmother. she walked 498.75 meters in all, mia walks on each trip is 71.25 m

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Answer:

5m + 5

Step-by-step explanation:

Combine like terms (terms with like & the same amount of the variable).

Set the equation:

(8m - 3m) + (9 - 4)

Simplify by combining like terms:

(8m - 3m)  = 5m

(9 - 4) = 5

5m + 5 is your answer.

~

Answer:

5m+5

Step-by-step explanation:

\( \sf \: -15 < x - 8 < -4 \)

\(\sf \longmapsto−15+8<x−8+8<−4+8\)

\(\sf \longmapsto−7<x<4\)

\( \boxed{\sf−7<x<4 }\)

Answer:

Whats the Question

Step-by-step explanation:

Answer:

I dont really understand the question but here what I got.

Step-by-step explanation:

1.73$+$0.13=1.86$

So he bought something else that cause 0.97$ to equal =2.83$

Answer:

Next 3 terms 031, N30, D31

Step-by-step explanation:

J30, J31, A31, S30

The above series represent the

first letter of the month and the days in it

J30= June which have 30 days

J31= July which has 31 days

A31= August which have 31 days

S30= September which have 30 days

30 days hath September, April, June and November. All the rest have 31 except February alone which has but 28 or 29 days

The next 3 terms of the series J30, J31, A31, S30 are 031, N30, D31

Answer:

$8.50 ? let me know if im wrong

Step-by-step explanation:

Answer:

4+(-5)=-1. C was just irrelevant here

Answer:

replace the variables in the equations with the values they give you, and solve it. it'll give you -1

Step-by-step explanation:

a + b becomes

4 + (-5), which is 4-5

so the answer is -1

Answer:

57,416

Step-by-step explanation:

50,000 + 7,000 + 400 + 10 + 6

50,000 is in the ten thousands place.

7,000 is in the thousands place.

400 is in the hundreds place.

10 is in the tens place.

6 is in the ones place.

a) The equation 1 has an infinite number of solutions, as both linear functions have the same slope and internet, hence Nyoko is incorrect.

b) Nyoko cannot find a value of b so that the equation has no solutions.

The equation 1 is given as follows:

6x - 5 + 2x = 4(2x - 1) - 1.

Combining the like terms and applying the distributive property, the simplified equations are given as follows:

8x - 5 = 8x - 4 - 1

8x - 5 = 8x - 5.

As they are linear functions with the same slope and intercept, the number of soltuions is of infinity.

The equation 2 is given as follows:

3x + 7 = bx + 7.

A system of linear equations will have zero solutions when:

As they have the same intercept for this problem, it is not possible to attribute a value of b such that the equation will have no solution.

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Answer:

What is the magnitude and direction of the PQ−→− with tail and head points P(-6, 0) and Q(2, 4)? magnitude = √(64+16) = √80 = appr 8.9 , ruling out c.

hope this helps

The magnitude of PQ is 8.9 and PQ is making an angle of 26.57° so 8.9 units, 26.6° north of east will be the correct answer.

A vector is a quantity in which direction and magnitude both matters called a vector.

A vector joining by two points (x₁,y₁) and (x₂,y₂) will be (x₂-x₁)\(\hat{i}\) + (y₂-y₁)\(\hat{j}\)

The magnitude of this vector will be  

\(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2\)  

The direction of this vector will be  

\(\\tan^{-1}(\frac{y_2-y_1}{x_2-x_1)}\)

Given P(-6,0) and Q(2,4) so vector joining by

\(\vec{PQ}\) = [2-(-6)]\(\vec{i}\)  + [4-0]\(\vec{j}\)

\(\vec{PQ}\) =  8\(\vec{i}\) + 4\(\vec{j}\)

now, the magnitude  \(\vec{PQ}\) will be

⇒  \(\sqrt{(8^2+4^2)}\)

⇒ 8.944

The direction of the vector \(\vec{PQ}\) will be

⇒  \(\tan^{-1}(\frac{4}{8})\)

⇒26.57°

So, the magnitude of PQ is around 8.944 and the direction is 26.57° from the positive x axis so it's north of the east.

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Answer:

56

Step-by-step explanation:  

Answer:

Diameter Length: ( About ) 5.4 km; Option B

Step-by-step explanation:

~ Let us apply the Area of the Circle formula πr^2, where r ⇒ radius of the circle ~

1. We are given that the area of the circle is 22.9 km^2, so let us substitute that value into the area of the circle formula, solving for r ( radius ) ⇒ 22.9 = π * r^2 ⇒ r^2 = 22.9/π ⇒ r^2 = 7.28929639361.... ⇒

radius = ( About ) 2.7

2. The diameter would thus be 2 times that of the radius by definition, and thus is: 2.7 * 2 ⇒ ( About ) 5.4 km

Diameter Length: ( About ) 5.4 km

Answer:

y = 9

Step-by-step explanation:

I think you might have written this problem wrong, but I think the question is asking me to find y in the equation:

\(-2y+6=-12\)

Lets start by subtracting 6 from both sides:

\(-2y+6-6=-12-6\)

Simplify:

\(-2y = -18\)

Multiply both sides by -1

\((-1)(-2y) = (-18)(-1)\)

Simplify(cancel duplicate negatives)

\(2y = 18\)

Divide both sides by 2

\(\frac{2y}{2}=\frac{18}{2}\)

Simplify(remove denominator)

\(y = 9\)

Therfore, y = 9

Answer:

1. ∠YZX = 37°

2. ∠YXZ = 53°

3. XZ = 30 meters

Step-by-step explanation:

1. To solve for an unknown angle, we need to utilize the inverse of a trigonometric function.

⭐What are the inverses of the trigonometric functions?

To know which inverse of the trigonometric functions we use, we have to see the type of side lengths we are given (opposite, adjacent, or hypotenuse)

We are given side length XY, which is opposite of ∠YZX, and we are given side length YZ, which is adjacent to ∠YZX. Therefore, we will use \(tan^-1 (opposite/adjacent) =\)

Substitute the values we are given into the function:

\(tan^-1 (18/24) =\)

Compute this equation using a scientific calculator. I recommend using the Desmos Scientific Calculator:

\(= 37\)

∴ ∠YZX = 37°

2. We already know 2 angles (∠YZX = 37°, and ∠ZYX = 90°) Therefore, to find ∠YXZ, we have to utilize the triangle sum theorem.

⭐ What is the triangle sum theorem?

Substitute the angles we know already (∠YZX and ∠ZYX), and solve for ∠YXZ.

\(< YZX + < ZYX + < YXZ = 180\)

\(37 + 90 + YXZ = 180\)

\(127 + YXZ = 180\)

\(< YXZ = 53\)

∴ ∠YXZ = 53°

3. We already know 2 side lengths (ZY = 24 meters, and XY = 18). Therefore, to find XZ, we have to utilize the Pythagoras' theorem.

⭐ What is the Pythagoras' theorem?

Substitute the values of the side lengths into the formula:

\((XZ)^2 = (XY)^2 + (ZY)^2\)

\((XZ)^2 = 18^2 + 24^2\)

Solve for XZ:

\((XZ)^2 = 324 + 576\)

\((XZ)^2 = 900\)

\(\sqrt{(XZ)^2} = \sqrt{900}\)

\(XZ = 30\)

∴ XZ = 30 meters

The line passes through the origin and if it has a constant of variation of 3, so it represents a direct variation , Option A is the correct answer.

When a quantity increases and the other quantity also increases , then those two quantities are said to be in direct variation.

It is asked in the question that which option depicts whether On a coordinate plane, a line goes through points (0, 0) and (1, 3) , represents a direct variation

As it is mentioned in the question that the line passes through the origin and if it has a constant of variation of 3, so it represents a direct variation.

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Answer:

a

Step-by-step explanation:

Answer: The values that make the denominator equal to zero for a rational expression are known as restricted values. How do you identify restrictions in rational expressions? The restriction is that the denominator can not be equal to zero. So in this problem, since 4x is in the denominator it can not equal zero.

The solution of the differential equation y′′(t) y(t) = g(t),

y(0) = 1, y′(0) = 1, for t ≥ 0 with

g(t) = (et sin(t), 0 ≤ t < π 0, t ≥ π] is:

y(t) = - t + \(c_4\) for 0 ≤ t < πy(t) = \(c_5\) for t ≥ π.

where \(c_4\) and \(c_5\) are constants of integration.

The solution of the differential equation

y′′(t) y(t) = g(t),

y(0) = 1,

y′(0) = 1, for t ≥ 0 with

g(t) = (et sin(t), 0 ≤ t < π 0, t ≥ π] is as follows:

The given differential equation is:

y′′(t) y(t) = g(t)

We can write this in the form of a second-order linear differential equation as,

y′′(t) = g(t)/y(t)

This is a separable differential equation, so we can write it as

y′dy/dt = g(t)/y(t)

Now, integrating both sides with respect to t, we get

ln|y| = ∫g(t)/y(t) dt + \(c_1\)

Where \(c_1\) is the constant of integration.

Integrating the right-hand side by parts,

let u = 1/y and dv = g(t) dt, then we get

ln|y| = - ∫(du/dt) ∫g(t)dt dt + \(c_1\)

= - ln|y| + ∫g(t)dt + \(c_1\)

⇒ 2 ln|y| = ∫g(t)dt + \(c_2\)

Where \(c_2\) is the constant of integration.

Taking exponentials on both sides,

we get |y|² = \(e^{\int g(t)}dt\ e^{c_2\)

So we can write the solution of the differential equation as

y(t) = ±\(e^{(\int g(t)dt)/ \sqrt(e^{c_2})\)

= ±\(e^{(\int g(t)}dt\)

where the constant of integration has been absorbed into the positive/negative sign depending on the boundary condition.

Using the initial conditions, we get

y(0) = 1

⇒ ±\(e^{\int g(t)}dt\) = 1y′(0) = 1

⇒ ±\(e^{\int g(t)}dt\) dy/dt + 1 = 0

The above two equations can be used to solve for the constant of integration \(c_2\).

Using the first equation, we get

±\(e^{\intg(t)\)dt = 1

⇒ ∫g(t)dt = 0,

since g(t) = 0 for t ≥ π.

So, the first equation gives us no information.

Using the second equation, we get

±\(e^{\intg(t)}dt\) dy/dt + 1 = 0

⇒ dy/dt = - 1/\(e^{\intg(t)dt\)

Now, integrating both sides with respect to t, we get

y = \(- \int1/e^{\intg(t)\)dt dt + c₃

Where c₃ is the constant of integration.

Using the second initial condition y′(0) = 1,

we get

1 = dy/dt = - 1/\(e^{\int g(t)}\)dt

⇒ \(e^{\int g(t)}\)dt = - 1

Now, substituting this value in the above equation, we get

y = - ∫1/(-1) dt + c₃

= t + c₃

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Answer:

-27

Step-by-step explanation:

(-3)^4 / (-3)^1 = 81/-3

                    = -27

Answer:

Exact Form: 3/2,Decimal Form: 1.5 ,Mixed Number Form:1 1/2

Step-by-step explanation:

Answer:

the y-intercept is 10

Step-by-step explanation:

The area of triangle AOB is 26 square units.

To determine the area of the triangle AOB formed by the line defined by the parametric equations x = -10 - 2s and y = 8 + s, where A is the point (4, 0), O is the origin (0, 0), and B is the point (0, b), we need to find the coordinates of point B.

Let's substitute the coordinates of point B into the equations of the line to find the value of b:

x = -10 - 2s

y = 8 + s

Substituting x = 0 and y = b:

0 = -10 - 2s

b = 8 + s

From the first equation, we have:

-10 = -2s

s = 5

Substituting s = 5 into the second equation:

b = 8 + 5

b = 13

So, the coordinates of point B are (0, 13).

Now, we can calculate the area of triangle AOB using the formula for the area of a triangle given its vertices:

Area = 0.5 * |x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2)|

Substituting the coordinates of points A, O, and B:

Area = 0.5 * |4(0 - 13) + 0(13 - 0) + (-10)(0 - 0)|

     = 0.5 * |-52|

     = 26

Therefore, the area of triangle AOB is 26 square units.

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The production rate of the assembly line is 6000 parts per month.The assembly line produces 6000 parts per month, with a weekly production rate of 1500 parts.

To calculate the production rate of the assembly line, we need to consider the number of weeks in a month. Since the plant operates for 4 weeks per month, we can divide the total production by the number of weeks to find the weekly production rate.

Given that the assembly operations are sequential, we can calculate the weekly production rate as follows:

6000 parts/month ÷ 4 weeks/month = 1500 parts/week.

Therefore, the assembly line produces 1500 parts per week.

The assembly line produces 6000 parts per month, with a weekly production rate of 1500 parts. This information can be used for planning and managing production schedules efficiently.

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The original number of friends planning the party was 25.

Let's assume the total number of friends originally planning the party is 'x'.

Initially, each friend would contribute an equal amount to raise $200. So the initial contribution per friend would be $200/x.

When five friends decided not to attend, the number of friends remaining is (x - 5). Now, each friend has to contribute $2.00 more than before.

So, the new contribution per friend is $200/(x - 5) + $2.

According to the given information, the new contribution is $2.00 more than the initial contribution:

$200/(x - 5) + $2 = $200/x

To solve this equation, we can eliminate the dollar signs and simplify:

200/(x - 5) + 2 = 200/x

Multiplying both sides of the equation by x(x - 5) to eliminate the denominators:

200x + 2x(x - 5) = 200(x - 5)

200x + 2x^2 - 10x = 200x - 1000

Rearranging the equation and simplifying:

2x^2 - 10x - 1000 = 0

Dividing the equation by 2:

x^2 - 5x - 500 = 0

Using the quadratic formula, we can find the values of x:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = -5, and c = -500.

x = (-(-5) ± √((-5)^2 - 4(1)(-500))) / (2(1))

x = (5 ± √(25 + 2000)) / 2

x = (5 ± √2025) / 2

x = (5 ± 45) / 2

Simplifying further:

x1 = (5 + 45) / 2 = 50 / 2 = 25

x2 = (5 - 45) / 2 = -40 / 2 = -20

Since the number of friends cannot be negative, we discard x2 = -20 as an extraneous solution.

Therefore, the original number of friends planning the party was 25.

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