Solve the right triangle. 5
6
F
E
D
30°
Write your answers in simplified, rationalized form. Do not round. DF =
DE =
10
m∠D =
60
°

P(1) = p³,S = 1  P(2) = 3p²(1 − p),S = 2  P(3) = 3p(1 − p)²,S = 3 P(4) = (1 − p)³p,S = 4 P(5) = 3(1 − p)²p,S = 5 E(S) = 4 − 3p Var(S) = 30p(1 − p)(1 − 2p)

(a)P(s) is the probability that set s is the deciding set, and it can be expressed as follows:

• P(1) = p³ : Ivan wins the first 3 sets.

• P(2) = 3p²(1 − p) : Ivan wins two of the first three sets and the match is decided in the fourth set.

• P(3) = 3p(1 − p)² : Ivan wins two of the first three sets and the match is decided in the fifth set.

• P(4) = (1 − p)³p : Roger wins the first 3 sets.

• P(5) = 3(1 − p)²p : Roger wins two of the first three sets and the match is decided in the fourth set. (Note that P(4) and P(5) are equal.)

(b)Let S be a random variable that represents the set that decides the match. We can see that:

• S = 1 with probability P(1) = p³

• S = 2 with probability P(2) = 3p²(1 − p)

• S = 3 with probability P(3) = 3p(1 − p)²

• S = 4 with probability P(4) = (1 − p)³p

• S = 5 with probability P(5) = 3(1 − p)²p

The expected value of S, which is E(S), is given by:

E(S) = 1 x P(1) + 2 x P(2) + 3 x P(3) + 4 x P(4) + 5 x P(5)

E(S) = p³ + 6p²(1 − p) + 9p(1 − p)² + 4(1 − p)³p + 15(1 − p)²p

E(S) = 4 − 3p

This is the expected value of the set that decides the match, expressed in terms of p.

Optional: Find Var(S), which is the variance of the set that decides the match.

Var(S) = E(S²) − [E(S)]² We can see that:

S² = 1² x P(1) + 2² x P(2) + 3² x P(3) + 4² x P(4) + 5² x P(5)S² = p³ + 12p²(1 − p) + 27p(1 − p)² + 16(1 − p)³p + 75(1 − p)²p Thus, Var(S) = p³ + 12p²(1 − p) + 27p(1 − p)² + 16(1 − p)³p + 75(1 − p)²p − (4 − 3p)²

Var(S) = 30p(1 − p)(1 − 2p)

This is the variance of the set that decides the match, expressed in terms of p.

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If the correlation coefficient r is -1.0, it means that there is a perfect negative linear relationship between X and Y.

a) As X scores increase, Y scores decrease in a perfectly consistent manner. The magnitude is strong.

b) If the correlation coefficient r is +0.32, it means that there is a positive linear relationship between X and Y, but it is a weak relationship. As X scores increase, Y scores tend to increase, but not in a perfectly consistent manner. The magnitude is weak.

c) If the correlation coefficient r is -0.10, it means that there is a weak negative linear relationship between X and Y. As X scores increase, Y scores tend to decrease, but not in a consistent manner. The magnitude is weak.

d) If the correlation coefficient r is -0.71, it means that there is a strong negative linear relationship between X and Y. As X scores increase, Y scores tend to decrease in a consistent manner. The magnitude is strong.

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the solution for the initial value problem is: u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t) where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

The given partial differential equation is:

au(x, t) = 4 * (d²u(x, t) / dt²) / (dx²)

a) BCs (Boundary Conditions):

We have u(0, t) = 0 and u(π, t) = 0.

IC (Initial Condition):

We have u(x, 0) = x.

To solve this initial value problem, we need to find a function f(x) that satisfies the given boundary conditions and initial condition.

The solution for f(x) can be found using the method of separation of variables. Assuming u(x, t) = X(x) * T(t), we can rewrite the equation as:

X(x) * T'(t) = 4 * X''(x) * T(t) / a

Dividing both sides by X(x) * T(t) gives:

T'(t) / T(t) = 4 * X''(x) / (a * X(x))

Since the left side only depends on t and the right side only depends on x, both sides must be equal to a constant value, which we'll call -λ².

T'(t) / T(t) = -λ²

X''(x) / X(x) = -λ² * (a / 4)

Solving the first equation gives T(t) = C1 * exp(-λ² * t), where C1 is a constant.

Solving the second equation gives X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) + C3 * cos(sqrt(-λ² * (a / 4)) * x), where C2 and C3 are constants.

Now, applying the boundary conditions:

1) u(0, t) = 0:

Plugging in x = 0 into the solution X(x) gives C3 * cos(0) = 0, which implies C3 = 0.

2) u(π, t) = 0:

Plugging in x = π into the solution X(x) gives C2 * sin(sqrt(-λ² * (a / 4)) * π) = 0. To satisfy this condition, we need the sine term to be zero, which means sqrt(-λ² * (a / 4)) * π = n * π, where n is an integer. Solving for λ, we get λ = ± sqrt(-4n² / a), where n is a non-zero integer.

Now, let's find the expression for u(x, t) using the initial condition:

u(x, 0) = X(x) * T(0) = x

Plugging in t = 0 and X(x) = C2 * sin(sqrt(-λ² * (a / 4)) * x) into the equation above, we get:

C2 * sin(sqrt(-λ² * (a / 4)) * x) * C1 = x

This implies C2 * C1 = 1, so we can choose C1 = 1 and C2 = 1.

Therefore, the solution for the initial value problem is:

u(x, t) = sin(sqrt(-λ² * (a / 4)) * x) * exp(-λ² * t)

where λ = ± sqrt(-4n² / a), and n is a non-zero integer.

Note: Please double-check the provided equation and ensure the values of a and the given boundary conditions are correctly represented in the equation.

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Answer:

Step-by-step explanation:

2) What type of angle is 4 and 8?

I What is the special angle pair relationship between < 8 and < 4?

Angle 4 and angle 8 are also alternate interior angles. Alternate exterior angles: Pairs of exterior angles on opposite sides of the transversal. Angle 2 and angle 7 are alternate exterior angles.

3)Twos are generally thoughtful, people-oriented, and caring, while Threes are resourceful, energetic, and determined.

4) Do not use a protractor. Use the properties of straight and vertical angles to help you.

b =(17,6x 8,9):19,7=7,95

Answer:

Area = base x height

Step-by-step explanation:

8X11=88sq

Answer:

88 units^2

Step-by-step explanation:

The area of a parallelogram can be found using he following formula:

a=b*h

where b is the base and h is the height.

We know that 11 units is the base and 8 units is the height (forms a right angle with one of the bases).

b= 11 units

h= 8 units

Substitute these values into the formula.

a= 11 units * 8 units

Multiply

a= 88 units^2

The area of the parallelogram is 88 square units.

Step-by-step explanation:

To find the angle between a line and the x-axis, we need to calculate the slope of the line first. The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:

m = (y₂ - y₁) / (x₂ - x₁)

Using the points (-2, 2) and (8, 9), we can substitute the values into the formula:

m = (9 - 2) / (8 - (-2))

= 7 / 10

= 0.7

The slope of the line is 0.7. The angle between the line and the x-axis can be found using the inverse tangent (arctan) function. The tangent of an angle is equal to the slope of the line, so we can write:

tan(θ) = 0.7

Taking the inverse tangent of both sides, we find:

θ = arctan(0.7)

Using a calculator, we can evaluate this to be approximately:

θ ≈ 35.87 degrees

Therefore, the angle between the line and the x-axis is approximately 35.87 degrees.

Answer:

Barry is 71 inches tall

Step-by-step explanation:

52 + 19 = 71

The minutes it would take Kari to get to the comic store is 15 minutes.

The average speed is the total distance per time. It is determined by dividing the total distance travelled by the total time travelled.

Average speed = total distance / total time

Kari's average speed is 12 miles per hour.

In order to determine the time it would take Kari to the comic store, divide the total distance by the average speed.

Division is the process used to determine the quotient of two or more numbers. Division entails grouping a number into equal groups using another number. The sign used to denote division is ÷

Time = total distance / average speed

3 /12 = 1/4 hour

1/4 x 60 = 15 minutes

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The kinematics equation s = 1/2at² can be used to determine the acceleration in the Second Law Experiment, where acceleration is calculated by measuring the time taken for the cart to move from the start point to the end point.

Let's derive how the equation is used to find the acceleration from this experiment:

Drivation of kinematics equation:

The kinematics equation is derived by integrating the acceleration function twice with respect to time (t).

Acceleration, a = d²s/dt², where s is displacement, and d²s is the second derivative of displacement with respect to time.

taking integral w.r.t time twice:

∫ a dt = ∫ d²s/dt² dt integrating once gives, v = at + C₁ (where C₁ is a constant of integration)

integrating twice gives, s = 1/2at² + C₁t + C₂ (where C₂ is a constant of integration)

Thus, the kinematics equation for an object moving with constant acceleration can be represented by

s = 1/2at² + vit + s₀,

where s is the displacement, a is the acceleration, t is time, vi is initial velocity, and s₀ is initial displacement.

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Answer:

349.94

Step-by-step explanation:

First u add all the square sides because they all are the same which gives u 196 then u can see there is two half circles which if they add up will be a circle and the diameter is the same as the square sides then if the diameter is 14 then the radius will be 7 becaue 14/2 therefore u use the area of the circle which is

\(\pi \times {7}^{2} \)

then u add all the numbers.

Answer:

4 books

Step-by-step explanation:

We are told that: Hugh bought some magazines that cost 3.95 each

Hugh bought 3 magazines.

Hence, total cost of magazines = 3 magazines × 3.95

= 11.85.

Total cost of magazines = 11.85

From the question, we know that, the total amount Hugh spent = 47.65

Hence, The total cost for books =

Total amount spent - Total cost of magazines

= 47.65 - 11.85

= 35.80

Therefore, the total cost of books = 35.80

We are told in the question that the cost of books is 8.95 each or 8.95 per book.

Hence, the number of books that Hugh bought is calculated as :

35.80/8.95

= 4 books

Therefore, Hugh bought 4 books

The initial number of white tokens in the bag is given as follows:

8.

The amount is obtained by a system of equations, for which the variables are given as follows:

A bag contains black and white tokens in a ratio of 3:2, hence:

x/y = 3/2.

2x = 3y

x = 1.5y.

8 black tickets and 2 white tokens are added to the bag. The ratio of black to white tokens are now 2:1, hence:

(x + 8)/(y + 2) = 2.

Replacing the first equation into the second, the initial amount of white tokens is given as follows:

(1.5y + 8)/(y + 2) = 2

2(y + 2) = 1.5y + 8

2y + 4 = 1.5y + 8

0.5y = 4

y = 8.

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Using it's concept, the scatter plot that accurately reflects the data given in the problem is scatter plot D.

A scatter plot is a graph that maps each input in the data-set to it's respective output, as points.

Considering this concept, an age of 12(x - axis) should be mapped to a time in minutes of 23(y-axis), and so on, until an age of 71 is mapped to a time of 33 minutes, which happens in option D.

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The p-value for the test is found as 0.05 for the given hypothesis.

Given,Sample mean = 59.1 km

Sample standard deviation = 2.31 km

Population is normally distributed

P-value for the test is to be determined.

To find the p-value, we need to perform a hypothesis test. Here, we have to test whether the null hypothesis is true or not.

Hypothesis statements:

Null hypothesis (H0): µ = 60 km (The population mean is 60 km)

Alternative hypothesis (Ha): µ ≠ 60 km (The population mean is not equal to 60 km)

Level of significance, α = 0.05

Z-score formula is given as,Z = (x - µ) / (σ/√n)

Where,x = Sample mean = 59.1 km

µ = Population mean

σ = Standard deviation of the population = 2.31 km

n = Sample size

We have,σ/√n = 2.31/√n

For α = 0.05, the two-tailed critical values are ±1.96

Now, the calculated Z-score is given as,

Z = (59.1 - 60) / (2.31/√n)

Z = - (0.9) * ( √n / 2.31)

P(Z < -1.96) = 0.025 and P(Z > 1.96) = 0.025

P-value = P(Z < -1.96) + P(Z > 1.96)

P-value = 0.025 + 0.025

P-value = 0.05

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Answer:

4/5 or 0.8 in decimal form

Step-by-step explanation:

slope is rise/run

4/5 as it goes 4 up and 5 to the right

Answer: table 2

Step-by-step explanation:

its because the rest of the tables do not have a pattern that ‘makes sense’ unlike table 2. for table 2, for every one 1 km there’s 10$

Answer:  137.25π ≈ 431.18 cm³

Step-by-step explanation:

Entire Cone - Tip of Cone = Frustrum

\(V_{cone}=\dfrac{1}{3}\pi r^2h\\\\\\V_{Entire}=\dfrac{1}{3}\pi (6)^2(12)\quad =\quad 144\pi\\\\V_{Tip}=\dfrac{1}{3}\pi \bigg(\dfrac{6}{4}\bigg)^2\bigg(\dfrac{12}{4}\bigg)\quad =\quad \dfrac{27}{4}\pi\\\\\\\\V_{Frustrum}=V_{Entire}-V_{Tip}\\\\.\qquad \qquad =144\pi-\dfrac{27}{4}\pi\\\\.\qquad \qquad =\dfrac{549}{4}\pi\\\\.\qquad \qquad =137.25\pi\\\\.\qquad \qquad=\large\boxed{431.18}\)

Answer: 137.25π ≈ 431.18 cm³

Step-by-step explanation:

Entire Cone - Tip of Cone = Frustrum

Answer:

x=9

Step-by-step explanation:

pythagorean theorem

\(a^2+b^2=c^2\\a=6 \\b=x\\c=\sqrt{117} \\6^2+x^2=\sqrt{117} ^2\\36+x^2=117\\x^2=81\\\sqrt{x^2} =\sqrt{81} \\x=9\)

The value for x is 7π/6 and 11π/6.

sin2xsecx+2cosx=0

(2sinxcosx)(1/cosx)+2cosx=0

sinxcosx/cosx+2cosx=0

So,

2sinx+2cosx=0

2(sinx+cosx)=0

(2(sinx+cosx))²=0

4(sin²x+cos²x+2sinxcosx)=0

hence,

(sin²x+cos²x+2sinxcosx)/4=0/4

sin2x+cos2x+2sinxcosx=0

1+sin2x=0

sinx=−1/2

=7π/6 and 11π/6

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Answer:

His answer was correct:

(200*0.09)=18

18*16=288

200+288=488

Answer:

4 pencils and 7 erasers

Step-by-step explanation:

36÷9=4

63÷9=7

Answer:

4 pencils and 7 erasers

Step-by-step explanation:

Each student will get: 36:9=4 (pencils)

                                    63:9=7 (erasers)

The fully expanded and simplified form of (x - 2)(2x + 4) is 2(x - 2)(x + 2).

Expanding a bracket means multiplying the terms inside the bracket by the term(s) outside the bracket. In the case of (x - 2)(2x + 4), we can use the FOIL method to expand the brackets:

F: Multiply the First terms in each bracket, which gives us x * 2x = 2x²

O: Multiply the Outer terms in each bracket, which gives us x * 4 = 4x

I: Multiply the Inner terms in each bracket, which gives us -2 * 2x = -4x

L: Multiply the Last terms in each bracket, which gives us -2 * 4 = -8

Putting these results together, we get:

(x - 2)(2x + 4) = 2x² + 4x - 4x - 8

Notice that the -4x and +4x terms cancel out, so we're left with:

(x - 2)(2x + 4) = 2x² - 8

So the expanded form of (x - 2)(2x + 4) is 2x² - 8.

To simplify this expression further, we can factor out a 2:

2x² - 8 = 2(x² - 4)

And then factor the quadratic expression inside the parentheses:

2(x² - 4) = 2(x - 2)(x + 2)

So the fully expanded and simplified form of (x - 2)(2x + 4) is 2(x - 2)(x + 2).

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sin(A)=2/5

trig identity:

* sin^2(A)+cos^2(A)=1

* tan(A)=sin(A)/cos(A)

find: tan(A)

Answer:

32/4 = 8 \(\frac{1}{2}\) = 8.5

Step-by-step explanation:

34/4 = 8 \(\frac{2}{4}\)

8 \(\frac{2}{4}\) = 8 \(\frac{1}{2}\) = 8.5

Answer:

Below

Step-by-step explanation:

34 ÷ 4 =  34/ 4 =  8 2/4 = 8 1/2     <===pick the one you need

The sine function is  y = 5 sin (\(16\pi x\) ) + 3.

The function y =\(sin x\) defines a sine wave as a geometric waveform that oscillates (moves up, down, or side to side) frequently.

It is an s-shaped, smooth wave that oscillates above and below zero, to put it another way.

In its most general form, the sine wave can be described using the function y=\(a sin (bx)\)

Given:

Amplitude = 3

midline= 5

As, we know

General sine function:  a sin b\((x- c)\) + d

and, amplitude = absolute value of a

b = 2π / period

c = phase shift (horizontal shift)

d = vertical shift  (i/e., midline is  y = d)

So,

a = 5, d = 3, and b = 2π / (1/8) = 16π, c = 0

Substituting the above value in Sine function we get

Thus, y = 5 sin (\(16\pi x\)) + 3

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Answer:

Smaller Trapezoid: 12

Bigger Paralleogram: 24

Bigger Triangle: 22/40

Smaller Triangle : 2.75/ 5

Step-by-step explanation:

cross multiplication

Bigger Paralleogram: 9 x 16= 144 / 6= 24

Bigger Triangle: 5 x 22= 110 / 2.75 = 40

hope this helps

The new temperature would be 8 when the temperature is -4 and rises by 12.

Addition is a fundamental mathematical operation that is used to join two or more numbers or quantities. It entails calculating the sum of two or more values.

The sign "+" represents the procedure.

For example, adding 2 and 3 yields a total of 5, which is expressed as:

2 + 3 = 5

As per the question, If the temperature starts at -4 and rises by 12, we need to add 12 to the starting temperature to find the new temperature.

So, the new temperature would be:

-4 + 12 = 8

Therefore, the new temperature would be 8.

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Answer:

To compare who saves the most of their salary each month among Amy, John, and Emily, we need to calculate the amount of salary each person saves.

Let's assume that the monthly salary of each person is 'S'. Then we can calculate the amount saved by each person as follows:

Amy:

Amount saved = 20% of S = 0.2S

Amount spent = S - 0.2S = 0.8S

John:

Amount spent = 2/5 of S = (2/5)S

Amount saved = S - (2/5)S = (3/5)S

Emily:

Let's assume that Emily saves '5x' and spends '8x' of her monthly salary.

Then, according to the question, we have:

Amount saved = 5x

Amount spent = 8x

We know that the ratio of the amount saved to the amount spent is 5:8, so we can write:

Amount saved / Amount spent = 5/8

Substituting the values of amount saved and amount spent, we get:

5x / 8x = 5/8

5x = (5/8) x 8x

5x = 5x

Therefore, the ratio of amount saved to amount spent is equal to 5:8. This means that Emily saves 5/13 of her monthly salary and spends 8/13 of her monthly salary.

So, the amount saved by each person is:

Amy: 0.2S

John: (3/5)S

Emily: 5/13 of S

Now, we need to compare these amounts to find out who saves the most.

To compare these amounts, we can write them in terms of a common denominator:

Amy: 0.2S

John: (3/5)S = (0.6)S

Emily: (5/13)S = (0.3846)S (approx.)

Therefore, we see that John saves the most of his salary each month, followed by Amy and then Emily.

Working out:

Let's assume that each person earns $1000 per month.

Amy:

Amount saved = 20% of $1000 = $200

Amount spent = $800

John:

Amount spent = 2/5 of $1000 = $400

Amount saved = $1000 - $400 = $600

Emily:

Let's assume that Emily saves $5x and spends $8x of her monthly salary.

Then, we have:

Amount saved = $5x

Amount spent = $8x

We know that the ratio of the amount saved to the amount spent is 5:8, so we can write:

$5x / $8x = 5/8

Solving for x, we get:

x = 8/13

Substituting the value of x, we get:

Amount saved = $5 x (8/13) x $1000 = $384.62 (approx.)

Amount spent = $8 x (8/13) x $1000 = $615.38 (approx.)

Therefore, we see that John saves the most of his salary each month, followed by Amy and then Emily.