the random variable w represents the value of the prize won for a single ticket minus the cost of the ticket. what is the expected value of w ?

After using the formula for the area of the triangle, we know that the area is 200 cm² when the height is 25 cm and the base is 16 cm respectively.

A polygon with three edges and three vertices is called a triangle. It is one of the fundamental geometric shapes.

Triangle ABC is the designation for a triangle with vertices A, B, and C. In Euclidean geometry, any three points that are not collinear produce a distinct triangle and a distinct plane.

The easiest approach I've found to know for sure if an angle is 90 degrees is to use the 3:4:5 triangle.

According to this criterion, a triangle is a right triangle if the diagonal between its two points measures 5, even though one of its sides measures 3 and the other measures 4.

So, the formula for the area of the triangle is:

A = 1/2 *  base  *  height

Now, calculate the area by inserting the values as follows:\

A = 1/2 *  base  *  height

A = 1/2 *  16 *  25

A = 8 * 25

A = 200 cm²

Therefore, after using the formula for the area of the triangle, we know that the area is 200 cm² when the height is 25 cm and the base is 16 cm respectively.

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The correct expression in the expanded form of \(log6(4(2^9^x))\) is log6(9x).

The given expression log6(4(\(2^9^x\))) represents the logarithm of the product of 4 and 2 raised to the power of 9x, all with base 6. To simplify this expression, we can apply the property of logarithms that states log(ab) = log(a) + log(b).

By using this property, we can separate the expression into two logarithms: log6(4) + log6(\(2^9^x\)). The first term, log6(4), remains unchanged.

However, the second term, log6(\(2^9^x\)), simplifies to 9x * log6(2) since the logarithm of an exponential expression with the same base results in the exponent multiplied by the logarithm of the base. Combining these terms, we obtain the expanded form expression of log6(4(\(2^9^x\))) as log6(4) + 9x * log6(2).

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Answer:

the answer is C.450 mm 2

Step-by-step explanation:

We can use this information to answer different questions, such as the probability that the total amount of impurity in 50 batches is below a certain value, or the probability that the average amount of impurity in the 50 batches is within a certain range.

The distribution of the amount of impurity in a batch of the chemical product follows a normal distribution with mean 4.0 g and standard deviation 1.5 g.

If 50 batches are independently prepared, the total amount of impurity in all 50 batches will also follow a normal distribution with mean 50*4.0 g = 200 g (the sum of the means of each batch) and standard deviation sqrt(50)*1.5 g = 3.74 g (the square root of the sum of the variances of each batch).

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Using the probability mass function, the probability of getting heads more than 12 times is 0.465.

In the given question,

The probability of landing on heads P(H)=0.455

We toss the coin 27 times. So n=27

We have to find the probability of getting heads more than 12 times.

Then, the probability mass function should be

P(X=x)=

where x=no of heads in 27 tosses.

\(\[f(x)=\begin{cases}\binom{27}{x}(0.455)^x(0.545)^{27-x},&\text{if }x=0,1,2,........,27\\0,&\text{elsewhere}\end{cases}\]\)

So the probability;

P(X>12)=1-P(x≤12)

P(X>12)=1- \(\Sigma_{x=0}^{12}\) P(X=x)

From the binomial table

P(X>12)=1-0.535

P(X>12)=0.465

Hence, the probability of getting heads more than 12 times is 0.465.

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Percentage of people who viewed version 1 and are likely to buy = 25 / 65 x 100% = 38.46%.

The way of expressing a number as a fraction of 100. It represents a proportion or rate per 100, and is often used to express changes, comparisons, and proportions in various fields such as mathematics, finance, and statistics.

The completed two-way table is:

Version 1 Version 2 Version 3 Total

Likely to 25 20 54 99

Buy    

Unsure or 40 10 21 71

Unlikely to    

Total 65 30 75 180

To find the number of people likely to buy version 2 or version 3, we add up the number of people who said they were likely to buy for those two versions:

Number of people likely to buy version 2 or version 3 = 20 + 54 = 74

To find the percentage of people who are unsure or unlikely to buy, we add up the number of people who said they were unsure or unlikely to buy, and divide by the total number of people surveyed, then multiply by 100%:

Percentage of people unsure or unlikely to buy = (40 + 10 + 21) / 180 x 100% = 38.89%

To find the percentage of people who viewed version 1 and are likely to buy, we divide the number of people who said they were likely to buy version 1 by the total number of people who viewed version 1, then multiply by 100%.

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The maximum rate of change always occurs in the direction of the gradient.

The directional derivative of a function f(x, y) in the direction of a unit vector v = ⟨a, b⟩ is defined as follows:

D_vf(x, y) = ∇f(x, y) · v

∇f(x, y) represents the gradient vector of f(x, y), which is defined as ∇f(x, y) = ⟨∂f/∂x, ∂f/∂y⟩. It is a vector that points in the direction of the steepest ascent of the function at a given point (x, y).

v = ⟨a, b⟩ is the unit vector that determines the direction in which we want to compute the derivative.

To show why the maximum rate of change of a function always occurs in the direction of the gradient, we can use the definition of the directional derivative.

Let's consider the dot product ∇f(x, y) · v, where v is a unit vector:

∇f(x, y) · v = ||∇f(x, y)|| ||v|| cosθ

In this equation, ||∇f(x, y)|| represents the magnitude (or length) of the gradient vector, ||v|| is the magnitude of the unit vector v (which is 1), and θ is the angle between ∇f(x, y) and v.

Since ||v|| = 1, the equation simplifies to:

∇f(x, y) · v = ||∇f(x, y)|| cosθ

The maximum value of the cosine function is 1, which occurs when the angle θ between the vectors is 0° (or when they are parallel). In this case, cosθ = 1, and the equation becomes:

∇f(x, y) · v = ||∇f(x, y)||

Therefore, the maximum rate of change of the function f(x, y) occurs when the angle between the gradient vector ∇f(x, y) and the direction vector v is 0°, or when they are parallel. In other words, the maximum rate of change happens when we move in the direction of the gradient.

This result is intuitive since the gradient vector points in the direction of the steepest ascent of the function. Moving in the direction of the gradient allows us to maximize the rate of change of the function at a given point.

If we were to move in a different direction, the dot product would be smaller than the magnitude of the gradient vector, resulting in a slower rate of change.

Therefore, the maximum rate of change always occurs in the direction of the gradient.

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Answer:

32+n

Step-by-step explanation:

32 more than the number n would be 32+n.

Hope this helps!

The Taylor series expansion of the function 141, centered at the point 1, is given by 141 + 141(x - 1) + 141(x - 1)^2 + 141(x - 1)^3 + ... The Taylor series expansion of cos x, centered at the point d = 2x, is given by cos(2x) - 2sin(2x)(x - 2x) + (2cos(2x)(x - 2x))^2/2! - (8sin(2x)(x - 2x))^3/3! + ...

141, centered at 1:

To find the Taylor series expansion of the function 141 centered at the point 1, we need to compute the derivatives of the function with respect to x and evaluate them at x = 1.

f(x) = 141

f'(x) = 0

f''(x) = 0

f'''(x) = 0

...

Since all the derivatives of the function are zero, the Taylor series expansion of the function 141 centered at 1 is simply the constant term 141.

Taylor series expansion of 141 centered at 1:

141

cos x, centered at 2x:

To find the Taylor series expansion of cos x centered at the point d = 2x, we need to compute the derivatives of cos x with respect to x and evaluate them at x = 2x.

f(x) = cos x

f'(x) = -sin x

f''(x) = -cos x

f'''(x) = sin x

...

Evaluating the derivatives at x = 2x:

f(2x) = cos(2x)

f'(2x) = -sin(2x)

f''(2x) = -cos(2x)

f'''(2x) = sin(2x)

...

Now we can use these derivatives to build the Taylor series expansion.

Taylor series expansion of cos x centered at 2x:

cos(2x) - 2sin(2x)(x - 2x) + (2cos(2x)(x - 2x))^2/2! - (8sin(2x)(x - 2x))^3/3! + ...

This is the Taylor series expansion of cos x centered at d = 2x.

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The possible ways to find the length of the hypotenuse of a triangle are by using the Pythagorean theorem, the midpoint formula, and the distance formula.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Given the vertices of the triangle, you can calculate the lengths of the sides using the distance formula, which measures the distance between two points in a coordinate plane. Once you have the lengths of the two sides, you can apply the Pythagorean theorem to find the length of the hypotenuse.

The midpoint formula can be used to find the midpoint of a line segment between two points. While it does not directly provide the length of the hypotenuse, it can be used to find the coordinates of the midpoint of the hypotenuse, which can then be used to calculate its length using the distance formula.

In order to calculate the slope of the hypotenuse, you would need the coordinates of two points on the hypotenuse. However, in the given information, the coordinates of the vertices are missing, so it is not possible to determine the slope.

In conclusion, the correct ways to find the length of the hypotenuse are by using the Pythagorean theorem and the distance formula, while the midpoint formula can be used to find the midpoint of the hypotenuse. However, finding the slope of the hypotenuse is not possible without additional information.

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a) We have shown that 4h - 64 = 2k is equivalent to 2h - 32 = k. b) Aaron is born on the 25th of the month and Belle is born on the 18th of the month.

In mathematics, two expressions or equations are said to be equivalent if they have the same value or solutions.

a) We are given that Aaron multiplies h by 4 and subtracts 64 to get the same result as Belle when she multiplies k by 2. Mathematically, we can write this as:

4h - 64 = 2k

We want to show that this is equivalent to 2h - 32 = k.

Starting with 4h - 64 = 2k, we can simplify it by adding 64 to both sides:

4h = 2k + 64

Dividing both sides by 2, we get:

2h = k + 32

Subtracting 32 from both sides, we get:

2h - 32 = k

Therefore, we have shown that 4h - 64 = 2k is equivalent to 2h - 32 = k.

b) We are given that Belle adds h to 5 times k and gets 115. Mathematically, we can write this as:

h + 5k = 115

We also know from part (a) that 2h - 32 = k. We can substitute this expression for k into the equation h + 5k = 115:

h + 5(2h - 32) = 115

Simplifying, we get:

h + 10h - 160 = 115

Combining like terms, we get:

11h = 275

Dividing both sides by 11, we get:

h = 25

Now we can use the expression 2h - 32 = k to solve for k:

2(25) - 32 = k

k = 18

Therefore, Aaron is born on the 25th of the month and Belle is born on the 18th of the month.

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Answer:

f(x) = -1

Step-by-step explanation:

f(x) = 5-3x

Put x = 2

f(x) = 5-3(2)

f(x) = 5-6

f(x) = -1

Answer:-1

Step-by-step explanation:

f(2)=5-3(2)

f(2)=5-6

f(2)=-1

Answer:

14. x = 42, the angles are 84°

15. x = 22, the angles are 109°

16. x = 55, the angles are 212°

17. x = 25, the angles are 125°

Step-by-step explanation:

angles across from eachother are congruent :)

Answer:

SN = √29 cm = 5.39 cm (2 dp)

Step-by-step explanation:

Using the given information, we can create a new right triangle with hypotenuse SN, then use Pythagoras' Theorem to find the length of SN.

Extend the line SU to the left until it is under point N → label this point P.  Connect point P to point N.  This creates a new right triangle (see attachment).

If BU = 4 cm, and M is the midpoint of BU, then UM = 2 cm

⇒ PN = UM = 2 cm

As NM = 2, then PU = 2 cm

⇒ PS = PU + US = 2 + 3 = 5 cm

Therefore, the two legs of the right triangle with SN as its hypotenuse are 2 cm and 5 cm.

Using Pythagoras' Theorem:

⇒ a² + b² = c²

⇒ 2² + 5² = SN²

⇒ 4 + 25 = SN²

⇒ SN² = 29

⇒ SN = √29 cm = 5.39 cm (2 dp)

In this case, since the eigenvalue λ2 = 4 is positive, the solution decays exponentially towards the origin along the line defined by the eigenvector [1; 1].

To solve the system dx/dt = [6, -2; 20, -6]x with x(0) = [-2; 2], we can find the eigenvalues and eigenvectors of the coefficient matrix [6, -2; 20, -6]. Let's denote the coefficient matrix as A.

The characteristic equation of A is given by det(A - λI) = 0, where λ is the eigenvalue and I is the identity matrix. So we have:

|6 - λ, -2|

|20, -6 - λ| = 0

Expanding the determinant, we get:

(6 - λ)(-6 - λ) - (-2)(20) = 0

(λ - 2)(λ - 4) = 0

Solving for λ, we find two eigenvalues: λ1 = 2 and λ2 = 4.

To find the corresponding eigenvectors, we substitute each eigenvalue back into the equation (A - λI)v = 0 and solve for v. Let's find the eigenvectors for each eigenvalue.

For λ1 = 2:

(A - 2I)v1 = 0

|4, -2|v1 = 0

|20, -8|v1 = 0

Simplifying, we get the equation 4v1 - 2v2 = 0, which gives us v1 = v2.

For λ2 = 4:

(A - 4I)v2 = 0

|2, -2|v2 = 0

|20, -10|v2 = 0

Simplifying, we get the equation 2v1 - 2v2 = 0, which gives us v1 = v2.

So, the eigenvectors for both eigenvalues are v = [1; 1].

Now we can express the general solution of the system as:

x(t) = c1 * e^(λ1 * t) * v1 + c2 * e^(λ2 * t) * v2

Substituting the values, we have:

x(t) = c1 * e^(2t) * [1; 1] + c2 * e^(4t) * [1; 1]

Since x(0) = [-2; 2], we can solve for the constants c1 and c2. Plugging t = 0 into the equation, we get:

[-2; 2] = c1 * e^0 * [1; 1] + c2 * e^0 * [1; 1]

[-2; 2] = c1 * [1; 1] + c2 * [1; 1]

[-2; 2] = [c1 + c2; c1 + c2]

From the first component of the vector equation, we have -2 = c1 + c2.

From the second component of the vector equation, we have 2 = c1 + c2.

Solving these equations, we find c1 = 0 and c2 = -2.

Therefore, the particular solution to the system dx/dt = [6, -2; 20, -6]x with x(0) = [-2; 2] is:

x(t) = -2 * e^(4t) * [1; 1]

The trajectory of the solution represents a line in the direction of the eigenvector [1; 1], with exponential growth/decay based on the eigenvalues.

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Answer: 1/3 books per month

Step-by-step explanation:

Th different triangles that can you make if three property of the triangle is known and one side must be known, will be one.

The polygonal shape of a triangle has a number of sides and three independent variables. Angles in the triangle add up to 180°.

To specify the triangle, atleast one of the sides must be known and any other two property like side-angle, side-side, and angle-angle.

Th different triangles that can you make if three property of the triangle is known and one side must be known, will be one.

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The trail that is closest in length to 10.5 kilometers is Spruce Trail. option D

From the information given, we have that;

The closest in length would be the trail with the lowest difference;

Oak trail = 10.653 - 10. 5 = 0. 153

Maple Trail = 10.592 - 10. 5 = 0. 092

Pine Trail = 10.732 - 10. 5 = 0. 232

Spruce Trail = 10.484 - 10. 5 = -0. 016

Thus, the trail that is closest in length to 10.5 kilometers is Spruce Trail. option D

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Answer:

All three products were already rounded to the nearest 5g.

"Algebra made Easy" 450g

“Terrible Trigonometry" 95g

Wrapping 120g

867.107844 Is the maximum Jemima might have to pay to post the parcel.

Step-by-step explanation:

510 divided by 79 = 6.455g per 510g

665g total of products so divide 665 by 510 now to get 1.30392157g per weight which brings us to the maximum total of 867.107844

The value of the expression a^3 is 3 3/8

The given parameters can be represented as

If a = 3, then 5a = 15.

Also, we have the value of the variable a to be

a = 1 1/2

Express the value of the variable a as an improper fraction

So, we have

a = 3/2

When a = 1 1/2 or a = 3/2, the value of a³ is calculated using the following formula

a³ = (1 1/2)³ = (3/2)³

Rewrite the formula as

a³ = (3/2)³

Evaluate the exponent in the above equation

So, we have

a³ = (27/8)³

Evaluate the quotient in the above equation

So, we have

a³ = 3 3/8

Hence, the value of the expression a³ is 3 3/8

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The numerical coefficient of  \(\frac{2b}{3}\)   is \(\frac{2}{3}\) . So here we know how to find numeric coefficient.

A numerical coefficient is defined as a fixed (stable) number that is multiplied to a variable.

When an algebraic expression consists of several (various) parts, each part that is added called a term. The numerical part of a term is called as its numerical coefficient (or coefficient).

Variable, It is a symbol that represents (means) a mathematical object. A variable may represent(mean) a number, a matrix, a function, a vector, the argument of a set, a function, or an element of a set.

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Answer:

obtuse

Step-by-step explanation:

It is greater than 90 degrees

Answer:

should be Obtuse

Step-by-step explanation:

right is like L. Acute is smaller than that. Obtuse is bigger than that. Straight is just a straight line.

Answer:

I think it's between +1 and +2

Step-by-step explanation:

Please follow me on brainy

Answer:

(C). {3}

Step-by-step explanation:

\(\frac{x}{2(x+1)}\) = \(\frac{-2x}{4(x+1)}\) + \(\frac{2x-3}{x+1}\) ( x ≠ - 1 )

\(\frac{x}{2(x+1)}\) - \(\frac{2x-3}{x+1}\) + \(\frac{x}{2(x+1)}\) = 0

\(\frac{2x-3}{x+1}\) - \(\frac{x}{(x+1)}\) = 0

\(\frac{x-3}{x+1}\) = 0 ⇒ x = 3