Triangle ABC~ Triangle DEF. Find the scale factor from triangle ABC to triangle DEF

To show that Sk - ku is a martingale, we need to show that it satisfies the two properties of a martingale:

Sk - ku is integrable for all k.

For any k, E[Sk+1 - ku+1 | E0,...,Ek] = Sk - ku.

First, we show that Sk - ku is integrable for all k:

E[|Sk - ku|] = E[|(X1 + ... + Xk) - k|]

= E[|X1 + ... + Xk| - k] (using linearity of expectation)

≤ E[|X1| + ... + |Xk|] + k (by triangle inequality)

= kE[|X1|] + ... + kE[|Xk|] + k (using linearity of expectation)

= kE[|X1|] + ... + kE[|X1|] + k (since Xi are IID)

= kE[|X1|] + k (since there are k terms)

< ∞ (since E[|X1|] is finite)

Therefore, Sk - ku is integrable for all k.

Next, we show that for any k, E[Sk+1 - ku+1 | E0,...,Ek] = Sk - ku:

E[Sk+1 - ku+1 | E0,...,Ek] = E[(X1 + ... + Xk+1) - (k+1) | E0,...,Ek]

= E[(X1 + ... + Xk) + Xk+1 - (k+1) | E0,...,Ek]

= E[(X1 + ... + Xk) - k | E0,...,Ek] + E[Xk+1 | E0,...,Ek]

= Sk - ku + E[Xk+1] (by the martingale property of Sk - ku)

= Sk - ku + u (since the mean of X1, X2, ... is u)

Therefore, Sk - ku is a martingale.

Now, we can use the optional stopping theorem to prove that E[ST] = uE[T]:

By the optional stopping theorem, we have:

E[ST] = E[(Sk - ku)(T - k) + Sku]

= E[(Sk - ku)(T - k)] + E[Sku]

= E[E[(Sk - ku)(T - k) | E0,...,Ek]] + E[Sku] (using law of total expectation)

= E[(Sk - ku)E[T - k | E0,...,Ek]] + E[Sku] (using linearity of expectation)

= E[(Sk - ku)(E[T | E0,...,Ek] - k)] + E[Sku] (using law of total expectation)

= E[(Sk - ku)(T - k)] + E[Sku] (since T is a bounded stopping time)

= E[SkT - kuT - kS + ku] + E[Sku]

= E[SkT] - uE[KT] - kE[S] + ku + E[Sku] (using linearity of expectation)

= uE[KT] - uE[KT] - kE[S] + ku + E[Sku] (using the optional stopping theorem)

= ku + E[S(T-k)]

= ku + E[Sk - k^2] (since T is a stopping time)

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Answer: $1,293.60

Step-by-step explanation:

It is not true that there exists a C > 0 such that for every sample S of n training examples with binary label, which are linearly separable, the hard-SVM and the soft-SVM (with parameter C) solutions will return exactly the same weight vector.

This can be shown by considering the case where n = 2, d = 1 and S = {(x1, yı), (x2, y2)}. Let a > 0 and consider x1 = a, yı = 1, X2 = –a, y2 = -1.

For the hard-SVM, the optimal solution is to find the weight vector w that maximizes the margin between the two classes while ensuring that all training examples are classified correctly. In this case, the margin is 2a and the weight vector is w = (0, 1) or w = (0, -1), depending on the choice of the positive and negative class labels.

For the soft-SVM with parameter C, the optimal solution is to find the weight vector w and the slack variables ξi that minimize the objective function:

min ||w||^2 + CΣξi
s.t. yi(wT xi) ≥ 1 - ξi, ξi ≥ 0

In this case, since the data is linearly separable, the slack variables will be zero for all training examples and the optimal solution will be the same as for the hard-SVM. However, if C is chosen to be a large value, the soft-SVM may allow some misclassifications by penalizing them with the slack variables. In this case, the optimal solution will be different from the hard-SVM.

Therefore, it is not always true that the hard-SVM and soft-SVM solutions will return exactly the same weight vector for linearly separable data, but they may be the same if the parameter C is chosen appropriately.

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Answer:

C

Step-by-step explanation:

It does not belong because like the other ones it is not side ways, as well as it's common sense.

Answer:

This graph is not a function.

The domain is -3<x<3.

The range is 1<y<3.

Gus will need to fill the bucket 3 times to completely fill the fish tank, assuming he doesn't spill any water.

To find how many times Gus will need to fill the bucket to completely fill the fish tank, we first need to calculate the volume of the bucket. The formula for the volume of a cylinder is:

V = πr²h

where V is the volume, r is the radius, and h is the height.

Substituting the given values into the formula gives:

V = 3.14 x 5² x 20

V = 3.14 x 25 x 20

V = 1570 inches³

Therefore, each time Gus fills the bucket, he will add 1570 inches³ of water to the fish tank. To calculate how many times he needs to fill the bucket to fill the entire fish tank, we can divide the total volume of the fish tank by the volume of the bucket:

4710 / 1570 = 3

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Complete question is:

Gus has a fish tank that holds 4710 inches³ of water. He is using a cylinder shaped bucket with a radius of 5 inches and a height of 20 inches to fill the tank. How many times will Gus need to fill the bucket to completely fill the fish tank if he doesn't spill a drop?

The measure of the angles are:

∠1 = 90

∠2 = 90

∠3 = 132

We have,

The angle between the tangent and the radius of a circle is 90.

So,

∠1 = 90

∠2 = 90

And,

48 + 90 + 90 + ∠3 = 360 ______(1)

Solve for ∠3 from (1).

48 + 90 + 90 + ∠3 = 360

48 + 180 + ∠3 = 360

∠3 = 360 - 228

∠3 = 132

Thus,

The measure of the angles are:

∠1 = 90

∠2 = 90

∠3 = 132

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Here are the steps to use the double number line diagram to find a percentage:

It should be noted that the double number line diagram is a visual tool that is used to represent two equivalent sets of values, and it can be useful to find percentages in a given context.

In this case, the steps are given above.

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Answer: C. 2

Step-by-step explanation: you add 2x and x to get 3x. Then add 6 to the other side to get 3x = 6. Then divide 6 by 3 to get x=2

The  scale factor of the dilation is 30:1.

We have,

A superhero action figure has a volume of 1500 cm³.

The volume of the dilated figure is 45,000 cm³

Now, the scale factor should be

= 45000/ 1500

= 30 /1

Thus, the scale factor of the dilation is 30:1.

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Answer:

2/3

Step-by-step explanation:

since a is 5 and b is 9.

so, 5+1( according to the expression given)/9

5+1/9 equals 6/9 and taking them down to the lowest terms that makes the answer 2/3

Answer:

Step-by-step explanation:

7x+2.50y=600

Sebastian can choose between 4,320 different sweaters.

to find the total number of different sweaters, we need to take the product between the numbers of options for each of the given selections, we have the numbers:

Color = 10 options.

Pattern = 4 options.

Letter type = 12 options.

Image = 9 options.

Then the total number of different options is:

D = 10*4*12*9 = 4,320

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Using the triangle proportionality theorem, the length of side PB in the given triangle is 10

From the question, we are to determine what the length of PB is in the given triangle

From the Triangle Proportionality Theorem which states that "If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally".

Thus,

From the given diagram, we can write that

AP / PB = AQ /QC

From the given information,

AP = 4

AQ = 6

QC = 15

4 / PB = 6 / 15

6 × PB = 4 × 15

6 × PB = 60

Divide both sides by 6

PB = 60 / 6

PB = 10

Hence, the length of PB is 10

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10 pounds of milk chocolate, 15 pounds of dark chocolate, and 25 pounds of almond chocolate.

We have,

A grocer sells milk chocolate at $2.60 per pound, dark chocolate at $4.80 per pound, and dark chocolate with almonds at $5.50 per pound.

Let x be the pounds of milk chocolate and y is the number of pounds of of dark chocolate.

So, x + y + (x+y)= 50

2(x+y)= 50

x+y = 25

x = 25 - y

Now, 2.6x + 4.8y + 5.5(x+y)= 50(4.71)

2.6x + 5.5x + 4.8y + 5.5y = 235.5

8.1x+ 10.3y = 235.5

8.1(25-y) + 10.3y = 235.5

202.5 - 8.1y + 10.3y = 235.5

2.2y = 33

y = 15

and, x = 10

Thus, 10 pounds of milk chocolate, 15 pounds of dark chocolate, and 25 pounds of almond chocolate.

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The calculated value of r ihe lateral area of the cylinder is 512π square units is 16 units

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

The lateral area of the cylinder is 512π square units

The lateral area of the cylinder is 2πrh

So, we have

2πrh = 512π

Substitute r for h

2πr^2 = 512π

Divide both sides by 2π

So, we have

r^2 = 256

Take the square roots

r = 16

Hence, the radius is 16 units

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The directed line segment PQ is divided at this point at a 3:1 ratio (1, 8)

According to the section formula, the coordinates of a point R may be determined by dividing a line segment PQ by that point in the ratio m:n as follows:

[tex]R_x = (n * Px + m * Qx) / (m + n)\\R_y = (n * Py + m * Qy) / (m + n)[/tex]

where [tex]P_x, P_y[/tex] are the coordinates of point P, and [tex]Q_x, Q_y[/tex] are the coordinates of point Q.

In this case, we want to divide the line segment PQ in a ratio of 3:1. This means that m = 3 and n = 1.

Using the formula, we can find the coordinates of the point R that divides PQ in a 3:1 ratio:

Rx = (1 * (-8) + 3 * 4) / (3 + 1) = 1

Ry = (1 * (-4) + 3 * 12) / (3 + 1) = 8

Therefore, the point that partitions the directed line segment PQ in a ratio of 3:1 is (1, 8).

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32 is the number of teams took less than 20minutes to complete the game

Given that Isaac measured the time it took for some teams to complete an escape room game.

The times were recorded in the table

We have to find the number of teams which are  less than 20minutes to complete the game

Number of teams took less than 20 minutes

=8+24

=32

Hence, 32 is the number of teams took less than 20minutes to complete the game.

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If u= (-10,-3) and v= (4,8) . Proj vu is  is (640/109, 192/109).

Using this formula to determine the projection of v onto u (proj vu)

proj vu = ((v dot u) / (||u||)^2) × u

Where:

v dot u = dot product of v and u

||u|| = magnitude of u

First step is to find  the dot product of v and u:

v dot u = (-104) + (-38)

v dot u = -40 - 24

v dot u = -64

Second step to to calculate the magnitude of u:

||u|| = √((-10)^2 + (-3)^2)

||u|| = √(100 + 9)

||u|| = √(109)

Let plug in the formula

proj vu = ((-64) / (√(109))^2)× (-10,-3)

proj vu = (-64 / 109) × (-10,-3)

proj vu = (640/109, 192/109)

Therefore  the correct option is A.

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Work Shown:

(A'C')/(AC) = 12/5 = 2.4

or you could say

(B'C')/(BC) = 16.8/7 = 2.4

or

(A'B')/(AB) = 9.6/4 = 2.4

The sides of triangle A'B'C' are 2.4 times longer compared to the corresponding sides of triangle ABC.

There are 125 sheets in the sale package.

If a regular package of graph paper contains 100 sheets, then a package with 25% more sheets would contain:

100 + (25/100) x 100 = 100 + 25 = 125

So a package with 25% more sheets would contain 125 sheets.

During the sale, the store is offering a package with 25% more sheets for no additional cost.

This means that the sale package contains the same number of sheets as a regular package, plus an additional 25% of that same number of sheets.

To find the number of sheets in the sale package, we can set up an equation:

Number of sheets in sale package = Number of sheets in regular package + 25% of the Number of sheets in the regular package

or

Number of sheets in sale package = 100 + 0.25 x 100

Simplifying,

Number of sheets in the sale package = 100 + 25

Number of sheets in the sale package = 125

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Yes, of course!

Ricardo has made 4 out of the 12 recipes in his cookbook.

To find out what fraction of the recipes he has made, you can write it as a fraction:

4/12

To simplify this fraction, you can divide both the numerator and denominator by 4, which gives:

4/12 = 1/3

Therefore, Ricardo has made 1/3 of the recipes in his cookbook.

The point on the number line that is 7/9 of the way from -3.6 to 10 is approximately 6.97777777778.

To find the point on the number line that is 7/9 of the way from -3.6 to 10, we can use the following formula:

Point = Starting point + (Fraction of distance) x (Total distance)

In this case, the starting point is -3.6, the total distance is the distance from -3.6 to 10 (which is 10 - (-3.6) = 13.6), and the fraction of distance is 7/9.

Plugging in these values, we get:

Point = -3.6 + (7/9) x (13.6)

Simplifying,

Point = -3.6 + (7/9) x (13.6)

Point = -3.6 + 9.9556

Point = 6.97777777778

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The line of reflection of the given image is:

A reflection across y = -3

A line of reflection is a line that lies between two identical mirror images, so the distance of any point of one figure from the line will equal the distance of the same point of the mirror image (flipped figure).

The definition above tells us that if we are given two images, such as mirror images of each other, the line of reflection can be determined by calculating the midpoint from any two points of the figures.

Looking at the given image, the center point of the mirror images is seen as y = -3

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Answer:

5.45

Step-by-step explanation:

literally do 10.9/2

The surface area of the prism is 70.44 yd²

A prism is a solid shape that is bound on all its sides by plane faces. There are different types of prism, triangular prism, rectangular prism e.t.c

The surface area of a prism is expressed as;

SA = 2B +pH

where B is the base area and p is the perimeter of the base and h is the height of the prism

Base area = 1/2 bh

= 1/2 × 3 × 3 = 4.5 yd²

The hypotenuse of the triangle = √3²+3²

= √9+9 = √18 = 4.24

perimeter of the base = 4.24+3+3 = 10.24yds

Therefore,

SA = 2×4.5 + 10.24 × 6

SA = 9 + 61.44

SA = 70.44 yd²

therefore the surface area of the prism is 70.44yd²

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The required two pairs of polar coordinates for the point (0, -10) are:

(10, π/2) and (10, -π/2)

To convert from Cartesian coordinates (x, y) to polar coordinates (r, θ), we can use the following formulas:

[tex]r = \sqrt{(x^2 + y^2)}\\\theta = \tan^{-1}(y/x)[/tex]

Since the x-coordinate of the point (0, -10) is 0, we can directly say that the angle θ is either π/2 or -π/2, depending on whether y is positive or negative.

For y = -10, we get:

[tex]r = \sqrt{(0^2 + (-10)^2)} = 10[/tex]

Therefore, two pairs of polar coordinates for the point (0, -10) are:

(10, π/2) and (10, -π/2)

Both of these pairs have the same distance from the origin (r = 10) but different angles (π/2 and -π/2).

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Answer:

Step-by-step explanation:

1

Therefore, for the equation y=x+5 , the slope is 1 , and the y-intercept is 5 .