help please its due in 2 hrs

Answer:

The marginal pdf of X is fX(x) = x + 1/2

We need to integrate the joint pdf over the given region. This can be done as follows:

P(X < 1/2, Y > 1/4) = ∫∫[x + y] dx dy over the region 0 ≤ x ≤ 1/2 and 1/4 ≤ y ≤ 1

= ∫[x + y] dy from y = 1/4 to 1 ∫ dx from x = 0 to 1/2

= ∫[x + y] dy from y = 1/4 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1 (1/2 - 0)

= ∫[x + y] dy from y = 1/4 to 1/2 + ∫[x + y] dy from y = 1/2 to 1/2

= ∫[x + y] dy from y = 1/4 to 1/2

= [(x + y)y] evaluated at y = 1/4 and y = 1/2

= [(x + 1/2)(1/2) - (x + 1/4)(1/4)]

= (1/2 - 1/4)(1/2) - (1/4 - 1/8)(1/4)

= (1/4)(1/2) - (1/8)(1/4)

= 1/8 - 1/32

= 3/32

Therefore, P(X < 1/2, Y > 1/4) = 3/32.

The marginal pdfs of X and Y can be done as follows:

For the marginal pdf of X:

fX(x) = ∫[x + y] dy over the range 0 ≤ y ≤ 1

= [xy + (1/2)y^2] evaluated at y = 0 and y = 1

= (x)(1) + (1/2)(1)^2 - (x)(0) - (1/2)(0)^2

= x + 1/2

Therefore, the marginal pdf of X is fX(x) = x + 1/2.

For the marginal pdf of Y:

fY(y) = ∫[x + y] dx over the range 0 ≤ x ≤ 1

= [xy + (1/2)x^2] evaluated at x = 0 and x = 1

= (y)(1) + (1/2)(1)^2 - (y)(0) - (1/2)(0)^2

= y + 1/2

Therefore, the marginal pdf of Y is fY(y) = y + 1/2.

To determine if X and Y are independent, we need to check if the joint pdf factors into the product of the marginal pdfs.

fX(x) * fY(y) = (x + 1/2)(y + 1/2)

However, this is not equal to the joint pdf f(x, y) = x + y. Therefore, X and Y are not independent.

To derive the conditional pdf of X given Y = y, we can use the formula:

f(xy) = f(x, y) / fY(y)

Here, we have f(x, y) = x + y from the joint pdf, and fY(y) = y + 1/2 from the marginal pdf of Y.

Therefore, the conditional pdf of X given Y = y is:

f(xy) = (x + y) / (y + 1/2)

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The incorrect phrases in the paragraph are as follows:

Since ∠AMC is a right angle,  ∠BMC and  ∠CMD are complementary.

Since  ∠BMD is a right angle,  ∠AMB and  ∠BMC are complementary.

To identify the wrong phrases, note that a right angle is an angle that forms 90° when inclined to another line. Also, complementary angles are those whose sum is 90°.

A close look at the angles will show that ∠BMD is a right angle but ∠BMC and  ∠CMD are complementary. Also, the following is correct: ∠AMC is a right angle,  ∠AMB and  ∠BMC are complementary.

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Answer:

? = 92°

Step-by-step explanation:

the chord- chord angle ? is half the sum of the measures of the arcs intercepted by the angle and its vertical angle, that is

? = [tex]\frac{1}{2}[/tex] (LM + AK) = [tex]\frac{1}{2}[/tex] (120 + 64)° = [tex]\frac{1}{2}[/tex] × 184° = 92°

Answer:

The sixth angle is 115°.

Step-by-step explanation:

Number of sides in a hexagon = n =6

Sum of interior angles = (n−2)180°

= (6−2)180 ∘

= 720

∴ Let the six angle of hexagon be x.

⇒ x + 123 + 124 + 118 + 130 + 110 = 720°

⇒ x + 605 = 720°

⇒ x = 720 - 605

⇒ x = 115

The examples match with the following properties:

1. C. Distributive Property

2. A. Commutative Property

3. B. Associative Property

4. B. Associative Property

The correct matching of the examples on the left with the corresponding properties on the right is as follows:

1. 3(x + 3) = 3x + 9 : C. Distributive Property

  This example demonstrates the distributive property, where we distribute the factor 3 to both terms inside the parentheses, resulting in 3 times x and 3 times 3, which simplifies to 3x + 9.

2. 2 + 3 + 4 = 4 + 3 + 2 : A. Commutative Property

  This example illustrates the commutative property of addition, which states that the order of adding numbers does not affect the sum. Here, the numbers are rearranged on both sides of the equation, but the sum remains the same.

3. 4(2 x 3) = (4 x 2)3 : B. Associative Property

  This example showcases the associative property of multiplication, where the grouping of factors does not affect the product. The expression on the left side is calculated as 4 times the product of 2 and 3, while the expression on the right side is calculated as the product of 4 and 2, followed by multiplying the result by 3. Both calculations yield the same result.

4. 6 + (7 + x) = (6 + 7) + x : B. Associative Property

  This example also demonstrates the associative property, but in the context of addition. The expression on the left side associates the addition of 7 and x first, while the expression on the right side associates the addition of 6 and 7 first. Both expressions ultimately add up to the same sum.

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The question probable may be:

Match the example on the left with the corresponding property on the right.

1. 3(x + 3) = 3x + 9

2. 2 + 3 + 4 = 4 + 3 +2.

3. 4(2 x 3) = (4 x 2)3

4. 6 + (7 + x) = (6 + 7) + x

A. Commutative Property

B. Associative Property

C. Distributive Property

Answer:

The table shows that there are a total of 142 books on Diana's bookshelf, with 77 books written by female authors and the rest by male authors. However, the number of non-fiction books written by female authors is missing from the table, so it is impossible to determine the exact number without more information.

Step-by-step explanation:

Answer:

104 books

Step-by-step explanation:

                 Fiction         Non-Fiction      Total

Male              36                     38

Female                                    x

Total              77                     142

----------------------------------------------------------------------------

x = 142 - 38 = 104

The volume of a cylinder is given by the equation presented as follows:

V = πr²h.

In which:

The volume of a cylinder of radius r and height h is given by the equation presented as follows:

V = πr²h.

Hence the measures must be identified and have it's values replaced into the formula to obtain the volume of a cylinder.

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The equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. option B.

The equation y = ax² represents a parabola with its vertex at the origin. In this case, if the coefficient 'a' is negative, it determines the direction in which the parabola opens.

When 'a' is negative, the parabola opens downward. This means that the vertex, which is at the origin (0, 0), represents the highest point on the graph, and the parabola curves downward on both sides.

To understand this concept, let's consider the basic equation y = x², which represents a standard upward-opening parabola. As 'a' increases, the parabola becomes narrower. Conversely, when 'a' becomes negative, it flips the parabola upside down, resulting in a downward-opening parabola.

For example, if we have the equation y = -x², the negative coefficient causes the parabola to open downward. The vertex remains at the origin, but the shape of the parabola is now inverted.

In summary, when the equation of a parabola with its vertex at the origin includes a negative coefficient 'a', the parabola opens downward. This can be visually represented as a U-shape curving downward from the origin. So Optyion B is correct.

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The system has no solution. Option C is correct.

To solve the given system of equations using row operations, we can write the augmented matrix and perform Gaussian elimination. The augmented matrix for the system is:

1  1 -1 |  6

3 -1  1 |  2

1  4  2 | -34

We'll use row operations to transform the augmented matrix into row-echelon form or reduced row-echelon form. Let's proceed with the row operations:

R2 = R2 - 3R1:

1  1 -1 |  6

0 -4  4 | -16

1  4  2 | -34

R3 = R3 - R1:

1  1 -1 |  6

0 -4  4 | -16

0  3  3 | -40

R3 = R3 + (4/3)R2:

1  1 -1 |  6

0 -4  4 | -16

0  0  0 |  -4

Now, we can rewrite the augmented matrix in equation form:

x +  y -  z =  6

      -4y + 4z = -16

              0 = -4

From the last equation, we can see that it leads to a contradiction (0 = -4), which means the system is inconsistent. Therefore, the system has no solution.

The correct answer is (C) This system has no solution.

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The property that is illustrated by the statements is Transitive. Option C

Using the principle of transitivity, if two objects are equal to a third, they are also equal to one another.

From the information given, we have that;

< ABC = <DEF

< DEF = < XYZ

This simply proves that < ABC and < XYZ are both equivalent to < DEF in this situation.

By using the transitive property, we can determine that A ABC and A XYZ are also equal. This attribute enables us to construct relationships between many elements based on their equality to a shared third element and to connect logically equalities.

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Answer:

-2(3x + 12y - 5 - 17x - 16y + 4)

-6x - 24y + 10 + 34x + 32y - 8

-6x + 34x - 24y + 32y + 10 - 8

28x + 8y + 2

The distance around the shape after the smaller semicircle is removed is 29 cm.The correct answer is option D.

To find the distance around the shape after the smaller semicircle is removed, we need to calculate the circumference of the larger semicircle and subtract the circumference of the smaller semicircle.

The circumference of a semicircle is given by the formula:

Circumference = π * radius + diameter/2

For the larger semicircle:

Radius = diameter/2 = 28/2 = 14 cm

Circumference of the larger semicircle = π * 14 + 28/2 = 22/7 * 14 + 14 = 44 + 14 = 58 cm

For the smaller semicircle:

Radius = diameter/2 = 14/2 = 7 cm

Circumference of the smaller semicircle = π * 7 + 14/2 = 22/7 * 7 + 7 = 22 + 7 = 29 cm

Therefore, the distance around the shape after the smaller semicircle is removed is:

58 cm - 29 cm = 29 cm

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The Probable question may be:
A metalworker cuts out a large semicircle with a diameter of 28 centimeters. Then the metalworker cuts a smaller semicircle out of the larger one and removes it. The diameter of the semicircular piece that is removed is 14 centimeters. What will be the distance around the shape after the smaller semicircle is removed? Use 22/7​ as an approximation for π.

A. 80cm

B. 82cm

C. 85cm

D. 86cm

Answer: [-2,7]

Step-by-step explanation: The range of a graph is just the range of minimum and maximum outputs of the y axis. The minimum y axis value is -2, while the maximum is 7. Putting your answer in brackets means that the endpoints (-2 and 7) are inclusive, which is the case since the dots are filled in. If the dots are hollow, the range does not include those endpoints and you would use parentheses instead.

[tex]\displaystyle \text{To solve this problem, let's assume the distance of the run is denoted by 'x' miles, and the distance of the bicycle race is denoted by '48 - x' miles.}[/tex]

[tex]\displaystyle \text{We can use the formula: time} = \text{distance/velocity to find the time taken for each segment of the race.}[/tex]

[tex]\displaystyle \text{For the run:}[/tex]

[tex]\displaystyle \text{Time taken} = \text{Distance/Velocity}[/tex]

[tex]\displaystyle t_1 = \frac{x}{5}[/tex]

[tex]\displaystyle \text{For the bicycle race:}[/tex]

[tex]\displaystyle \text{Time taken} = \text{Distance/Velocity}[/tex]

[tex]\displaystyle t_2 = \frac{48 - x}{23}[/tex]

[tex]\displaystyle \text{Given that the total time for the race is 6 hours, we can write the equation:}[/tex]

[tex]\displaystyle t_1 + t_2 = 6[/tex]

[tex]\displaystyle \text{Substituting the expressions for } t_1 \text{ and } t_2, \text{ we get:}[/tex]

[tex]\displaystyle \frac{x}{5} + \frac{48 - x}{23} = 6[/tex]

[tex]\displaystyle \text{To solve this equation, we can simplify it by multiplying through by the common denominator, which is 115:}[/tex]

[tex]\displaystyle 23x + 5(48 - x) = 6 \times 115[/tex]

[tex]\displaystyle \text{Simplifying further:}[/tex]

[tex]\displaystyle 23x + 240 - 5x = 690[/tex]

[tex]\displaystyle 18x = 450[/tex]

[tex]\displaystyle x = \frac{450}{18}[/tex]

[tex]\displaystyle x = 25[/tex]

[tex]\displaystyle \text{Therefore, the distance of the run is 25 miles, and the distance of the bicycle race is } 48 - 25 = 23 \text{ miles.}[/tex]

The vertex for the graph of the equation [tex]v - 3 = - (x+2)^2 is (-2, 3).[/tex]

To find the vertex of the graph of the equation [tex]v - 3 = - (x+2)^2,[/tex] we can rewrite it in the standard vertex form: [tex]v = a(x - h)^2 + k,[/tex]

where (h, k) represents the vertex coordinates.

First, let's rearrange the given equation:

[tex]v - 3 = - (x+2)^2[/tex]

[tex]v = - (x+2)^2 + 3[/tex]

Comparing this with the standard vertex form, we can see that h = -2 and k = 3.

Therefore, the vertex of the graph is (-2, 3).

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The correct linear inequality represented by the graph is:

y < -2x + 3. Option D

To determine which linear inequality is represented by the graph of the dashed straight line with a negative slope and going through (0, 3) and (2, -1), we can start by finding the slope of the line.

The slope of a line can be calculated using the formula:

m = (y2 - y1) / (x2 - x1).

Using the coordinates (0, 3) and (2, -1), we have:

m = (-1 - 3) / (2 - 0),

m = -4 / 2,

m = -2.

So, we know that the slope of the line is -2.

Next, we need to determine the y-intercept of the line. To do this, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.

Using the point (0, 3), we can substitute the coordinates into the equation and solve for b:

3 = -2(0) + b,

3 = b.

Therefore, the y-intercept is 3.

Now that we have the slope and y-intercept, we can write the equation of the line in slope-intercept form:

y = -2x + 3.

Since we are shading everything to the left of the line, we want the region where y is less than the line. Option D is correct.

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Answer:

Step-by-step explanation:

Given x=3, [tex]\sqrt{2}[/tex], and -4i

y= (x-3)([tex]x^{2}[/tex]-2)([tex]x^{2}[/tex]+16)

Answer:

x^4 - 3x^3 - 16√2x^2 + (16√2 + 16)x - 48 = 0

Step-by-step explanation:

If the roots of a polynomial equation are 3, √2 and -4i, then the factors of that polynomial are (x - 3), (x - √2) and (x + 4i), since each factor represents one of the roots.

However, since -4i is a complex number, its conjugate 4i is also a root of the polynomial. So we also need the factor (x - 4i).

Thus, the polynomial equation is:

(x - 3) (x - √2) (x + 4i) (x - 4i) = 0

To simplify this equation, we can use the fact that (a + bi)(a - bi) = a^2 - b^2i^2 = a^2 + b^2:

(x - 3) (x - √2) (x^2 + 16) = 0

Expanding this equation yields:

x^4 - 3x^3 + 16x - 16√2x^2 + 48√2x - 48 = 0

So the polynomial equation with roots 3, √2, and -4i is:

x^4 - 3x^3 - 16√2x^2 + (16√2 + 16)x - 48 = 0

Answer:

We have to multiply everything to find the amount of different possible outcomes so 3*7*8*4 = 672 unique outfits

Step-by-step explanation:

Hope this helps!

Using Gauss-Jordan method, the value of x, y and z are 0, -2 and 1

To solve the system of equations using the Gauss-Jordan method, we'll perform row operations to transform the augmented matrix into row-echelon form and then further transform it into reduced row-echelon form. Here are the steps:

1. Write the augmented matrix for the system of equations:

  [1  -2   4  |  20]

  [1   1  13  | -31]

  [-2   6  -1 | -69]

2. Perform row operations to transform the matrix into row-echelon form:

  R2 = R2 - R1

  R3 = R3 + 2R1

  [1  -2   4  |  20]

  [0   3   9  | -51]

  [0   2   7  | -29]

3. Perform row operations to further transform the matrix into reduced row-echelon form:

  R2 = R2 / 3

  R3 = R3 - 2R2

  [1  -2   4  |  20]

  [0   1   3  | -17]

  [0   0   1  |   1]

4. Perform row operations to obtain a diagonal of 1s from left to right:

  R1 = R1 + 2R2 - 4R3

  R2 = R2 - 3R3

  [1  0  0  |   0]

  [0  1  0  |  -2]

  [0  0  1  |   1]

The resulting matrix corresponds to the system of equations:

x = 0

y = -2

z = 1

Therefore, the solution to the given system of equations is x = 0, y = -2, and z = 1.

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The volume of the square based pyramid would be =60cm³.

To calculate the volume of a square based pyramid, the formula that should be used would be given below as follows;

Volume = 1/3× base²× height

where base length = 6cm

height = 5cm

Volume = 1/3× 6×6×5

= 60cm³

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Answer:

15

Step-by-step explanation:

5x = 4x + 3

x = 3

BC = 5x = 5(3) = 15

Answer: 15

The correct statement is D. Hannah's unit rate is 1 fewer word per minute than Becky's unit rate.

To compare the unit rates, we need to determine the rate at which each person types words per minute.

For Hannah, the equation y = 11x represents her typing speed, where y is the number of words and x is the number of minutes. This means that Hannah types 11 words per minute (11 words/minute).

Looking at Becky's graph, we can determine her unit rate by calculating the change in the number of words divided by the change in time.

The change in words is 48 - 2 = 46, and the change in time is 3 - 2 = 1. So, Becky's unit rate is 46 words per minute (46 words/minute).

Comparing the unit rates:

Hannah's unit rate: 11 words/minute

Becky's unit rate: 46 words/minute

Therefore, Hannah's unit rate is 35 words per minute less than Becky's unit rate.

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The exact value of sec(-135°) is 1.

To find the exact value of sec(-135°), we need to use the relationship between secant and cosine functions.

The secant function is defined as the reciprocal of the cosine function:

sec(theta) = 1 / cos(theta).

We know that the cosine function has a period of 360°, which means that cos(theta) = cos(theta + 360°) for any angle theta.

In this case, we want to find sec(-135°). Since the cosine function is an even function (cos(-theta) = cos(theta)), we can rewrite sec(-135°) as sec(135°).

Now, let's focus on finding the value of cos(135°). The cosine function is negative in the second and third quadrants.

In the second quadrant, the reference angle is 180° - 135° = 45°. The cosine of 45° is equal to √2/2.

Therefore, cos(135°) = -√2/2.

Now, we can find sec(135°) using the reciprocal property:

sec(135°) = 1 / cos(135°).

Substituting the value of cos(135°), we have:

sec(135°) = 1 / (-√2/2).

To simplify further, we multiply the numerator and denominator by -2/√2:

sec(135°) = -2 / (√2 * -2/√2).

Simplifying the expression:

sec(135°) = -2 / -2,

sec(135°) = 1.

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Answer:

a) 50

Step-by-step explanation:

The variance will not change as all the observations are increased uniformly.

Proof:

Variance formula:

[tex]s^{2} = \frac{\sum x_i^{2} }{n} -\frac{(\sum x_i)^{2} }{n^{2} }[/tex]

When the obervations are inc by 20,

[tex]s_1^{2} = \frac{\sum (x_i + 20)^{2} }{n} -\frac{(\sum (x_i + 20))^{2} }{n^{2} }\\\\=\frac{\sum(x_i^{2} + 2*20*x_i + 20^{2} )}{n} - \frac{(\sum x_i +20n)^{2} }{n^{2} } \\\\=\frac{\sum x_i^{2} + 40\sum x_i + 20^{2}n }{n} - \frac{(\sum x_i)^{2} +2*20n\sum x_i + 20^{2} n^{2} }{n^{2} } \\\\= \frac{\sum x_i^{2}}{n} - \frac{(\sum x_i)^{2}}{n^{2} } +\frac{40\sum x_i}{n} + 20^{2} - \frac{40\sum x_i}{n} - 20^{2}\\\\s_1^{2}= \frac{\sum x_i^{2}}{n} - \frac{(\sum x_i)^{2}}{n^{2} }\\\\=s^{2}[/tex]

Therefore variance doesn't change

The concept of equidistant points or lines is fundamental in geometry and plays a significant role in many geometric constructions and properties.

In geometry, an "equidistant" point is a point that is at the same distance from other points or lines. This concept is often used in various geometric constructions and proofs.

For example, in a circle, the center of the circle is equidistant from all points on the circumference. This property is what defines a circle.

In terms of lines, an equidistant point can be found by drawing perpendicular bisectors. A perpendicular bisector of a line segment is a line that is perpendicular to the segment and divides it into two equal parts. The point where the perpendicular bisectors of a triangle intersect is called the circumcenter, and it is equidistant from the vertices of the triangle.

Another example is the concept of an equidistant curve. In some cases, there may be a curve or path that is equidistant from two fixed points. This curve is called the "locus of points equidistant from two given points" and is often referred to as a "perpendicular bisector" when dealing with line segments.

All things considered, the idea of equidistant points or lines is essential to geometry and is important to many geometrical constructions and features.

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The surface area of the cone is: 213.66 cm²

The volume of a cone is: 7.33 cm³

The formula for the surface area of a cone is:

T.S.A = πrl + πr²

where:

r is radius

l is slant length

From the diagram and using Pythagoras theorem,we have:

l = √(7² + 5²)

l = √74

Thus:

TSA =  (π * 5 * √74) + (π * 5²)

TSA = 213.66 cm²

Formula for the volume of a cone is:

V = ¹/₃πr²

V = ¹/₃π * 7

V = 7.33 cm³

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Answer:

JLK = PRQ

Step-by-step explanation:

We already know that JL = QR and KL = PR.

If we know the angle between the two sides is equal or if the other side lengths are equal, then that would prove the two triangles to be equal.

Since the option for the two other side lengths (JK = PQ) to be equal is not listed, then the option that shows the angles are equal is the correct answer.

Since both JKL and PRQ are between the line segments already said to be equal, the answer is JLK = PRQ

Answer:

C. This would be a true statement due to the fact the to sides touch are equal. The sides are going to come at the same angle giving you a SAS which is possible to show for congruent triangles.

Rose's age should be a whole number, we can round 7.8 to the nearest whole number, which is 8.

Let's assume the number of primroses in the vase is p, and the number of celandines is c.

Each primrose has 5 petals, so the total number of primrose petals is 5p.

Each celandine has 8 petals, so the total number of celandine petals is 8c.

According to the given information, the total number of petals is 39. Therefore, we can set up the equation:

5p + 8c = 39 (Equation 1)

Aunt Lily mentions that the total number of flowers is exactly Rose's age. Since Rose's age is not provided, we'll represent it with the variable r.

The total number of flowers is p + c, which is also equal to Rose's age (r). Therefore, we have another equation:

p + c = r (Equation 2)

We need to find the value of r (Rose's age). To do that, we'll solve the system of equations by eliminating one variable.

Multiplying Equation 2 by 5, we get:

5p + 5c = 5r (Equation 3)

Now we can subtract Equation 1 from Equation 3 to eliminate the p term:

(5p + 5c) - (5p + 8c) = 5r - 39

This simplifies to:

-3c = 5r - 39

Now, let's rearrange Equation 2 to solve for p:

p = r - c (Equation 4)

Substituting Equation 4 into the simplified form of Equation 3, we have:

-3c = 5r - 39

Substituting r - c for p, we get:

-3c = 5(r - c) - 39

Expanding, we have:

-3c = 5r - 5c - 39

Rearranging the terms, we get:

2c = 5r - 39

Now we have a system of two equations:

-3c = 5r - 39 (Equation 5)

2c = 5r - 39 (Equation 6)

To solve this system, we can eliminate one variable by multiplying Equation 5 by 2 and Equation 6 by 3:

-6c = 10r - 78 (Equation 7)

6c = 15r - 117 (Equation 8)

Now, let's add Equation 7 and Equation 8 to eliminate c:

-6c + 6c = 10r + 15r - 78 - 117

This simplifies to:

25r = 195

Dividing both sides by 25, we get:

r = 7.8

Since Rose's age should be a whole number, we can round 7.8 to the nearest whole number, which is 8.

Therefore, Rose is 8 years old.

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Answer:

880g of flour

Step-by-step explanation:

A ratio has two or more numbers that symbolize relation to each other. Ratios are used to compare numbers, and you can compare them using division.

The ratio the baker uses (strawberries: flour) is:

Adding the two of these together gives us:

This means that for 1 batch of cakes, 260g of flour and strawberries is needed in total.

Now that we know that, we can take 1040g and divide that by 260g.

1040g ÷ 260g = 4

This means that the baker made 4 batches of cakes using 1040g total.

Taking the amount of flour and multiplying that by 4 to get the amount of flour used:

220g × 4 = 880g

Therefore the baker uses 880g of flour for every 1040g total.

A rotation is a geometric transformation that preserves the shape and size of a figure while changing its orientation in space. It is a fundamental concept in geometry and is used in various fields, including art, design, and engineering.

A rotation of a figure can be considered as a transformation that rotates the figure around a fixed point, known as the center of rotation. During the rotation, each point of the figure moves along an arc around the center, maintaining the same distance from the center.

To perform a rotation, we specify the angle of rotation and the direction (clockwise or counterclockwise). The center of rotation remains fixed while the rest of the figure rotates around it. The resulting figure is congruent to the original figure, meaning they have the same shape and size but may be in different orientations.

Rotations are commonly described using positive angles for counterclockwise rotations and negative angles for clockwise rotations. The magnitude of the angle determines the amount of rotation. For example, a 90-degree rotation would result in the figure being turned a quarter turn counterclockwise.

In general, a rotation is a geometric transformation that keeps a figure's size and shape while reorienting it in space. It is a fundamental idea in geometry that is applied in a number of disciplines, including as engineering, design, and the arts.

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