help please its due in 50 minutes ill mark brainliest answer too and no need to show work

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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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!

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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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 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 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

To find the local and absolute maximum and minimum points of the function f(x) = (3/5)x^5 - 9x^3 + 2 on the closed interval [-4,5], we need to follow these steps:

a) Find all critical numbers (x-coordinates only):

To find the critical numbers, we need to identify where the derivative of the function is zero or undefined. Let's find the derivative of f(x) first:

f'(x) = 3x^4 - 27x^2

Now, set the derivative equal to zero and solve for x:

3x^4 - 27x^2 = 0

Factoring out a common factor of 3x^2, we get:

3x^2(x^2 - 9) = 0

This equation is satisfied when either 3x^2 = 0 or x^2 - 9 = 0.

For 3x^2 = 0, we have x = 0.

For x^2 - 9 = 0, we have x = -3 and x = 3.

Therefore, the critical numbers (x-coordinates) are 0, -3, and 3.

b) Find the intervals on which the graph is increasing (mark critical numbers):

To determine the intervals of increasing, we need to analyze the sign of the derivative on each side of the critical numbers. We create a sign chart for f'(x):

Interval (-∞, -3): Choose a test point x < -3, e.g., x = -4

f'(-4) = 3(-4)^4 - 27(-4)^2 = 768 > 0

The derivative is positive, indicating the graph is increasing.

Interval (-3, 0): Choose a test point x between -3 and 0, e.g., x = -1

f'(-1) = 3(-1)^4 - 27(-1)^2 = -24 < 0

The derivative is negative, indicating the graph is decreasing.

Interval (0, 3): Choose a test point x between 0 and 3, e.g., x = 1

f'(1) = 3(1)^4 - 27(1)^2 = -24 < 0

The derivative is negative, indicating the graph is decreasing.

Interval (3, ∞): Choose a test point x > 3, e.g., x = 4

f'(4) = 3(4)^4 - 27(4)^2 = 768 > 0

The derivative is positive, indicating the graph is increasing.

Therefore, the graph is increasing on the intervals (-∞, -3) and (3, ∞).

c) Find the intervals on which the graph is decreasing (mark critical numbers):

From the analysis above, we can see that the graph is decreasing on the intervals (-3, 0) and (0, 3).

d) Find all local maximum points:

To find the local maximum points, we need to examine the points where the graph changes from increasing to decreasing. By observing the sign changes in the derivative, we can identify potential local maximum points.

From our analysis in part b, we can see that the graph changes from increasing to decreasing at x = -3 and x = 0. Therefore, these are the local maximum points.

e) Find all local minimum points:

To find the local minimum points, we need to examine the points where the graph changes from decreasing to increasing. By observing the sign changes in the derivative, we can identify potential local minimum points.

From our analysis in part c, we can see that the graph changes.

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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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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:

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

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

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

28x + 8y + 2

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: [-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.

Answer:

15

Step-by-step explanation:

5x = 4x + 3

x = 3

BC = 5x = 5(3) = 15

Answer: 15

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.

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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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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The ordered pair that is a solution for both inequalities is (-2, 2).

Here we have the system of inequalities:

y < -x + 1

y > x

And we want to see which one of the given points makes both of them true.

To find that, just replace the values in both inequalities and see if both become true or not.

For example, for the first point:

(-3, 5)

We will get:

5 < -(-3) + 1 = 4

5 > -3

The first one is false, and the second one is true.

The correct option is the second point:

2 < -(-2) +1 = 3

2 > -2

Both are true.

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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 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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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

The vertex of the parabola is located at (-1, -2).To identify the vertex of the parabola given the points (-2, -1), (-3, 2), (-1, -2), and (1, 2), we can use the fact that the vertex of a parabola is the highest or lowest point on the graph.

From the given points, we can observe that the y-coordinate of the vertex will be either the highest or lowest among the y-coordinates of the points.By examining the y-coordinates, we see that the point (-3, 2) has the highest y-coordinate of 2, while the point (-1, -2) has the lowest y-coordinate of -2.

Therefore, the vertex of the parabola is the point (-1, -2), as it is the lowest point on the graph. The x-coordinate of the vertex is -1, and the y-coordinate is -2.

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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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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°

"1/49," accurately represents the result of Urvi's work.The correct answer is option D.

Urvi's work is accurate and follows the correct rule of "multiply by the reciprocal" to solve the fraction division problem.

In the given problem, she is dividing 14 by the fraction 2/7. According to the rule, to divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction.

In Urvi's work, she first takes the reciprocal of 2/7, which is 7/2. Then, she multiplies 14 by the reciprocal, which gives us (14 * 7/2).

Simplifying the multiplication, we get 98/2, which simplifies further to 49. Therefore, the correct answer to the fraction division problem is 1/49.

Option D, which states "1/49," accurately represents the result of Urvi's work. This option is the most accurate description of her work because it correctly shows the final simplified fraction after applying the "multiply by the reciprocal" rule.

Overall, Urvi's work demonstrates a correct understanding and application of the rule for solving fraction division problems.

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The Probable question may be:
Urvi solved a fraction division problem using the rule “multiply by the reciprocal.” Her work is shown below.

A StartFraction 14 divided by StartFraction 2 Over 7 EndFraction.

B. StartFraction 1 Over 14 EndFraction times

C. StartFraction 2 Over 7 EndFraction = StartFraction 2 Over 98

D. EndFraction or StartFraction 1 Over 49 EndFraction

Which is the most accurate description of Urvi’s work?

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

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.

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 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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[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 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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