There are 6 horses in a race. How many ways can the first three positions of the order of the finish occur assume there are no ties

Step-by-step explanation:

Let the statement be:

The converse of a statement is:

Let's try it for each statement and see if the converse is true.

a) The converse is:

It is true.

b) The converse  is:

It is true.

c) The converse is:

It is false, since not all rhombus have four right angles.

d) The converse is:

It is false, since it could be a square.

a. Converse: If the interior angles sum to 180°, then my shape is a triangle. (Converse is not true)

b. Converse: If alternate interior angles are equal, then the lines are parallel. (Converse is true)

c. Converse: If I have a Rhombus, then I also have a Square. (Converse is not true)

d. Converse: If I have a shape with four right angles, then I have a rectangle. (Converse is true)

a. Converse: If the interior angles of a shape sum to 180°, then the shape is a triangle.

The converse of this statement is not true. There are many shapes other than triangles whose interior angles sum to 180°, such as quadrilaterals and polygons with more than four sides.

b. Converse: If alternate interior angles of two lines are equal, then the lines are parallel.

The converse of this statement is true. If the alternate interior angles of two lines are equal, then the lines are parallel. This is a property of parallel lines and can be proven using the corresponding angles postulate.

c. Converse: If I have a Rhombus, then I also have a Square.

The converse of this statement is not true. While all squares are rhombuses (a square is a special type of rhombus), not all rhombuses are squares. A rhombus is a quadrilateral with all sides of equal length, but its angles can be anything other than 90°.

d. Converse: If I have a shape with four right angles, then I have a rectangle.

The converse of this statement is true. If a shape has four right angles, then it is a rectangle. This is a defining characteristic of rectangles, as all rectangles have four right angles.

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The complete question is :

Write the converse of each statement and determine if the converse is true.

a. If my shape is a triangle, then the interior angles sum to 180°.

b. If two lines are parallel, then alternate interior angles are equal.

c. If I have a Square, then I also have a Rhombus.

d. If I have a rectangle, then I have a shape with four right angles.

The difference quotient of the function f(x) = 12x + 1 is 12.

The difference quotient of a function measures the average rate of change of the function between two points. To find the difference quotient of the function f(x) = 12x + 1, we can follow these steps:

Select two points on the function, let's call them x and x + h, where h is a small positive value.

Evaluate the function at those two points to get the corresponding y-values. For f(x) = 12x + 1, we have:

f(x) = 12x + 1

f(x + h) = 12(x + h) + 1

Calculate the difference quotient by subtracting the values and dividing by h:

[f(x + h) - f(x)] / h

= [(12(x + h) + 1) - (12x + 1)] / h

= [12x + 12h + 1 - 12x - 1] / h

= (12h) / h

= 12

In this case, since the function is linear with a slope of 12, the difference quotient is constant and equal to the slope of the function. This means that for every unit increase in x, the function f(x) increases by 12.

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Answer:

Not parallelogram

Step-by-step explanation:

points A(-4, 1), B(1,3), C(8, -1) and D(4, -3).

The slope of a line can be found using the following formula:

Slope  [tex]m= = \frac{(y_2 - y_1) }{(x_2 - x_1)}[/tex]

where (x1, y1) and (x2, y2) are the coordinates of two points on the line.

Using the formula above,

Side AB: Slope =[tex]\frac{3-1}{1-(-4)}=\frac{2}{5}[/tex]

Side BC: Slope =[tex]\frac{-1-3}{8-1}=\frac{-4}{7}[/tex]

Side CD: Slope =[tex]\frac{-3 - (-1)}{4 - 8}= \frac{1}{2}[/tex]

Side DA: Slope =[tex]\frac{-3 - 1}{4 - (-4)}= \frac{-1}{4}[/tex]

Since The slopes of opposite sides are equal,

Therefore ABCD is not Parallelogram:

Step-by-step explanation:

1. Answer is 2

2. Answer is -3

3. Answer is 4

Answer:

i) -5x + 3/2

ii) x = 11

iii) x + 9/2

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The solutions to the equation (cosec(x)+2)(2cos(x)-1) are x = -π/6 + 2πn, 7π/6 + 2πn, x = π/3 + 2πn, 5π/3 + 2πn.

To solve the given equation, we can make use of zero-product property. The zero-product property states that if the product of the two factors is equal to zero, then one of the factors has to be equal to zero. So, now we will equate each factor to zero to find the solution.

1. (cosec(x)+2) = 0

 = cosec(x) = -2

 Taking reciprocal on both the sides:

 = 1/cosec(x) = -1/2

 = sin(x) = -1/2

The solutions for sin(x) = -1/2 occur when x = -π/6 + 2πn and 7π/6 + 2πn, where 'n' is an integer.

2. (2cos(x)-1) = 0

 = 2cos(x) = 1

 = cos(x) = 1/2

The solutions for  cos(x) = 1/2 occur when  x = π/3 + 2πn and 5π/3 + 2πn, where 'n' is an integer.

Therefore, the solutions to the equation (cosec(x)+2)(2cos(x)-1) are x = -π/6 + 2πn, 7π/6 + 2πn, x = π/3 + 2πn, 5π/3 + 2πn.

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The complete question is: Find out the possible solutions of the equation (cosecx+2)(2cosx-1)=0.

The central angle of the sector is 540°.

The area of the sector formula is A = (1/2) r² θ Where, A = Area of sector r = Radius of sectorθ = Central angle of sector Given that the radius of the sector is 10mm and area is 40/3 pie mm²

To find the measure of central angle θ, plug the given values in the formula as shown; A = (1/2) r² θ40/3 pie = (1/2)(10)² θ40/3 pie = (1/2)100 θ40/3 pie = 50θ (multiply by 3/40)θ = (3/40) × 40πθ = 3π.

So, the measure, in degrees, of the central angle of the sector is;θ = (180/π) × 3πθ = 540°Therefore, the central angle of the sector is 540°.

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The numerical value of x in the minor arc KU of the circle is 7.

The inscribed angle theorem states that an angle x inscribed in a circle is half of the central angle 2x that subtends the same arc on the circle.

It is expressed as:

Internal angle = 1/2 × ( major arc + minor arc )

From the diagram:

Internal angle = 122 degrees

Major arc = 194 degrees

Minor arc = ( 6x + 8 )

Plug these values into the above formula and solve for x:

Internal angle = 1/2 × ( major arc + minor arc )

122 = 1/2 × ( 194 + ( 6x + 8 ) )

Multiply both sides by 2:

122 × 2 = 2 × 1/2 × ( 194 + ( 6x + 8 ) )

122 × 2 =( 194 + ( 6x + 8 ) )

244 = 194 + 8 + 6x

244 = 202 + 6x

6x = 244 - 202

6x = 42

x = 42/6

x = 7

Therefore, the value of x is 7.

Option B) 7 is the correct answer,

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The average rate of change for the number of pounds of metal per person per day between 1980 and 1988 is -0.00125 pounds per year.

To calculate the average rate of change for the number of pounds of metal per person per day between 1980 and 1988, we need to find the difference in the values and divide it by the number of years.

In 1980, the pounds of metal per person per day was 0.35, and in 1988, it was 0.34. The difference between these values is -0.01.

The number of years between 1980 and 1988 is 1988 - 1980 = 8 years.

Now, we can calculate the average rate of change:

Average rate of change = (Change in pounds of metal) / (Number of years)

= (-0.01) / 8

= -0.00125

The average rate of change for the number of pounds of metal per person per day between 1980 and 1988 is -0.00125 pounds per year.

Interpretation:

The negative value of the average rate of change (-0.00125) indicates that there was a decrease in the number of pounds of metal per person per day from 1980 to 1988.

Specifically, on average, there was a decrease of approximately 0.00125 pounds per year.

This suggests that there was a declining trend in the use or disposal of metal waste during this period.

It could indicate improvements in recycling or waste management practices, or a shift in consumer behavior towards reducing metal waste.

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Answer:

x = 1 or

x = 1/2

Step-by-step explanation:

2x² - 5 = 3x - 6

⇒ 2x² - 5 - 3x + 6 = 0

⇒ 2x² - 3x + 1 = 0

Using quadratinc formula, the roots are

[tex]\frac{-b \pm\sqrt{b^{2} -4ac } }{2a}[/tex]

where, a = 2, b = -3 and c = 1

sub in the formula,

[tex]\frac{-(-3) \pm\sqrt{(-3)^{2} -4(2)(1) } }{2(2)}\\\\= \frac{3 \pm\sqrt{9 -8 } }{4}\\\\= \frac{3 \pm\sqrt{1 } }{4}\\\\= \frac{3 \pm 1 }{4}\\\\= \frac{3+1}{4} \;or\; \frac{3-1}{4}\\ \\=\frac{4}{4} \; or\; \frac{2}{4} \\\\= 1 \; or \; \frac{1}{2}[/tex]

Answer:

7.80 ounces can be bought for $1.95

Step-by-step explanation:

Step 1:  Determine how many ounces is in a pound:

Thus, 16 ounces cost $4.

Step 2:  Create a proportion to determine how many ounces can be bought for $1.95.

Since you can get 16 ounces for $4, we can create a proportion to determine how many ounces can be bought for $1.95:

16 ounces / $4 = x ounces / $1.95

Step 3:  Simplify on the left-hand side of the equation:

16/4 = x/1.95

4 = x/1.95

Step 4: multiply both sides by 1.95 to determine how many ounces can be bought for $1.95:

(4 = x/1.95) * 1.95

7.80 = x

Thus, 7.80 ounces can be bought for $1.95.

The pixel resolution is 4.896 x 10^5 pixels; in megapixels, it's approximately 0.490; and the aspect ratio is 18:17.

First, let's calculate the resolution in pixels: Resolution is simply the width times the height of the sensor. Specifically for this question, multiply 720 by 680. This gives us 489600 pixels for the resolution - in scientific notation to 4 significant figures we have 4.896 x 10^5.

Next, to calculate the resolution in megapixels, we need to divide the resolution by 1 million (since one megapixel is equal to 1 million pixels). So, 489600/1,000,000 is approximately 0.490 megapixels when rounded to the nearest thousandth of a megapixel.

Finally, the aspect ratio is the relationship of the width to the height of an image or screen. For a 720 x 680 sensor, if we divide 720 by 680, the result would be approximately 1.0588. However, we want this ratio in its simplest form, so we should use the greatest common divisor (GCD). In this case, the GCD of 720 and 680 is 40, so by dividing both dimensions by 40, we get an aspect ratio of 18:17.

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Answer:

  76 units

Step-by-step explanation:

You want the length of AC in the given triangular pyramid.

The Pythagorean theorem can be used to find the lengths of AD and CD.

  AD² + 40² = 41²

  AD² = 41² -40² = 81 . . . . . = 9²

and

  CD² +40² = 85²

  CD² = 85² -40² = 5625 . . . . . = 75²

It can also be used to find AC:

  AD² + CD² = AC²

  81 + 5625 = AC²

  AC = √5706 = 3√634 ≈ 76

The length of side AC is about 76 units.

__

Additional comment

The Pythagorean theorem tells you the square of the hypotenuse is the sum of the squares of the legs of a right triangle.

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Answer:

7

Step-by-step explanation:

In an 30-60-90 right triangle, the longer leg is the shorter leg multiplied by [tex]\sqrt{3}[/tex]

So [tex]7\sqrt{3} =x*\sqrt{3[/tex]

[tex]7=x[/tex]

x=7

Answer:

A. 21

Step-by-step explanation:

This is a 30-60-90 triangle, where you have one 30° angle, one 60°, and one 90° (aka right) angle.  

The sides of a 30-60-90 triangle adhere to the following rules:

Since 7√3 is the length of the side opposite the 30° angle, the entire expression represents x.

Since the length of the side opposite the 60° angle is given by x * √3, the length of this side is (7√3)(√3).

Simplifying gives us 7*3, which is 21.

Thus, the value of x in the triangle is 21 (answer choice A.)

The value of n is approximately -0.327.

To determine the value of n, we can substitute the given solution (x = 2.1) into the system of equations:

4x² - my = 10 ...(1)

mx² + ny = 20 ...(2)

Substituting x = 2.1 into equation (1), we get:

4(2.1)² - my = 10

8.4 - my = 10

-my = 10 - 8.4

-my = 1.6

Now, substituting x = 2.1 into equation (2), we get:

m(2.1)² + ny = 20

4.41m + ny = 20

Since we already have an expression for -my from equation (1), we can substitute it into equation (2):

4.41m + (-1.6) = 20

4.41m - 1.6 = 20

4.41m = 20 + 1.6

4.41m = 21.6

To solve for m, we divide both sides by 4.41:

m = 21.6 / 4.41

m ≈ 4.9

Now that we have the value of m, we can substitute it back into equation (1) to solve for n:

4(2.1)² - 4.9y = 10

8.4 - 4.9y = 10

-4.9y = 10 - 8.4

-4.9y = 1.6

Finally, to solve for y, we divide both sides by -4.9:

y = 1.6 / -4.9

y ≈ -0.327

Consequently, n has a value of roughly -0.327.

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4/21 of the books in the library are both fiction and paperbacks.

To determine the fraction of books in the library that are both fiction and paperback, we need to multiply the fractions representing each condition.

Let's start with the fraction of books in the library that are fiction. If 4/7 of the books are fiction, then this fraction represents the number of fiction books.

Next, we want to find the fraction of fiction books that are also paperbacks. Since 1/3 of the fiction books are paperbacks, we multiply 4/7 (fiction books) by 1/3 (paperback fraction).

Multiplying fractions is done by multiplying the numerators together to get the new numerator and multiplying the denominators together to get the new denominator.

Thus, the fraction of books in the library that are both fiction and paperbacks is:

(4/7) * (1/3) = (4 * 1) / (7 * 3) = 4/21

Therefore, 4/21 of the books in the library are both fiction and paperbacks.

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The solution to the system is (b) (-2, 3, z) where z is any real number

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

The augmented matrix

Where, we have

[tex]\left[\begin{array}{ccc|c}1&0&0&-2\\0&1&0&3\\0&0&0&3\end{array}\right][/tex]

From the above, we have the first two diagonals to be 1

And other elements to be 0

This means that

x = -2 and y = 3

For z, the value is infinitely many

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The extreme of the function is Absolute minimum

y = -2

x-intercept = 3

A discontinuous function is a mathematical function that has one or more points in its domain where the function fails to be continuous.

In other words, at these specific points, the function exhibits a discontinuity, indicating a break or jump in the graph.

In the graph, there was discontinuities at points

x = -2, and x = 2

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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:

First, we multiply 2×2 then the answer we obtain we add to it 6 which in total gives us 10

The graph that shows a pair of lines representing the equations with the solution (4, -1) is option D.

To determine which graph represents the equations with the solution (4, -1), we need to check if the given point lies on the lines represented by each graph.

Let's examine each option:

A) The lines intersect 1 unit to the right and 4 units down. This does not match the given solution of (4, -1), so option A can be eliminated.

B) The lines intersect 4 units to the right and 1 unit down. Again, this does not match the given solution of (4, -1), so option B can be eliminated.

C) The lines intersect 4 units to the left and 1 unit up. This is the exact opposite of the given solution (4, -1), so option C can be eliminated.

D) The lines intersect 1 unit to the left and 4 units up. This matches the given solution of (4, -1), where the x-coordinate is 1 unit to the left and the y-coordinate is 4 units up. Therefore, option D represents the equations with the solution (4, -1).

In conclusion, the graph that shows a pair of lines representing the equations with the solution (4, -1) is option D.

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Answer:

1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755
0.0755 * 100 = 7.55%

Step-by-step explanation:

To find the percentage of 1 - 3√(5/35), we need to first evaluate the expression.

1 - 3√(5/35) = 1 - 3√(1/7) = 1 - 3*(1/sqrt(7)) ≈ 0.0755

To convert this decimal to a percentage, we simply multiply by 100:

0.0755 * 100 = 7.55%

Answer:

(5, 3 )

Step-by-step explanation:

under a reflection in the x- axis

a point (x, y ) → (x, - y ) , then

D (5, - 3 ) → (5, - (- 3) ) → (5, 3 )

Answer:

(5,3)

Step-by-step explanation:

got it right on edg

Answer:

A, C

Step-by-step explanation:

x² + 3x - 5 = 0

[tex] x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} [/tex]

[tex] x = \dfrac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)} [/tex]

[tex] x = \dfrac{-3 \pm \sqrt{29}}{2} [/tex]

Answer:

The valid prediction about the continuous function f(x) is : f(x) ≥ 0 over the interval [5, ∞).

Step-by-step explanation:

The expected return on the portfolio is approximately 13.465%.

To calculate the expected return on the portfolio, we need to consider the weights of each stock in the portfolio.

Let's denote the weight of Stock A as wA and the weight of Stock B as wB. The weight of a stock is the proportion of the total portfolio value that is invested in that stock.

Given that $3,000 is invested in Stock A and $4,100 is invested in Stock B, we can calculate the weights as follows:

wA = $3,000 / ($3,000 + $4,100) = $3,000 / $7,100

wB = $4,100 / ($3,000 + $4,100) = $4,100 / $7,100

Next, we need to calculate the weighted average of the expected returns of the two stocks using their respective weights:

Expected return on the portfolio = (wA * Return on Stock A) + (wB * Return on Stock B)

Expected return on the portfolio = (wA * 10%) + (wB * 16%)

Substituting the calculated weights into the equation:

Expected return on the portfolio = ($3,000 / $7,100 * 10%) + ($4,100 / $7,100 * 16%)

Simplifying the equation:

Expected return on the portfolio = (0.4225 * 10%) + (0.5775 * 16%)

Expected return on the portfolio = 0.04225 + 0.0924

Expected return on the portfolio = 0.13465 or 13.465%

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The equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.

From the given options, the equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.

To determine the equation of a hyperbola, we examine the standard form:

For a hyperbola centered at (h, k), with vertical transverse axis, the standard form is:

(y - k)²/a² - (x - h)²/b² = 1

From the given graph, we can observe that the center of the hyperbola is (-2, 3). This corresponds to the values of (h, k) in the standard form.

Next, we need to determine the values of a and b, which are the lengths of the transverse and conjugate axes, respectively. Looking at the graph, we see that the transverse axis has a length of 2a = 4, so a = 2. The conjugate axis has a length of 2b = 10, so b = 5.

Plugging these values into the standard form, we obtain:

(y - 3)²/4 - (x + 2)²/25 = 1

The equation that represents the hyperbola shown in the graph is (y - 3)²/4 - (x + 2)²/25 = 1.

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The correct equation representing the hyperbola shown in the graph is:

(x + 2)²/25 - (y - 3)²/4 = 1.

The equation that represents the hyperbola shown in the graph is:

(x + 2)²/25 - (y - 3)²/4 = 1

Let's analyze the options provided:

(x - 2)²(v + 3)² = 1:

This equation is not a valid representation of a hyperbola because it contains a term (v + 3)², which is not consistent with the variable used in the graph.

(x + 2)²/25:

This equation represents a horizontal parabola, not a hyperbola.

(x - 2)²/25:

This equation represents a horizontal parabola, not a hyperbola.

(y - 3)²/1:

This equation represents a vertical line, not a hyperbola.

(y - 3)²/4:

This equation represents a hyperbola with a vertical transverse axis and a conjugate axis length of 2b = 4 (b = 2).

The equation is in the standard form for a hyperbola with a vertical transverse axis.

The equation is provided as a standard form assuming the given coordinates and graph match the standard form representation of a hyperbola.

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Answer:

3.34 * 10^-21 g

Step-by-step explanation:

1.67*10^-24 * 2000 = 1.67 * 10^-24 * 2 *10^3 = 3.34 * 10^-21 g

Given the equation f(x) = a · bx where a and b are constants. So, the answer to the given problem is g(x) = a · bx + h, and the explanation of the trigonometric function.

To find the equation of transformed function g in terms of function f is explained below: If f(x) = a · bx, then the transformed function g(x) can be represented by g(x) = a · bx + h, where h is the vertical shift (if h > 0, the graph shifts upward, and if h < 0, the graph shifts downward).

Now, we have to replace the value of 'a' to complete the equation of g(x). But, we don't have any value of 'a' provided in the question. Hence, we can't determine the equation of transformed function g in terms of function f for the given information.

Next, let's move to the trigonometric function. It is given that: R S 9 sin cos tan sin cos tan-¹ /ASin, Cos, Tan, Cosec, Sec, and Cot are six trigonometric functions. Let's see their definitions and their corresponding inverse functions:

1. Sine: It is defined as the ratio of the length of the side opposite the given angle to the length of the hypotenuse in a right-angled triangle. Its corresponding inverse function is sin⁻¹.

2. Cosine: It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. Its corresponding inverse function is cos⁻¹.

3. Tangent: It is defined as the ratio of the length of the side opposite the given angle to the length of the adjacent side in a right-angled triangle. Its corresponding inverse function is tan⁻¹.

4. Cosecant: It is defined as the ratio of the length of the hypotenuse to the length of the side opposite the given angle in a right-angled triangle. Its corresponding inverse function is cosec⁻¹.

5. Secant: It is defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. Its corresponding inverse function is sec⁻¹.

6. Cotangent: It is defined as the ratio of the length of the adjacent side to the length of the side opposite the given angle in a right-angled triangle. Its corresponding inverse function is cot⁻¹.

Hence, the answer to the given problem is g(x) = a · bx + h, and the explanation of the trigonometric function.

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