) Quantifier negation.
Form the negation of the following statements. Then apply De Morgan’s law and/or conditional law, when
applicable. Negation should appear only within predicates, i.e., no negation should be outside a quantifier
or an expression involving logical connectives. Show all steps.
a) ∀x (P(x) ∧ R(x))
b) ∀y∃z(¬P(y) → Q(z))
c) ∃x (P(x) ∨ (∀z (¬R(z) → ¬Q(z))))

The function f(x) = x + 10√(9 - x) is increasing on the interval (-∞, 9) and decreasing on the interval (9, ∞).

To determine the intervals on which the function is increasing or decreasing, we need to find the derivative of the function and analyze its sign.

Let's find the derivative of the function f(x) = x + 10√(9 - x) with respect to x.

f'(x) = 1 + 10 * (1/2) * (9 - x)^(-1/2) * (-1)

= 1 - 5√(9 - x) / √(9 - x)

= 1 - 5 / √(9 - x).

To analyze the sign of the derivative, we need to find the critical points where the derivative is equal to zero or undefined.

Setting f'(x) = 0:

1 - 5 / √(9 - x) = 0

5 / √(9 - x) = 1

(√(9 - x))^2 = 5^2

9 - x = 25

x = 9 - 25

x = -16.

The critical point is x = -16.

We can see that the derivative f'(x) is defined for all x values except x = 9, where the function is not differentiable due to the square root term.

Now, let's analyze the sign of the derivative f'(x) in the intervals (-∞, -16), (-16, 9), and (9, ∞).

For x < -16:

Plugging in a test value, let's say x = -17, into the derivative:

f'(-17) = 1 - 5 / √(9 - (-17))

= 1 - 5 / √(9 + 17)

= 1 - 5 / √26

≈ 1 - 0.97

≈ 0.03.

Since f'(-17) is positive, the function is increasing in the interval (-∞, -16).

For -16 < x < 9:

Plugging in a test value, let's say x = 0, into the derivative:

f'(0) = 1 - 5 / √(9 - 0)

= 1 - 5 / √9

= 1 - 5 / 3

≈ 1 - 1.67

≈ -0.67.

Since f'(0) is negative, the function is decreasing in the interval (-16, 9).

For x > 9:

Plugging in a test value, let's say x = 10, into the derivative:

f'(10) = 1 - 5 / √(9 - 10)

= 1 - 5 / √(-1)

= 1 - 5i,

where i is the imaginary unit.

Since the derivative is not a real number for x > 9, we cannot determine the sign.

Combining the information, we conclude that the function f(x) = x + 10√(9 - x) is increasing on the interval (-∞, 9) and decreasing on the interval (9, ∞).

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The required formula to find the missing angles in the triangle is x + x + 10 = 130

The sum of angles in a triangle gives us 180°, Mathematically, this gives :

50 + x + (x + 10) = 180

50 + x + x + 10 = 180

x + x + 60 = 180

x + x + 10 = 180 - 50

x + x + 10 = 130

Therefore, the required formula is x + x + 10 = 130

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The angle measures for this problem are given as follows:

The sum of the measures of the internal angles of a triangle is of 180º.

The triangle in this problem is ABC, hence the measure of a is obtained as follows:

a + 68 + 50 = 180

a = 180 - (68 + 50)

a = 62º.

c and d are corresponding angles to angle a, as they are on the same position relative to parallel lines, hence their measures are given as follows:

Angle b is a corresponding interior angle with angle a, hence they are supplementary and it's measure is given as follows:

a + b = 180

62 + b = 180

b = 118º.

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Pedro's book weight approximately 2 pound more than Blanca's book.

The correct answer is B.

Weight of Pedro's book = [tex]\frac{19}{4}[/tex]

                                        = [tex]4.7[/tex]

Weight of Blanca's book = [tex]\frac{17}{7}[/tex]

                                         = [tex]2.4[/tex]

We have to determine how much more Pedro's book weight than Blanca's using estimation

Difference between weight of book of Pedro and Blanca = [tex]4.7 - 2.4[/tex]

= [tex]2.3[/tex]

Difference  between weight of book of Pedro and Blanca= [tex]2.3 \approx 2Ibs[/tex]

Hence, the weight of Pedro's book is 2 lbs more than the weight of Blanca's book.

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Answer:

39 3/8 ft ^2

Step-by-step explanation:

To find the area of a rectangle, multiply the length times the width.

A = l*w

  = 7 1/2  * 5 1/4

Change the mixed numbers to improper fractions.

A = 15/2 * 21/4

    =315/8

Change back to a mixed number.

8 goes into 315 39 times with 3 left over.

 = 39 3/8

Answer:

(you can round to 39.4)

Step-by-step explanation:

Find the area of the rectangle shown

7 1/2 ft

5 1/4ft

7 1/2 = 7.5 ft

5 1/4 = 5.25 ft

Area = L x W

Area = 7.5 x 5.25

Area = 39.375

If two different signs are multiplied, the product obtained should be multiplied by another negative sign. We would then get the last sign as positive.

When two different signs are multiplied, the product obtained should be multiplied by another negative sign. When two different signs are multiplied, we obtain either a positive or negative product depending on the signs. If both signs are positive, the product is positive. If both signs are negative, the product is also positive.

However, if two different signs are multiplied, the product is negative. This is the first step in answering the question.

After we obtain a negative product by multiplying two different signs, we are to multiply this product with another negative sign. Multiplying a negative product with another negative sign would yield a positive result. This is because the product of two negatives is positive.

To understand this concept, we can use numbers as examples. Let us say we want to multiply -5 and 7. The product of -5 and 7 is -35 since we are multiplying two different signs. Now we are to multiply this product with another negative sign, say -1. Multiplying -35 and -1, we obtain +35, which is the answer to the problem.
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Since line a and line c are parallel and line c and line b are perpendicular, then the value of angle 2 is 90°

A line can be defined as a straight set of points that extend in opposite directions.

Parallel lines are lines that never intersect, and they form the same angle when they cross another line.

When a straight vertical line crosses another straight horizontal line we say the two lines are perpendicular.

Perpendicular lines intersect at a 90-degree angle, forming a square corner.

Therefore the value of angle 2 is 90°.

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Angle a + Angle d = 180 degreesAngle b + Angle c = 180 degrees.

In a parallelogram, adjacent angles are supplementary. As a result, we can express c in terms of the angles adjacent to it as follows: Angle a + Angle c = 180 degreesAngle c + 46 degrees = 180 degreesAngle c = 180 degrees - 46 degrees = 134 degreesTherefore, the size of angle c is 134 degrees.

A parallelogram is a four-sided quadrilateral that has opposite sides parallel and equal in length. Each of the opposite angles is also equal in size.In general, if we label the angles of a parallelogram as follows: Angle a, Angle b, Angle c, and Angle d,.

we have the following:Angle a = Angle c (opposite angles are equal)Angle b = Angle d (opposite angles are equal)Adjacent angles in a parallelogram are supplementary, which means they add up to 180 degrees.

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Therefore, the answer is A. {5, 5√3, 10}.

The set of values {5, 5√3, 10} could be the side lengths of a 30-60-90 triangle.

In a 30-60-90 triangle, the sides are in a specific ratio. The ratio is 1 : √3 : 2, where the shortest side (opposite the 30-degree angle) has length x, the side opposite the 60-degree angle has length x√3, and the hypotenuse (opposite the 90-degree angle) has length 2x.

Let's check if the given set of values satisfies this ratio:

   If x= 5, then the side opposite the 60-degree angle should be 5√3, and the hypotenuse should be 10. These values match the ratio, so the set {5, 5√3, 10} could be the side lengths of a 30-60-90 triangle.

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You will pay S/.80 for 2 bananas, 3 peaches, and a pineapple.

Let the price of a banana be b, a peach p, a medlar m, and a pineapple P in Peruvian soles.

From the information, 8b=6p, 4p=10m, 12m=2P, and 10P=S/.320.

From these equations, we have p=4b/3, m=p/2=2b/3, and P=6m=4b.

Substituting for P, 10(4b)=320, so b=8. Hence, p=32/3, m=16/3, and P=32.

For 2 bananas, 3 peaches and a pineapple, the total cost is 2b+3p+P = 2(8) + 3(32/3) + 32 = 16 + 32 + 32 = S/.80.

Therefore, you will pay S/.80 for 2 bananas, 3 peaches, and a pineapple.

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The Question in English

At a fair they sell 8 bananas at the same price as 6 peaches, 4 peaches the same as 10 medlars. A dozen medlars at the same price as 2 pineapples, if 10 pineapples cost S/.320, how much will I pay for 2 bananas, 3 peaches and a pineapple?

If the water snake is initially 30 meters below the ground level and then climbs 20 meters upward, it will be 30 - 20 = 10 meters below the ground level. However, if it then slips down 10 meters, it will be 10 + 10 = 20 meters below the ground level.

Answer:

hikers can travel before they have to turn around be "d" days. Also, let's represent the supplies the hikers have with "s."Given that after three days the hikers used 2/5 of their supplies, we can say that they have 3/5 of their supplies left. Hence,3/5s = Supplies remaining after 3 days, which means that2/5s = Supplies used in the first 3 daysWe need to find the number of days the hikers can travel before they have to turn around. So, we can say that the ratio of supplies used to the number of days traveled will be equal to the ratio of supplies remaining to the number of days remaining.2/5s ÷ 3 = (3/5s) ÷ dWe can simplify the above expression to find d as:d = (3/2) × 3/5 = 9/10 daysTherefore, the hikers can go forward for 9/10 days before they have to turn around.

Since function, g, has a domain of -20 ≤ x ≤ 5 and a range of -5 ≤ g(x) ≤ 45 and that g(0) = -2 and g(-9) = 6, the statement that could be true for g is: A. g(-13) = 20.

Based on the information provided in this scenario, the domain and range of this function g(x) in interval notation are as follows:

Domain = -20 ≤ x ≤ 5 or [-20, 5]

Range = -5 ≤ g(x) ≤ 45 or [-5, 45]

Function g(0) = 2 is simply a false statement because it was explicitly stated that g(0) = -2. Additionally, g(-4) = -11 is a false statement because it is outside the scope of the given range for this function.

Furthermore, the function g(7) = -1 is a false statement because it is not within the domain for the given function. However, we can logically deduce that the function g(-13) = 20 is a true statement because it is within the domain for the given function and it would produce an output that is within the scope of the given range for this function.

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Answer: The caret symbol (^) represents a power.

Step-by-step explanation:

In short, a power or an exponent indicates the number of times a number needs to be multiplied by itself. The caret can be used to represent a number that is to be used as an exponent.

For example, 5^2 is the same thing as [tex]5^2[/tex]. [tex]5^{2} =[/tex] 5 x 5 = 25.

In the example you provided, 1200 x (1.0041666667) ^ 48, the caret means that you need to raise 1.0041666667 to the power of 48.

Answer:

3.280

Step-by-step explanation:

When your rounding your answer the thousandths place is the 3rd number away from the decimal point so that means that 3.28 would be 3.280 rounded to the nearest thousandth.

The correct option is the third one.

The new vertices are:

(-4, -1), (0, -1). (-4, -3)

For a point (x, y), a rotation of 90° counterclockwise changes a quadrant, and changes the coordinates to (-y, x)

So, for the point (-1, 4) which is on the second quadrant, it will move it to the third quadrant, so the new point is (-4, -1)

For the second point:

(-1, 0)

It is on the second quadrant, and we go to the third one again, the new point is (0, -1)

Finally_

(-3, 4) --> (-4, -3)

Then the correct option is the third option.

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The total distance of the obstacle course can be calculated by finding the distance between each pair of consecutive points and adding them up. The distance between two points (x1, y1) and (x2, y2) can be calculated using the formula: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2).

Using this formula, we can calculate the distances between each pair of consecutive points as follows:

Start to Tire Race: distance = sqrt((-40 - (-40))^2 + (-30 - (-10))^2) = 20 yards

Tire Race to Rope Climb: distance = sqrt((40 - (-40))^2 + (-30 - (-30))^2) = 80 yards

Rope Climb to Monkey Bars: distance = sqrt((40 - 40)^2 + (20 - (-30))^2) = 50 yards

Monkey Bars to Finish: distance = sqrt((10 - 40)^2 + (20 - 20)^2) = 30 yards

Adding up all these distances, we get a total distance of 20 + 80 + 50 + 30 = 180 yards for the obstacle course.

The cost to produce the 10-th unit of that item is 6512

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

C(p) = 8p³ -16p² + 4p + 72

For the 10-th unit of that item, we have

p = 10

substitute the known values in the above equation, so, we have the following representation

C(10) = 8 * 10³ -16 * 10² + 4 * 10 + 72

Evaluate

C(10) = 6512

Hence, the cost is 6512

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Answer:

23 gallons per minute

Step-by-step explanation:

The Volume of Water = 1380 gallons

Time = 1 hour

Since we want to determine the rate at which the water is flowing out of the pipe in gallons per minute, we convert our time from hour to minutes.

1 hour = 60 Minutes

Rate= Volume ÷ Time

 = 1380 ÷ 60

=23 gallons per minute

The water is flowing out of the pipe at a rate of 23 gallons per minute.

im propably wrong but yu can try this out!! :)

The area of the triangle opening, with a base of 4 feet and a height of 5.25 feet, is 10.5 square feet.

To find the area of the triangle opening, we can use the formula for the area of a triangle, which is given by A = (1/2) * base * height. In this case, the base and height of the triangle are represented by the variables C and h, respectively.

Given that C = 4 feet and h = 5.25 feet, we can substitute these values into the formula:

A = (1/2) * 4 feet * 5.25 feet

Simplifying the expression, we have:

A = 2 feet * 5.25 feet

A = 10.5 square feet

Therefore, the area of the triangle opening is 10.5 square feet.

In conclusion, with a base of 4 feet and a height of 5.25 feet, the triangle opening has an area of 10.5 square feet.

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The inverse statement is false.

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Step-by-step explanation:

Scatter plot is the correct answer

The domain for the function is the interval [0, 14], which represents the feasible values for the length of one side of the rectangular garden.

To determine the appropriate values of the domain for the graph of the function [tex]A(t) = -t^2 + 14t[/tex], we need to consider the situation and the constraints given.

The function A(t) represents the area of the rectangular garden as a function of the length of one of its sides, which is denoted by t.

We are also told that Marama plans to fence the garden with 28 feet of wired fencing.

Now, let's break down the problem and find the appropriate values for the domain.

We know that the perimeter of a rectangle is the sum of all its sides. In this case, since we have a rectangular garden, the perimeter can be represented as:

[tex]Perimeter = 2t + 2w[/tex],

where t is the length of one side (the width) and w is the length of the other side (the width).

The problem states that Marama plans to use 28 feet of wired fencing. Therefore, the perimeter of the garden must equal 28 feet:

[tex]2t + 2w = 28[/tex].

Simplifying this equation, we have:

[tex]t + w = 14[/tex].

We can express w in terms of t as [tex]w = 14 - t[/tex].

The area of a rectangle is given by the product of its length and width:

[tex]Area = t \times w[/tex].

Substituting the expression for w from step 2, we have:

[tex]A(t) = t \times (14 - t)[/tex].

Simplifying further:

[tex]A(t) = 14t - t^2[/tex].

To determine the appropriate values of the domain, we need to consider the context of the problem. Since we are dealing with a physical garden, both the length and width must be positive numbers. Additionally, the values of t must be feasible given the constraints of the perimeter.

We know that [tex]t + w = 14[/tex], so [tex]t + (14 - t) = 14[/tex], which simplifies to [tex]14 = 14[/tex].

This shows that the value of t can range from 0 to 14, inclusive.

Therefore, the appropriate values of the domain for the graph of the function [tex]A(t) = -t^2 + 14t[/tex] are [tex]t \epsilon [0, 14][/tex].

To illustrate this graphically, we can plot the function [tex]A(t) = -t^2 + 14t[/tex] and mark the appropriate values of the domain on the x-axis (representing t):

   ^

   |

A(t)|

   |

   |_______________________________

   0              t             14

The domain for the function is the interval [0, 14], which represents the feasible values for the length of one side of the rectangular garden.

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The system of linear inequalities that represent the graph is y ≥ -4x + 1 and y > -x - 2

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

The graph

Where, we have

Graph 1: (1, -3) and (0, 1)

A linear equation is represented as

y = mx + c

Where

c = y when x = 0

So, we have

y = mx + 1

Using the other points, we have

m + 1 = -3

So, we have

m = -4

So, we have

y = -4x + 1

As an inequality, we have

y ≥ -4x + 1

Also, we have

Graph 2: (1, -3) and (0, -2)

So, we have

y = mx - 2

Using the other points, we have

m - 2 = -3

So, we have

m = -1

So, we have

y = -x - 2

As an inequality, we have

y > -x - 2

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Answer:

Step-by-step explanation:

Given,

Total number of working days in a year = 212

No of absent days = 50

No. of present days = Total number of working days - No of absent days

                                 = 212-50

                                 = 162

Attendance Percentage =   (No.of present days)/(Total number of working days in a year) * 100

                                        = 162*100/212

                                        = 76.415 %

The exponential equation to model Puerto Rico's population in millions after t years is:

[tex]P(t) = 3.72 \times e^(^-^0^.^1^1^8^t^).[/tex]

To model Puerto Rico's population in millions after t years using an exponential equation, we can use the following formula:

[tex]P(t) = P₀ \times e^(^r^t^),[/tex]

where:

P(t) is the population at time t,

P₀ is the initial population (in 2010),

e is the base of the natural logarithm (approximately 2.71828), and

r is the growth rate.

Given that the population of Puerto Rico in 2010 was approximately 3.72 million people, we can substitute P₀ = 3.72 into the equation.

Now, to determine the growth rate (r), we need to consider the population decrease of 11.8% since 2010. We can convert this percentage into a decimal by dividing it by 100, so 11.8% becomes 0.118.

Since the population decreased, we use a negative growth rate, so r = -0.118.

Therefore, the exponential equation to model Puerto Rico's population in millions after t years is:

[tex]P(t) = 3.72 \times e^(^-^0^.^1^1^8^t^).[/tex]

This equation takes into account the initial population in 2010 and the negative growth rate to project the population in millions for any given year (t) after 2010.

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9 inches of wire will cost 54 cents.

To convert liters to gallons, we'll use the given conversion rate:

1 gallon is equal to 3.8 liters.

To find the number of gallons in 285 liters, we divide 285 by 3.8:

285 liters / 3.8 liters per gallon = 75 gallons

Therefore, 285 liters is equal to 75 gallons.

Now, let's calculate the cost of 9 inches of wire using the given rate of 12 inches costing 72 cents.

The cost per inch of wire can be found by dividing 72 cents by 12 inches:

72 cents / 12 inches = 6 cents per inch

To find the cost of 9 inches of wire, we multiply the cost per inch by 9:

6 cents per inch * 9 inches = 54 cents

Therefore, 9 inches of wire will cost 54 cents.

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Answer:

The answer is $3.91 hope it is help you

Answer:

5x + 7x = 1,224

12x = 1,224

x = 102

5(102) = 510 men, 7(102) = 714 women