Use the logical equivalences p → q ≡ ~p∨ q and p ↔ q ≡ (~p∨ q) ∧ (~ q∨ p) to rewrite the given statement forms without using the symbol → or ↔.
a) (p → r) ↔ (q → r)

The formula for the circumference of a circle is given by 2 * pi * r, where r is the radius. If the circumference is 9.42 inches, then we can write the equation:

9.42 = 2 * pi * r

Solving for r, we get:

r = 9.42 / (2 * pi)

The diameter of the circle is equal to 2 * r, so:

d = 2 * r = 2 * (9.42 / (2 * pi)) = 9.42 / pi

So, the radius of the circular donut is approximately 3 inches, and the diameter is approximately 9.42 inches.

For a set of data in which the sample mean is 83.3 and the sample standard deviation is 6.3 the z-score given that x = 79.8 would be -0.555

We know that the formula for calculating a z-score is :

z = (x - μ) / σ

where z is the standard z - score

x is the observed value

μ is the mean of the sample

σis the standard deviation of the sample

here, set of data in which the sample mean is 83.3 and the sample standard deviation is 6.3

i.e., μ = 83.3,  σ = 6.3

We calculate the z-score for x = 79.8

Using above formula,

z = (x - μ) / σ

z = (79.8 - 83.3) / (6.3)

z = (-3.5) / (6.3)

z = -0.555

Therefore, the z-score is:  -0.555

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The solutions are:

a. x = 7

b. x = 7

The tape diagrams are given in the image below.

The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.

Given:

Two equations,

a. x + 9 = 16

b. 4x = 28

Solving the first equation,

x + 9 = 16

x = 7

Solving the second equation,

4x = 28

x = 7

The tape diagrams are given in the image below.

Hence, all the unknown values are given above.

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Answer:

see below

Step-by-step explanation:

we need to write the inverse of ,

[tex]\longrightarrow f(x) = 3x -1 \\[/tex]

Substitute [tex] f(x)=y[/tex] ,

[tex]\longrightarrow y = 3x-1 \\[/tex]

Interchange x and y ,

[tex]\longrightarrow x = 3y -1 \\[/tex]

now solve for y ,

[tex]\longrightarrow 3y = x + 1 \\[/tex]

[tex]\longrightarrow y =\dfrac{x+1}{3} \\[/tex]

replace [tex] y[/tex] with [tex] f^{-1}(x)[/tex] as ,

[tex]\longrightarrow \underline{\underline{ f^{-1}(x) =\dfrac{x+1}{3}}}\\[/tex]

And we are done!

Answer:

[tex]f^{-1}(x)=\log_3(x)+1[/tex]

Step-by-step explanation:

Given function:

[tex]f(x)=3^{x-1}[/tex]

To find the inverse of the given function, swap y and x:

[tex]\implies x=3^{y-1}[/tex]

Take the log base 3 of each side of the equation:

[tex]\implies \log_3(x)=\log_{3}3^{y-1}[/tex]

[tex]\textsf{Apply the log power law}: \quad \log_ax^n=n\log_ax[/tex]

[tex]\implies \log_3(x)=(y-1)\log_{3}3[/tex]

[tex]\textsf{Apply log law}: \quad \log_aa=1[/tex]

[tex]\implies \log_3(x)=(y-1)(1)[/tex]

[tex]\implies \log_3(x)=y-1[/tex]

Add 1 to both sides of the equation:

[tex]\implies \log_3(x)+1=y[/tex]

[tex]\implies y=\log_3(x)+1[/tex]

Replace y for f⁻¹(x):

[tex]\implies f^{-1}(x)=\log_3(x)+1[/tex]

For the system of equations, y = 10 + 2z and y = 38 - 5z the value of y and z are 18 and 4 respectively.

What is an equation?

A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("=").

The first equation is : y = 10 + 2z

The second equation is : y = 38 - 5z

Subtract equation (1) and (2) -

y - y = 10 + 2z - (38 - 5z)

0 = 10 + 2z - 38 + 5z

Collect the like terms -

2z + 5z = -10 + 38

7z = 28

z = 28/7

z = 4

Substitute the value of z in equation (1) -

y = 10 + 2(4)

y = 10 + 8

y = 18

Therefore, the value of y = 18 and z = 4.

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The earning of the store is $62.55.

What is an equation?

An equation is expression with one or more variables. It can also have more than 1 degree. it is a linear combination of constants and variables.

We are given that a box contains 74.25 ounces of pecans

These pecans are divided in 8.125 ounces of containers

First let us see how many such containers can be made

To find that we divided the two we get

74.25/ 8.125 = 9 such containers can be made

Now Each container is being sold for $6.95

Hence the total earnings are

9 * 6.95 = $62.55

The earning of the store is $62.55

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The solutions for all the equations are:

1) x = 4

2) x = ±7

3) x = 5²´³

4) x = ±5³´²

The first equation is:

x³ = 64

If we apply the cubic root in both sides we will get:

∛x³ = ∛64

x = ∛64 = 4

2) now we have:

x² = 49

If we apply the square root:

√x² = ± √49

(remember the plus minus sign when you use a even degree root)

x = ±7

In this case we have two solutions.

3) x³ =25

Like in the first case:

x = ∛25 = ∛5² = 5²´³

Now we can't simplify this, so we leave it in this way.

4) x² = 125

Remember that 125 = 5³, then:

x² = 5³

x = ±√5³ = ±5³´²

These are all the solutions.

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The triangle is congruent by the Angle side Angle property.

In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other. The meaning of congruency is that the two shapes are similar as well as equal in size.

The Angle side Angle Congruence Theorem (ASA) defines two triangles to be congruent to each other if the included two angles and one side of one are congruent to the included angle and corresponding two sides of the other triangle.

The three parameters that are congruent are:-

BV ≅ BV ( common side)

∠1 ≅ ∠2  ( Bisector)

∠3 ≅ ∠4  ( Bisector)

Hence, the two triangles are congruent with each other.

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The last

M angle 3 = 68 degrees

Step-by-step explanation:

interior angles between parallel lines

sum =180

The relationship between the price of balloons and the amount earned is 56f + 120 dollars

The price of the balloons if 400 dollars is earned on friday and saturday is 5 dollars on friday and 8 dollars on saturday

a) If f is the price of a balloon on Friday, then the price of a balloon on Saturday would be f + 3 dollars. The total amount Thomas earns on Friday would be 16 balloons * f dollars/balloon = 16f dollars. The total amount Thomas earns on Saturday would be 40 balloons * (f + 3 dollars/balloon) = 40f + 120 dollars. Hence, the total amount earned on Friday and Saturday would be 16f + 40f + 120 dollars = 56f + 120 dollars.

Therefore, the equation to describe the relationship between the price of a balloon on Friday and the total amount earned is: t = 56f + 120 dollars.

b) We are given that Thomas earns 400 dollars on Friday and Saturday, so we can set t = 400 dollars in the equation derived above:

400 = 56f + 120

Solving for f, we have:

280 = 56f

f = 5 dollars

The price of a balloon on Friday is $5, and the price of a balloon on Saturday would be f + 3 = 5 + 3 = $8.

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The number of the 3 / 5 in a tape diagram is 3.

A fraction is written in the form of a numerator and a denominator where the denominator is greater that the numerator. Any number of equal parts is represented by a fraction, which also represents a portion of a whole.

A fraction is a number that is a component of a whole. By breaking a whole into a number of parts, it is evaluated.

Given that the fraction is 3 / 5. When we add the fraction three times we will get the value that is closest to 2.

E = 3 / 5 + 3 / 5 + 3 / 5

E = ( 3 + 3 + 3 ) / 5

E = 9 / 5

E = 1.8 ≅ 2

Hence, in a tape diagram, the 3/5 is represented by the number 3.

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12n+250

because you don't know the amount of tickets

The following results are:

a. Average height: 58.84

b. Median height: 58

c. Mode height: 55

d. Range of data: 73 - 44 = 29

e. Standard deviation: 8.23

f. Z-score for 65 inches: (65 - 58.84) / 8.23 = 0.83

Statistics is a process in which we collect, interpret, study and understand the data. We took a small sample from the large quantity and then we did study of the sample then according to results, we did our predictions for events which can happen in future.

Given that,

The height of 25 trees,

following results (recorded in inches).

60 57 62 69 46 54 64 60 59 58 75 51 49

67 65 44 58 55 48 62 63 73 52 55 50

a.   Average = Sum of  height of all the trees/Total number of trees

                     = 58.84

b.  Median =  (N + 1)/2th term after arranging into ascending order

                  = 58

c.  Mode = Highest frequency

               = 55

d. range = Highest value - Lowest value

              = 73 -44

              = 29

e. the standard deviation of the given data, you would need to perform the following steps:

Calculate the mean of the data:

(60 + 57 + 62 + 69 + 46 + 54 + 64 + 60 + 59 + 58 + 75 + 51 + 49 + 67 + 65 + 44 + 58 + 55 + 48 + 62 + 63 + 73 + 52 + 55 + 50) / 25 = 58.84

Find the deviation of each data point from the mean:

60 - 58.84 = 1.16, 57 - 58.84 = -1.84, 62 - 58.84 = 3.16, and so on...

Square each deviation:

1.16^2 = 1.3536, (-1.84)² = 3.3876, 3.16^2 = 9.9876, and so on...

Sum all the squared deviations:

1.3536 + 3.3876 + 9.9876 + ... = 519.24

Divide the sum by the number of data points minus 1 (for a sample) or by the number of data points (for a population):

519.24 / 24 = 21.63

Take the square root of the result:

√21.63 = 4.65

The standard deviation of the given data is 4.65.

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A'(15) = 2 * 15 = 30.The meaning of A'(15) in this situation is that if the side length of the wafer changes by 1 mm, the area will change by 30 mm^2 when the side length is 15 mm.

The area A(x) of a square wafer is given by A(x) = x², where x is the side length of the wafer. The derivative of A(x) with respect to x, denoted A'(x), represents the rate of change of the area with respect to the side length. To find A'(15), we need to find the derivative of A(x) at x = 15.

A'(x) = 2x

Therefore, A'(15) = 2 * 15 = 30.

The meaning of A'(15) in this situation is that if the side length of the wafer changes by 1 mm, the area will change by 30 mm² when the side length is 15 mm. This means that for every 1 mm increase in the side length, the area of the wafer will increase by 30 mm², and for every 1 mm decrease in the side length, the area of the wafer will decrease by 30 mm².

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The sum of the squares of the differences between each data value and the mean is given as follows:

828.

The mean of the data-set in this problem is given as follows:

33.

The data-set is given as follows:

28, 34, 27, 42, 52 and 15.

Hence the sum of the squares of the differences between each data value and the mean is obtained as follows:

(28 - 33)² + (34 - 33)² + (27 - 33)² + (42 - 33)² + (52 - 33)² + (15 - 33)² = 828.

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Answer:

84

Step-by-step explanation:

Let's call the father's present age "F" and the son's present age "S".

Five years ago, the father's age was F - 5 and the son's age was S - 5.

According to the first statement, F - 5 = 6(S - 5), so F = 6S - 35.

Two years in the future, the father's age will be F + 2 and the son's age will be S + 2.

According to the second statement, F + 2 = 3(S + 2) + 1, so F = 3S + 7.

We can now substitute the first equation into the second:

3S + 7 = 6S - 28,

3S - 6S = -35 - 7 = -42,

-3S = -42,

S = 14.

Finally, the father's present age is F = 6S - 35 = 84.

Answer:

The dad is 35 and the son is 10

Step-by-step explanation:

Let f =  the father's age

let s - the son's age

In the equations f-5 and s-5 represents the father and son's ages 5 years ago.   f + 2 and s +2 represent the father and son's ages two years in the future.

Answer:

he item should be put in the oven at 12:25 (in 24-hour format) to be ready for lunch at 13:00.

Step-by-step explanation:

To calculate what time the item should be put in the oven, you need to work backwards from the desired serving time of 13:00 and subtract the cooking time of 35 minutes. Here's how you can do it:

Convert the serving time to 24-hour format: 13:00 is the same as 1:00 PM in 12-hour format. In 24-hour format, it is 13:00.

Subtract the cooking time: Subtract 35 minutes from the serving time of 13:00 to get the time when the item should be put in the oven.

13:00 - 00:35 = 12:25

So, the item should be put in the oven at 12:25 (in 24-hour format) to be ready for lunch at 13:00.

Answer: i don't know I just did this so i can get points to ask a question super sorry.

Step-by-step explanation:

Step-by-step explanation:

Outliers: In this data set, the value of 501 is an outlier as it is significantly higher than the other values.

Median: To find the median, we arrange the data in order and find the middle value. In this case, since there are an odd number of values, the median is the middle value: 290.

Mean: To find the mean, we add up all the values and divide by the number of values:

(248 + 260 + 267 + ... + 311 + 501) / 21 = 293.67

10% Trimmed Mean: To find the 10% trimmed mean, we delete the bottom 10% of the values (2 values) and the top 10% of the values (2 values) and then find the mean of the remaining values:

(278 + 282 + 284 + 284 + 285 + 287 + 287 + 290 + 290 + 293 + 295 + 296 + 299 + 311) / 15 = 286.87

20% Trimmed Mean: To find the 20% trimmed mean, we delete the bottom 20% of the values (4 values) and the top 20% of the values (4 values) and then find the mean of the remaining values:

(284 + 284 + 285 + 287 + 287 + 290 + 290 + 293 + 295 + 296 + 299 + 311) / 12 = 288.92

The median, mean, 10% trimmed mean, and 20% trimmed mean can be compared to see how resistant each measure is to outliers. The median is the most resistant measure as it only considers the middle value, whereas the mean is the least resistant measure as it considers all values in the data set. The trimmed means are intermediate measures that are more resistant to outliers than the mean but less resistant than the median. In this case, we can see that the 10% trimmed mean is more resistant to outliers than the mean but less resistant than the 20% trimmed mean.

Answer:

Radius = 5 units

Diameter = 10 units

Circumference = 10π units

Area = 25π square units.

Step-by-step explanation:

i choose diameter.

Let's say the diameter of the circle is 10 units.

The radius of the circle can be found by dividing the diameter by 2:

Radius = Diameter / 2 = 10 / 2 = 5 units

The circumference of the circle can be found using the formula:

Circumference = 2πr = 2π * 5 = 10π units

The area of the circle can be found using the formula:

Area = πr^2 = π * 5^2 = 25π square units.

So, for a circle with a diameter of 10 units:

Radius = 5 units

Diameter = 10 units

Circumference = 10π units

Area = 25π square units.

The double integral [tex]\int\limits^a_b {(x+y)} \, dx dy[/tex]over the rectangle r with vertices (0,1), (1,0), (3,4) and (4,3) is equal to 22.

Integration can be thought of as the reverse of differentiation, which is the process of finding the rate of change of a function at a given point.

There are two main types of integration: definite and indefinite integration.  The process of integration can be represented symbolically as ∫ (the integral symbol) and the result of an integration is typically represented as an antiderivative. The fundamental theorem of calculus states that differentiation and integration are inverse operations, so the derivative of an antiderivative is the original function.

To evaluate the double integral  [tex]\int\limits^a_b {} \, dx dy[/tex]over the rectangle r with vertices (0,1), (1,0), (3,4) and (4,3), we first need to find the limits of integration for x and y.

The x-coordinates of the vertices of the rectangle range from 0 to 4, so the limits of integration for x are 0 to 4. The y-coordinates of the vertices of the rectangle range from 1 to 4, so the limits of integration for y are 1 to 4.

So, the double integral becomes:

[tex]\int\limits^a_b {(x+y)} \, dx dy[/tex] =  (2 × 4² + 8 × 4) - (2 × 1² + 8 × 1) = 32 - 10 = 22.

Therefore, the double integral [tex]\int\limits^a_b {(x+y)} \, dx dy[/tex] over the rectangle r with vertices (0,1), (1,0), (3,4) and (4,3) is equal to 22.

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In linear equation, The effect of the time of day a science class is taught on test scores being studied in an experiment will be the length of the class period.

What in mathematics is a linear equation?

An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.

                                 The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times. Equations with variables of power 1 are referred to as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.

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Based on the volume of the solid, the value of K is 8960.

The volume of the solid = volume of the cone + volume of the hemisphere

The volume of the cone = ¹/₃hπr²

The volume of the hemisphere = ²/₃πr³

where;

r is radius = 20 cm

h is the height of the cone = ?

The curved surface area of the cone = πrl

The curved surface area of the cone = 580π cm²

πrl =  580π

l = 580/20

l = 29 cm

h = √(29² - 10²)

h = 27.2 cm

The volume of cone = ¹/₃ * 27.2 * 20² * π

The volume of the cone = 3626.7π cm³

The volume of the hemisphere = ²/₃ * π * 20³

The volume of the hemisphere = 5333.3 cm³

The volume of the solid = 5333.3π cm³ + 3626.7π cm³

The volume of the solid = 8960π cm³

The volume of the solid is kπcm³ = 8960π cm³

K =  8960

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As a result, the length is 8.5 cm and the breadth is 3.5 cm, and the total perimeter is 22 cm.

what is perimeter ?

An enclosed line that surrounds, distinguishes, or incorporates a two-dimensional contour or a three-dimensional length is referred to as a boundary. The edge of an ellipse or circle refers to its broadest area. The perimeter calculation has many useful properties. The rim of a form is its edge's radius. Understand how to determine the perimeter by adding the lengths of such sides of various forms. You can always measure the perimeter of a shape by multiplying its side lengths. The perimeter of an object is the space around it. At your house, any enclosed terrace is one illustration. The border of anything is the area encircling it. For a 50 feet x 50 feet yard, a 200 foot fence is required.

given

let the length of the rectangular wallet be x

the width be x - 5

perimeter = sum of all the sides = 22

x + x - 5 = 22

2x - 5 = 22

2x = 17

x = 8.5

length is 8.5 and width is 8.5 - 5 = 3.5

As a result, the length is 8.5 cm and the breadth is 3.5 cm, and the total perimeter is 22 cm.

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Answer:

p = (39x +30 -5q)/13x

Step-by-step explanation:

16px + 10q + 10px = 78x + 60

Solve for p.

Subtract 10 q from  each side.

16px + 10q-10q + 10px = 78x + 60 -10q

16px + 10px = 78x + 60-10q

Combine like terms.

26px = 78x + 60-10q

Factor out p.

p(26x) =  78x + 60-10q

Divide each side by 26x.

p =  (78x + 60-10q)/(26x)

Simplify.

p = (39x +30 -5q)/13x

The  work required to pump the water out of the spout is 70.9 * 10⁶ joules which is 71 M Joules approx.

Let the spherical tank be divided into a series of horizontal disks.

Let x be the height of each disk, g = gravity and p = density

Volume of each disk = πr2dx

Where r is the radius of each disk and dx is the thickness

By Pythagoras r² = 6² - x²= 36 -x²

Therefore Volume = π*(36-x²)*dx cubic metres

The distance each disk has to be lifted to escape the top of the spout = 2 + 6-x meters = (8-x)

Work needed to lift each disk to outlet = distance * mass where

mass = volume * density * gravity

        = pi(36-x²)*p*g = 9800pi(36-x²)dx

dW =  (8-x)* 9800pi(36-x²)dx

Since we have already accounted for the height of the spout, the work required is obtained by integrating the above expression from the bottom of the tank to the top. In other words, - 6 to + 6.

W = ⌠(8-x)*9800pi(36-x2)dx = 9800pi⌠[288 - 36x - 8x² +x³]dx

The odd terms evaluate to zero and we can rewrite the above as 2* I from 0 to 6.

W = 9800pi*2 [288x - 8x³/3] at x = 6 = 1152*19600π Joules = 22579200π joules = 70963200j

Total work required = 70.9 * 10⁶ joules

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Answer:

28 cm^2

Step-by-step explanation:

The area will be the height (4)   times the average of the bases (8 and 6 , average = 7 )

4 x 7 = 28 cm^2

The lines are skew is Lines AD and BF.

Skew lines are two lines that do not intersect, are not parallel, and are not coplanar. This means that the skew lines never cross and are not parallel to each other. For lines to be two-dimensional or coplanar, they can intersect or be parallel. This property does not apply to diagonal lines, which are always non-coplanar and exist in three or more dimensions.

In real life, cities have different types of roads, such as highways and overpasses. These streets are supposed to be on different levels. Lines drawn on such roads never intersect, creating skew lines instead of being parallel to each other.

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