12.) Show that each conditional statement in Exercise 10 is a tautology without using truth tables.
b) [(p → q) ∧ (q → r)] → (p → r)

All the solutions are,

g⁻¹ = { (7, - 7), (- 7, - 6), (6, - 1), (9, 3) }

h⁻¹ (x) = 5x - 13

⇒ (h⁻¹ о h ) (- 3) = 3

We have to given that,

Functions g and f are defined as,

g = {(- 7, 7), (- 6, - 7), (- 1, 6), (3, 9)

And, h (x) = (x + 13) / 5

Now, We can simplify for inverse function as,

For inverse function of g,

g = {(- 7, 7), (- 6, - 7), (- 1, 6), (3, 9)

g⁻¹ = { (7, - 7), (- 7, - 6), (6, - 1), (9, 3) }

And, Inverse function of h is,

h (x) = (x + 13) / 5

y = (x + 13) / 5

Solve for x,

5y = x + 13

x = 5y - 13

h⁻¹ (x) = 5x - 13

So, We get;

⇒ (h⁻¹ о h ) (- 3)

⇒ (h⁻¹ (h (- 3))

⇒ (h⁻¹ (- 3 + 13)/5)

⇒ (h⁻¹ (2))

⇒ 5 (2) - 13

⇒ 10 - 13

⇒ - 3

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Hello!

A = 115° => opposite

B = 115° => corresponding angles

C = 65° => 180° - 115°

Answer:

A = 115 degrees

B = 115 degrees

C = 65 degrees

Step-by-step explanation:

A is opposite of 115 so it is 115 degrees.

B is 115 because it is on a parellel line to the one of 115

C is 180 minus B, which is 180 - 115 = 65

Answer:

Step-by-step explanation:

Answer:

For month 1, the current balance is $2750.00, the interest is $45.38, and the payment is $75.00. The amount applied to principal is $29.62.

For the remaining months, the interest and payment amount will stay the same, but the current balance and amount applied to principal will change based on the previous month's numbers.

Point of view:

Here's your answer but I prefer you to focus and study hard because school isn't that easy. But i'm glad I could help you!

:)

The solution set to the system of inequalities will be the overlapping region or the intersection of the shaded regions from all three inequalities.

The system of inequalities consists of three inequalities:

y ≤ 2x + 2

(1/2)x + y < 7

y - 3 ≥ 0

Let's analyze each inequality:

y ≤ 2x + 2 represents a shaded region below the line with a slope of 2 and a y-intercept of 2.

(1/2)x + y < 7 represents a shaded region below the line with a slope of -1/2 and a y-intercept of 7.

y - 3 ≥ 0 represents a shaded region above the line with a slope of 0 and a y-intercept of 3.

The solution set to the system of inequalities will be the overlapping region or the intersection of the shaded regions from all three inequalities.

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The value of x in triangle STU is given as follows:

b) 8.

The Pythagorean Theorem states that in the case of a right triangle, the square of the length of the hypotenuse, which is the longest side,  is equals to the sum of the squares of the lengths of the other two sides.

Hence the equation for the theorem is given as follows:

c² = a² + b².

In which:

Applying the Pythagorean Theorem for this problem, the value of x is obtained as follows:

x² + 15² = 17²

x² = 17² - 15²

x² = 64.

x = 8.

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Answer:

100°

Step-by-step explanation:

tThe sum of all arc measures that make up that circle is 360 degrees.

QS + RQ + RS = 360

QS = 360 - 120 - 140 = 100

An arc angle is the degree measurement of that angle inside the circle, opposite the arc

m∠R = arc QS = 100°

Answer:

∠ R = 50°

Step-by-step explanation:

the inscribed angle R is half the measure of its intercepted arc QS

the sum of the arcs on a circle is 360° , that is

RQ + QS + SR = 360°

120° + QS + 140° = 360°

QS + 260° = 360° ( subtract 260° from both sides )

QS = 100°

Then

∠ R = [tex]\frac{1}{2}[/tex] × 100° = 50°

By using the table of values, a graph of the exponential function is shown in the image below.

In Mathematics and Geometry, an exponential function can be modeled by using this mathematical equation:

[tex]f(x) = a(b)^x[/tex]

Where:

Based on the information provided above, we can logically deduce the following exponential function;

[tex]y = 2^x[/tex]

Next, we would create a table of values based on the exponential function;

when x = 0, the y-value is given by;

y = 2⁰

y = 1

when x = 1, the y-value is given by;

y = 2¹

y = 2

x                     y____

-2                 0.25

-1                  0.5

0                   1

1                    2

2                   4

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There is 90% confidence that the population mean number of books people read is between 11.55 and 13.25.

To construct a 90% confidence interval for the mean number of books people read, we can use the following formula:

Confidence Interval = x ± (Z * (s / √n))

Where:

x = sample mean (12.4 books)

s = sample standard deviation (16.6 books)

n = sample size (1017)

Z = Z-score

Since the sample size is large (n > 30), we can assume the sampling distribution is approximately normal.

We can use the standard normal distribution to find the Z-score for a 90% confidence level.

The Z-score for a 90% confidence level is approximately 1.645.

Now we can calculate the confidence interval:

Confidence Interval = 12.4 ± (1.645  (16.6 / √1017))

Confidence Interval ≈ 12.4 ± (1.645  (16.6 / √1017))

Confidence Interval ≈ 12.4 ± (1.645  (16.6 / 31.95))

Confidence Interval ≈ 12.4 ± (1.645 x 0.518)

Confidence Interval ≈ 12.4 ± 0.85

There is 90% confidence that the population mean number of books people read is between 11.55 and 13.25.

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Option C is correct, the value of x in the triangle is 10.28 units.

The given triangle is right angled triangle.

We know that the sine function is a ratio of opposite side and hypotenuse.

To the angle 40 degrees the opposite side is x.

The hypotenuse is 16.

Sin 40 degrees=x/16

0.6428=x/16

Apply cross multiplication:

x=16×0.6428

x=10.28

Hence, the value of x in the triangle is 10.28 units.

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Incorrect,  the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation

Neither Pierre nor Scott has the correct exponential growth equation.

The exponential growth equation represents a relationship where a quantity increases or grows exponentially over time. It is typically represented as y = a(1 + r)^x, where "a" represents the initial or starting value, "r" represents the growth rate (expressed as a decimal), "x" represents the time or number of periods, and "y" represents the resulting value after the growth.

In this case, Pierre's equation is y =[tex]300(1.02)^x.[/tex]This equation suggests a growth rate of 2% (0.02 as a decimal), which means that the quantity would increase by 2% with each period. This aligns with the given growth rate of 2%. Thus, Pierre's equation is correct.

On the other hand, Scott's equation is y = [tex]300(1.2)^x[/tex]. This equation suggests a growth rate of 20% (0.2 as a decimal), which means that the quantity would increase by 20% with each period. However, the given growth rate is 2%, not 20%. Therefore, Scott's equation is incorrect.

To summarize, Pierre's equation, y = 300(1.02)^x, is the correct exponential growth equation as it accurately represents a starting value of 300 and a growth rate of 2%. Scott's equation, y = 300(1.2)^x, does not match the given growth rate and is therefore incorrect.

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Answer: There are 16 fluid ounces in 1 pint. To determine the number of pints in the jug, we need to divide the total number of fluid ounces by 16.

Given that the jug contains 36 fluid ounces of apple juice, we divide 36 by 16:

36 fluid ounces ÷ 16 fluid ounces/pint = 2.25 pints

Therefore, the jug contains 2.25 pints of apple juice.

Answer:

1/4+1/3+1/8

=17/24

Step-by-step explanation:

1-17/24

=7/24

7/24×240

=70

This is a simple one.

All you need to do is find the scale factor of the shapes. (How much has it decreased by?)

Wecan find this by doing 15 ÷ 20 (not the other way round because we want to find the scale factor for the bigger rectangle to the smaller)

15 ÷ 20 = 0.75

So to find the area all you need to do is times the area with the scale factor which is 0.75

500 x 0.75 = 375cm²

So the answer is

(A) 375cm²

Answer: the question doesn't give that much information but you would have 7/4 or 1.75 green counters.

Step-by-step explanation: 7/4 as a decimal is 1.75

The average utility price in Orlando for the 6 month period, and the way it compares to the national average is d. $ 308. 83 ; higher than the national average .

The information for Orlando is April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.

The average is:

= ( 288 + 310 + 325 + 294 + 293 + 343 ) / 6

= 1, 853 / 6

= $ 308. 83

This average is higher than the national average of $ 308. 83.

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Full question is:

The average national utility price is $270.48. Over a 6-month period, what is the average utility price in Orlando? How does this compare with the national average?

A graph titled Orlando, Florida, has month on the x-axis and utility price on the y-axis. In April, the cost was 288 dollars; May, 310 dollars; June, 325 dollars; July, 294 dollars; August, 293 dollars; September, 343 dollars.

a. $370.60; higher than the national average

b. $292.17; higher than the national average

c. $38.35; lower than the national average

d. $308.83; higher than the national average

The remaining credit after 65 minutes of calls is approximately $15.45.

To find the remaining credit after 65 minutes of calls, we can use the given information to determine the linear function that relates the remaining credit to the total calling time.

Let's assume the total calling time in minutes is represented by the variable "x," and the remaining credit in dollars is represented by the variable "y."

We are given two data points:

When x = 43, y = $19.41.

When x = 56, y = $17.72.

We can use these data points to form a system of linear equations.

Let's solve it to find the equation of the linear function.

Using the point-slope form of a linear equation:

y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope, we can substitute the values:

For the first data point:

x1 = 43

y1 = 19.41

Using the second data point:

x2 = 56

y2 = 17.72

The slope (m) can be calculated as:

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

m = (17.72 - 19.41) / (56 - 43)

m = -1.69 / 13

m ≈ -0.13

Now, we can use the point-slope form with one of the data points to find the equation of the linear function:

Using (x1, y1) = (43, 19.41):

y - 19.41 = -0.13(x - 43)

Simplifying the equation:

y - 19.41 = -0.13x + 5.59

y = -0.13x + 24

Now that we have the equation of the linear function, we can substitute x = 65 to find the remaining credit after 65 minutes:

y = -0.13(65) + 24

y ≈ $15.45

Therefore, the remaining credit after 65 minutes of calls is approximately $15.45.

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Answer: To calculate the distance the beetle walks diagonally across the table, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, the two sides of the right triangle formed by the table are 160 cm and 90 cm. Let's call the hypotenuse (the distance the beetle walks) "d."

So, applying the Pythagorean theorem, we have:

d^2 = 160^2 + 90^2

Simplifying:

d^2 = 25600 + 8100

d^2 = 33700

Taking the square root of both sides:

d ≈ √33700

d ≈ 183.54 cm

Therefore, the beetle walks approximately 183.54 cm diagonally across the kitchen table.

==============================================

Explanation:

The formula to use is n*(n-1)/2

Plug in n = 12 to get 12*(12-1)/2 = 12*11/2 = 132/2 = 66

That formula is from the nCr combination formula when r = 2.

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

Here's a visual way to see why that rule works.

Imagine there are only 4 teams. We'll make a table that has 4 rows and 4 columns. The team names are A,B,C,D.

[tex]\begin{array}{|c|c|c|c|c|} \cline{1-5} & A & B & C & D\\\cline{1-5}A & & & & \\\cline{1-5}B & & & & \\\cline{1-5}C & & & & \\\cline{1-5}D & & & & \\\cline{1-5}\end{array}[/tex]

In the table, cross out the cells where one team pairs up with itself. That's along the main diagonal pointing to the northwest (or southeast).

[tex]\begin{array}{|c|c|c|c|c|} \cline{1-5} & A & B & C & D\\\cline{1-5}A & X & & & \\\cline{1-5}B & & X & & \\\cline{1-5}C & & & X & \\\cline{1-5}D & & & & X\\\cline{1-5}\end{array}[/tex]

Let's also cross out any cell below the main diagonal.

[tex]\begin{array}{|c|c|c|c|c|} \cline{1-5} & A & B & C & D\\\cline{1-5}A & X & & & \\\cline{1-5}B & X & X & & \\\cline{1-5}C & X & X & X & \\\cline{1-5}D & X & X & X & X\\\cline{1-5}\end{array}[/tex]

We do this because a mirrored copy is found above the diagonal.

We started with 4*4 = 16 cells. Then subtracted off the diagonal to get 16-4 = 12 cells left over. Then divide in half because we crossed off the lower half to get 12/2 = 6 different matchups among the 4 teams.

Those 6 matchups are:

The order doesn't matter. A matchup like "A vs B" is the same as "B vs A".

We'll take this concept to extend it to n teams. We have n*n = n^2 cells to start with. Subtract off the n cells along the diagonal to get n^2-n = n(n-1) cells remaining.

Then we split that in half to get the formula n(n-1)/2.

Given statement solution is :- The Inverse line that represents [tex]f^(-1[/tex]) is [tex]f^(-1)[/tex] (x) = 3x + 4.

To find the inverse function of f(x) = (x - 4)/3, we need to swap the roles of x and y and solve for y. Let's start by writing the equation with y instead of f(x):

y = (x - 4)/3

Now, let's interchange x and y:

x = (y - 4)/3

To solve for y, we can isolate it by multiplying both sides of the equation by 3:

3x = y - 4

Next, let's add 4 to both sides:

3x + 4 = y

Finally, we can write the inverse function as:

[tex]f^(-1)(x)[/tex] = 3x + 4

So, the Inverse line that represents [tex]f^(-1[/tex]) is [tex]f^(-1)[/tex] (x) = 3x + 4.

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A square and a traingle are present in a large rectangle with given dimensions in the figure.

[tex]\qquad\displaystyle \tt \dashrightarrow \: 12 \times 8[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 96 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 4 \times 4[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 16 \: \: unit {}^{2} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: \frac{1}{2} \times 4 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 2 \times 5[/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: 10 \: \: unit {}^{2} [/tex]

[ area of square / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{16}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(inside \: \: the \: \: square) = \frac{1}{6} [/tex]

[ total area except area of triangle / total area ]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{96 - 10}{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{86 }{96} [/tex]

[tex]\qquad\displaystyle \tt \dashrightarrow \: p(outside \: the \: triangle) = \frac{43}{48} [/tex]

The solution is the multiplication is: 5x × (3x-2)  = 15x² - 10x.

Here, we have,

given that,

the expression is:

5x(3x-2)

now, we have to Multiply out

so, we have,

5x × (3x-2)

= 5x×3x - 5x×2

=15x² - 10x

Hence, The solution is the multiplication is: 5x × (3x-2)  = 15x² - 10x.

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Answer:

32 inches Aprox

Step-by-step explanation:

To find the size of a rectangular television screen given its height and width, we can use the Pythagorean theorem. The diagonal measurement (size) is the hypotenuse of a right triangle formed by the height and width.

Given:

Height of the television screen (h) = 20 inches

Width of the television screen (w) = 25 inches

Using the Pythagorean theorem:

diagonal² = height² + width²

diagonal² = 20² + 25²

diagonal² = 400 + 625

diagonal² = 1025

Taking the square root of both sides:

diagonal ≈ √1025

diagonal ≈ 32.02

Rounding to the nearest inch, the size of the television screen is approximately 32 inches.

Therefore, the size of the rectangular television screen, rounded to the nearest inch, is 32 inches.

Answer:

[tex]\textsf{a)} \quad x = 17[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-3n-1[/tex]

[tex]\textsf{a)} \quad x = 20[/tex]

[tex]\textsf{b)} \quad T_n=3n^2-4n+5[/tex]

Step-by-step explanation:

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=-1\\a+b+c&=-1\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=5\\4a+2b+c&=5\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=35\\16a+4b+c&=35\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b-1[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b-1)&=5\\3a+b&=6\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b-1)&=35\\15a+3b&=36\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+6[/tex]

[tex]\begin{aligned}15a+3(-3a+6)&=36 \\15a-9a+18&=36\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+6\\&=-3(3)+6\\&=-9+6\\&=-3\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b-1\\&=-3-(-3)-1\\&=-3+3-1\\&=-1\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-3n-1}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-3(3)-1\\&=3(9)-3(3)-1\\&=27-9-1\\&=18-1\\&=17\end{aligned}[/tex]

Therefore, the value of x is 17.

[tex]\hrulefill[/tex]

Given quadratic number pattern:

To find the equation for the nth term, we can use the general form of a quadratic equation:

[tex]\boxed{T_n=an^2 + bn + c}[/tex]

where n is the position of the term.

Let's substitute the values of T₁, T₂ and T₄ into the quadratic equation: to create three equations:

[tex]\begin{aligned}T_1=a(1)^2+b(1)+c&=4\\a+b+c&=4\end{aligned}[/tex]

[tex]\begin{aligned}T_2=a(2)^2+b(2)+c&=9\\4a+2b+c&=9\end{aligned}[/tex]

[tex]\begin{aligned}T_4=a(4)^2+b(4)+c&=37\\16a+4b+c&=37\end{aligned}[/tex]

Rearrange the first equation to isolate c:

[tex]c=-a-b+4[/tex]

Substitute this into the second and third equations:

[tex]\begin{aligned}4a+2b+(-a-b+4)&=9\\3a+b&=5\end{ailgned}[/tex]

[tex]\begin{aligned}16a+4b+(-a-b+4)&=37\\15a+3b&=33\end{ailgned}[/tex]

Solve the equations simultaneously by rearranged the first equation to isolate b and substituting this into the second equation and solving for a:

[tex]b=-3a+5[/tex]

[tex]\begin{aligned}15a+3(-3a+5)&=33 \\15a-9a+15&=33\\6a&=18\\a&=3 \end{aligned}[/tex]

Substitute the found value of a into the equation for b and solve for b:

[tex]\begin{aligned}b&=-3a+5\\&=-3(3)+5\\&=-9+5\\&=-4\end{aligned}[/tex]

Finally, substitute the found values of a and b into the equation for c and solve for c:

[tex]\begin{aligned}c&=-a-b+4\\&=-3-(-4)+4\\&=-3+4+4\\&=5\end{aligned}[/tex]

Therefore, the equation for the nth term is:

[tex]\boxed{T_n=3n^2-4n+5}[/tex]

The value of x is the 3rd term. Therefore, to find the value of x, substitute n = 3 into the equation for the nth term:

[tex]\begin{aligned}T_3&=3(3)^2-4(3)+5\\&=3(9)-4(3)+5\\&=27-12+5\\&=15+5\\&=20\end{aligned}[/tex]

Therefore, the value of x is 20.

The value of AB is,

⇒ AB = 5.9

(rounded to nearest tenth)

We have to given that,

A right triangle ABC is shown.

Now, By trigonometry formula,

we get;

⇒ cos 33° = Base / Hypotenuse

Substitute all the values, we get;

⇒ cos 33° = AB / 7

⇒ 0.84 = AB / 7

⇒ AB = 0.84 × 7

⇒ AB = 5.88

⇒ AB = 5.9

(rounded to nearest tenth)

Thus, We get;

AB = 5.9

(rounded to nearest tenth)

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The measure of unknown angles,

∠1  = ∠4 = 110 degree

∠2 = ∠3 = 70 degree

Labeling the figure,

Then from figure we have,

⇒ x + 40 + x = 180

⇒     2x + 40 = 180

Subtract 40 both sides,

⇒            2x  = 140

Divide both sides by 2

⇒              x  = 70

From figure,

⇒ ∠1 + 70 = 180

⇒         ∠1 = 180 - 70

⇒         ∠1 = 110 degree

And from figure,

⇒ ∠1 + ∠2 = 180

⇒ 110 + ∠2 = 180

⇒          ∠2 =  70 degree

Since,

∠2 = ∠3 = 70 degree

And  ∠1  = ∠4 = 110 degree

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Answer:

x = 12

Step-by-step explanation:

sin45 = x/17         (sin = opposite/hypotenuse)

x = (sin45)(17) = 12.02 ≈ 12

Answer: A

Step-by-step explanation:

The maximum number of CYN Dewi can buy is 4268.8 CYN and cost in pounds is  £453.20.

Maximum number of CYN Dewi can buy:

Since £1 buys 9.28 CYN, Dewi's £460 can be converted to Chinese yuan by multiplying it by the exchange rate:

Maximum CYN = £460 × 9.28 CYN/£1

To calculate this, we multiply the pound amount by the exchange rate:

Maximum CYN = £460 × 9.28 CYN/£1 ≈ 4268.8 CYN

Cost in pounds to the nearest penny:

To calculate the cost in pounds, we divide the desired amount of Chinese yuan by the exchange rate:

Cost in pounds = 4268.8 CYN ÷ 9.42 CYN/£1

To calculate this, we divide the Chinese yuan amount by the exchange rate:

Cost in pounds = 4268.8 CYN ÷ 9.42 CYN/£1

= £453.20

Therefore, the cost in pounds, to the nearest penny, will be £453.20.

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Complete question

Buying chinese yuan (CYN)             £1 buys 9.28 CYN

Selling chinese yuan (CYN)           9.42 CYN buys  £1

(a) Dewi has £460 to buy Chinese yuan.

Calculate the maximum number of CYN Dewi can buy, and

how much, to the nearest penny, this will cost him.