Cameron aces all her exams. However, she blows off the homework, quizzes and class participation her score for the semester were:

Exam 1: 93/100
Exam 2: 95/100
Exam 3: 90/100
Exam 4: 89/100
Homework: 10/100
Quizzes: 14/100
Participation: 50/100

Cameron‘s teacher assigns final grades using a standard scale ( 90% is A, 80% is a B, 70% is a C, 60%is a D) if Cameron’s teacher uses the points system for averaging final grades where you add up all of the points earned and divide by the points possible in the course what grade did she get in the class?

A, B, C, D

Answer:

Step-by-step explanation:

To find the limit of (n!)^(1/n) as n approaches infinity, we can use the Stirling's approximation for n!, which is:

n! ≈ (n/e)^n √(2πn)

where e is the mathematical constant e ≈ 2.71828, and π is the mathematical constant pi ≈ 3.14159.

Using this approximation, we can rewrite (n!)^(1/n) as:

(n!)^(1/n) = [(n/e)^n √(2πn)]^(1/n) = (n/e)^(n/n) [√(2πn)]^(1/n)

Taking the limit as n approaches infinity, we have:

lim (n!)^(1/n) = lim (n/e)^(n/n) [√(2πn)]^(1/n)

Using the fact that lim a^(1/n) = 1 as n approaches infinity for any constant a > 0, we can simplify the second term as:

lim [√(2πn)]^(1/n) = 1

For the first term, we can rewrite (n/e)^(n/n) as [1/(e^(1/n))]^n and use the fact that lim a^n = 1 as n approaches infinity for any constant 0 < a < 1. Thus, we have:

lim (n/e)^(n/n) = lim [1/(e^(1/n))]^n = 1

Therefore, combining the two terms, we have:

lim (n!)^(1/n) = lim (n/e)^(n/n) [√(2πn)]^(1/n) = 1 x 1 = 1

Hence, the limit of (n!)^(1/n) as n approaches infinity is 1.

Answer:1

Step-by-step explanation:

Answer:

the value of r that makes the slope of the line passing through (3, 5) and (-3, r) equal to 3/4 is r = 1/2

Step-by-step explanation:

We can use the formula for the slope of a line passing through two points, which is:

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

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

In this case, we have two points (3, 5) and (-3, r), and we know that the slope m is 3/4. So we can write:

3/4 = (r - 5)/(-3 - 3)

Simplifying this equation, we get:

3/4 = (r - 5)/(-6)

Multiplying both sides by -6, we get:

-9/2 = r - 5

Adding 5 to both sides, we get:

r = -9/2 + 5

Simplifying, we get:

r = 1/2

The measure of the arc intercepted by the angle and the vertical angles make up the angle subtended at the center. As a result, XYZ has a value of 35°.

Two lines intersect at a location, creating an angle.

An "angle" is the term used to describe the width of the "opening" between these two rays. The character is used to represent it.

Angles are frequently expressed in degrees and radians, a unit of circularity or rotation.

In geometry, an angle is created by joining two rays at their ends. These rays are referred to as the angle's sides or arms.

An angle has two primary components: the arms and the vertex. T

he two rays' shared vertex serves as their common terminal.

Hence, The measure of the arc intercepted by the angle and the vertical angles make up the angle subtended at the center. As a result, XYZ has a value of 35°.

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The coordinates of A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin are A'(5, 10) and B'(-10, -5), respectively.

To find the coordinates of the points A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin, we can use the following formula:

[tex]$$(x', y') = (5(x - 0), 5(y - 0)) = (5x, 5y)$$[/tex]

where (x, y) are the original coordinates of the point, and (x', y') are the new coordinates after the dilation.

For point A(1, 2), the new coordinates A' are:

[tex]$$(x_A', y_A') = (5(1), 5(2)) = (5, 10)$$[/tex]

Therefore, the coordinates of point A' are (5, 10).

For point B(-2, -1), the new coordinates B' are:

[tex]$$(x_B', y_B') = (5(-2), 5(-1)) = (-10, -5)$$[/tex]

Therefore, the coordinates of point B' are (-10, -5).

Therefore, the coordinates of A' and B' after a dilation with a scale factor of 5 and a center point of dilation at the origin are A'(5, 10) and B'(-10, -5), respectively.

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The probability that the computer generates a number between 1 and 4 is 4/11.

Since X has a uniform distribution, the probability of generating any number between 0 and 10 is equally likely. Therefore, the probability of generating a number between 1 and 4 is the same as the probability of generating a number between 0 and 4 minus the probability of generating 0

P(1 ≤ X ≤ 4) = P(X ≤ 4) - P(X = 0)

Since X has a uniform distribution, the probability of generating any specific number is 1/11. Therefore

P(X ≤ 4) = (4 + 1)/11 = 5/11

and

P(X = 0) = 1/11

So,

P(1 ≤ X ≤ 4) = (5/11) - (1/11) = 4/11

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I have solved the question in general, as the given question is incomplete,

The complete question is:

A random number generator generates numbers between 0 and 10. Let random variable X be the number generated. Suppose X has a uniform distribution. What is the probability that the computer generates a number between 1 and 4?

The sequence, [tex]a_n =e ^{\frac{{-8}}{\sqrt{n}}}[/tex], is convergent sequence because the limit of an exists, that is as n approaches infinity, so the sequence an approaches 1 ( finite value).

The sequence can be convergent if the limit is zero, or if the limit is finite. The divergent sequence is one whose limit is not finite. The limit can be found suing the limit properties or by simplification method, as applicable. We have, an sequence, [tex]a_n =e ^{\frac{{-8}}{\sqrt{n}}}[/tex]. We have to check whether the sequence converges or diverges. Using limits, [tex]lim_ {n->\infty } a_n = lim_{n-> oo} e^{\frac{-8}{\sqrt{n}}} [/tex]

n approaches infinity, so square root of n approaches infinity,

= e⁻⁰

= 1/e⁰ = 1 ( finite )

Therefore, it is a convergent sequence.

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Multiple linear regression is a statistical technique used to model the relationship between a dependent variable and multiple independent variables.

In multiple linear regression,

The dependent variable is modeled as a linear function of several independent variables.

Regression coefficient that quantifies the strength and direction of the relationship between independent and dependent variable.

Coefficient standard deviation refers to the standard deviation of the estimated regression coefficients in multiple linear regression.

Provides a measure of the variability of the estimated coefficients and can be used to assess the precision of the estimates.

New multiple regression model,

Collect data on the dependent variable and multiple independent variables of interest.

Perform a multiple linear regression analysis, which involves fitting a linear equation to the data using the method of least squares.

Estimates the regression coefficients and their standard deviations.

Used to assess the significance of each independent variable and the overall fit of the model.

Therefore,  multiple regression model consider adding or removing independent variables, transforming the data, or machine learning algorithms.

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The above question is incomplete , the complete question is:

What is multiple linear regression with coefficient standard deviation and mean to get new multiple regression ?

So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.

The measurement and study of random events are the focus of the mathematic branch known as probability. It entails calculating the probability of an event happening, with a scale from 0 (impossible) to 1. (certain). The fundamental meaning of chance is: Number of favourable outcomes minus the total number of outcomes equals the probability of an occurrence.

given

There are a total of the following painted stones in the box:

Total painted stones equals Claire's painted stones plus Laura's painted stones.

Total number of decorated stones: 35 + 40 = 75

Number of yellow stones equals the sum of Claire's and Laura's yellow pebbles.

5 Plus 20 = 25 yellow pebbles total.

As a result, the likelihood of getting a yellow pebble on any given draw is:

Number of yellow pebbles / total painted pebbles represents the likelihood of painting a yellow pebble.

25/75 is the likelihood of getting a yellow pebble.

The likelihood of drawing a yellow pebble on each draw is the same because we are drawing with substitution.

In 75 pulls, there should be an average of:

Number of draws times the likelihood of getting a yellow pebble yields the expected number of yellow pebbles.

Number of yellow pebbles anticipated Equals 75 * (1/3)

Expected quantity of golden stones: 25

So, out of 75 pulls, we would anticipate choosing a yellow pebble 25 times.

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Therefore , the solution of the given problem of function comes out to be f(x) = sec(x) has a period of 2. 2 is the answer.

The midterm test questions will cover all of the topics, including actual as well as fictitious locations and arithmetic variable design. a diagram showing the relationships between different elements that cooperate to create the same result. A service is composed of numerous distinctive components that cooperate to create distinctive results for each input. Every mailbox has a particular spot that might be used as a haven.

Here,

=> F(x) = sec(x) has a 2 phase.

Because the secant function is periodic, its values recur after a predetermined amount of time.

This interval's length is equal to the secant function's duration.

The formula for the secant function is

=> sec(x) = 1/cos.(x).

The cosine function repeats its values every 2 units of x, which is known as its period.

Consequently, the secant function has a period of 2 as well.

Therefore, f(x) = sec(x) has a period of 2. 2 is the answer.

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The function of tiles with no empty spaces is n(t) = t² + 2t, the function of tiles removed to create empty spaces is r(t) = t.

We have a sequence of tiles represented by the numbers 3, 8, and 15.

This sequence can be described as a quadratic sequence, which can be expressed using the formula

n(t) = at² + bt + c.

By substituting the values of the first three terms into this formula, we can find the values of a, b, and c.

a + b + c = 3

4a + 2b + c = 8

9a + 3b + c = 15

These values can also be determined by graphing the sequence. In this case, a = 1, b = 2, and c = 0.

Therefore, the quadratic sequence can be written as n(t) = t² + 2t.

The given sequence of tiles can be represented as a linear sequence in which the current term is equal to the number of spaces.

Therefore, the function r(t) can be defined as r(t) = t.

The given equation is expressed as:

A(t) = n(t) + r(t)

Therefore, substituting t² + 2t + t for A(t), we get:

A(t) = t² + 2t + t

Consequently, the function can be represented as

A(t) = t² + 3t

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The steps for the perpendicular bisector construction when ordered are :

Identify the two endpoints of the given segment AB and labeling them as A and B. This is important for the subsequent steps of the construction. Then draw two circles: one centered at A with a radius equal to the length of AB, and the other centered at B with a radius equal to the length of BA (which is the same as AB).

The two circles drawn in Step 2 intersect at two points. These points are labeled as C and D. Note that these points are equidistant from A and B, since they lie on the circles with radii AB and BA.

Finally, we draw a line through points C and D. This line is the perpendicular bisector of the segment AB, meaning it intersects AB at its midpoint and is perpendicular to AB.

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The height οf the cylinder is apprοximately 6.38 centimetres.

The cylinder's altitude is a perpendicular segment frοm the plane οf οne base tο the plane οf the οther, and the cylinder's height is the length οf the altitude. A cylinder's axis is the segment cοntaining the centres οf the twο bases.

The vοlume οf a right circular cylinder is calculated as fοllοws:

[tex]V = \pi r^2h[/tex]

where V is the vοlume, r is the radius οf the base, and h is the cylinder's height.

Substituting the fοllοwing values:

[tex]1280 = \pi(8^2)h[/tex]

Simplifying:

[tex]1280 = 64\pi h[/tex]

By dividing bοth sides by 64, we get:

[tex]h = 1280/(64\pi)[/tex]

Simplifying and rοunding tο the nearest twο decimal places:

h ≈ 6.38

As a result, the cylinder's height is apprοximately 6.38 centimetres.

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20. Took the quiz, and received a 100 percent.

(a) 12 cm 40° : Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​: Area of shaded segments = 777.04 sq. cm.

Two radii that meet at the center to form a sector define a circle. The sector is the portion of the circle created by these two radii. Knowing a circle's central angle calculation and radius measurement are both crucial for solving circle-related difficulties.

Area of sector of circle = Ф/360 * πr²

π = 3.14

r  is the radius

Ф is the angle subtended.

(a) 12 cm 40°

Area of shaded segments = 40/60 * 3.14* 12²

Area of shaded segments = 40/60 * 452.16

Area of shaded segments = 301.44 sq. cm.

(b) 58° 16 cm ​

Area of shaded segments = 58/60 * 3.14* 16²

Area of shaded segments = 58/60 * 803.84

Area of shaded segments = 777.04 sq. cm.

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The diagram for the question is attached.

Answer:

The common difference between consecutive terms in the sequence is 8 (since -17 + 8 = -9, -9 + 8 = -1, -1 + 8 = 7, and so on). Therefore, the recursive formula for this arithmetic sequence is:

a(1) = -17

a(n) = a(n-1) + 8 for n >= 2

This formula says that the first term in the sequence is -17, and each subsequent term is found by adding 8 to the previous term.

(please mark my answer as brainliest)

Answer: -1/6

Step-by-step explanation:

Put the equation into y = mx + b (slope) form.

3x + 18y = -486

18y = -3x - 486

y = -1/6x - 27

If the line is parallel to this line, the slope must be the same.

Answer:

To subtract 1/9 - 1/14, we need to find a common denominator. The smallest number that both 9 and 14 divide into is 126.

So, we will convert both fractions to have a denominator of 126:

1/9 = 14/126

1/14 = 9/126

Now we can subtract them:

1/9 - 1/14 = 14/126 - 9/126

Simplifying the right-hand side by subtracting the numerators, we get:

5/126

Therefore, 1/9 - 1/14 = 5/126 as an improper fraction.

Answer:

1/9-1/14

=14-9/9*14

=5/126

= 25 1/5

therefore, we have shown that: [tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex] assuming that x2 ≠ x1.

Slope refers to the measure of steepness of a line or a curve. In mathematics, slope is usually denoted by the letter "m" and is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on a line.

The formula for calculating the slope between two points (x1, y1) and (x2, y2) on a line is:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]

by the question.

To finds the slope of a line passing through two points (x1, y1) and (x2, y2), we use the slope formula:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})[/tex]

This formula represents the change in y divided by the change in x between the two points.

Now, assuming that x2 ≠ x1, we can simplify the formula as follows:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(1/1)[/tex]

Multiplying the numerator and denominator by 1, which in this case is (x2 - x1) / (x2 - x1), we get:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})*(x_{2}-x_{1})/(x_{2}-x_{1})[/tex]

Simplifying the numerator, we have:

[tex]m= (y_{2} -y_{1} )/(x_{2}-x_{1})/[(x_{2}-x_{1})*1][/tex]

The term (x2 - x1) cancels out, leaving us with:

[tex]m=(y_{2}-y_{1} /1[/tex]

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Answer: 3 cm

Step-by-step explanation:

The formula for the area of a circle is A = πr², where A is the area and r is the radius. We are given that the area of the circle is 28½ cm².

So, 28½ = πr²

We need to solve for r. Dividing both sides by π, we get:

r² = 28½/π

r² = 9

Taking the square root of both sides, we get:

r = 3√1 = 3 cm

Therefore, the radius of the circle is 3 cm.

Step-by-step explanation:

try this option (see the attachment), answers are marked with red colour.

The solution using matrix inversion is x = -1 and y = 0.

To solve this system of equations using matrix inversion, we first need to write the system in matrix form:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}-4\\-4\end{array}\right][/tex]

Next, we need to invert the coefficient matrix on the left-hand side of the equation by finding its inverse. The inverse of a 2x2 matrix is given by:

[tex]\left[\begin{array}{cc}a&b\\c&d\end{array}\right]^{-1} =\frac{1}{(ad - bc)} \left[\begin{array}{cc}-d&b\\c&-a\end{array}\right][/tex]

Using this formula, we can find the inverse of the coefficient matrix [4 1; 4 -3]:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1} =\frac{1}{(-12 - 4)} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]=\frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right][/tex]

Now we can solve for [x y] by multiplying both sides of the equation by the inverse of the coefficient matrix:

[tex]\left[\begin{array}{cc}4&1\\4&-3\end{array}\right]^{-1}\left[\begin{array}{cc}4&1\\4&-3\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{cc}3&1\\4&-4\end{array}\right]\left[\begin{array}{c}-4\\-4\end{array}\right][/tex]

[tex]\left[\begin{array}{cc}1&0\\0&1\end{array}\right] \left[\begin{array}{c}x\\y\end{array}\right] = \frac{-1}{16} \left[\begin{array}{c}-12-4\\-16+16\end{array}\right] = \left[\begin{array}{c}-1\\0\end{array}\right][/tex]

Therefore, x = -1 and y = 0, which is the same solution we obtained using row reduction.

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Answer:

see step by step.

Step-by-step explanation:

The THEORETICALLY probability is 0.5000 (this is, 50% each one) since each one have equal chance of being in list A or List B, but this is still a random probability, so this is not the same that if you have 60 students, 30 of them are going to be in the list A and the other 30 in B.

The experimental result show:

[tex]P (ListA) = \frac{27}{60} =0.45[/tex]

[tex]P (ListB) = \frac{33}{60} =0.55[/tex]

This is 45% of being in list A and 55% of being in list B, notice these values are close the the experimental value (50%) but still are lightly higher or lower. Additionally notice if you add both of them you still have 100% (0.45+0.55=1.00)

To solve this problem, we can use Bayes' theorem. Let's define:

- A: two story home

- B: owning a computer

We want to find the probability of B given A, which we can write as P(B|A). Using Bayes' theorem, we have:

P(B|A) = P(A|B) * P(B) / P(A)

We know that 50% of two story homes own computers, so P(B) = 0.5. We also know that 75% of homes in Dawn's neighborhood are two story, so P(A) = 0.75.

To find P(A|B), we need to use the conditional probability formula:

P(A|B) = P(A and B) / P(B)

We don't have the probabilities for A and B happening together, but we know that:

P(A and B) = P(B|A) * P(A)

Substituting this into the conditional probability formula gives:

P(A|B) = (P(B|A) * P(A)) / P(B)

Putting all the values together, we get:

P(B|A) = (P(A|B) * P(B)) / P(A)

= ((P(B|A) * P(A)) / P(B)) * P(B) / P(A)

= (P(B and A) / P(B)) * P(B) / P(A)

= P(A and B) / P(B)

Substituting the values gives:

P(B|A) = (0.5 * 0.75) / 0.5

= 0.75

Therefore, the probability that a two story home in Dawn's neighborhood owns a computer is 0.75.

V is not a vector space, since at least one of the vector space axioms fails. Specifically, the axiom of closure under addition fails, since for any positive real numbers x and y, their sum xy may not be positive.

V is not a vector space, since some of the vector space axioms fail with the given operations.

Closure under addition fails For example, taking x=2 and y=3, we have xy = 6, but xy is not a positive real number, so xy is not in V.

Distributivity of scalar multiplication over vector addition fails For example, taking x=2, y=3, and c=-1, we have c(x+y) = cxy = -6, but cx + cy = -2 -3 = -5, which is not in V.

Other vector space axioms, such as associativity of addition, existence of additive identity and inverse, and compatibility of scalar multiplication with field multiplication, are still satisfied with the given operations.

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_____The given question is incomplete, the complete question is given below:

Let V be the set of all positive real numbers. Determine whether V is a vector space with the following operations. x + y = xy Addition cx = x Scalar multiplication If it is, then verify each vector space axiom; if it is not, then state all vector space axioms that fail. STEP 1: Check each of the 10 axioms. (1) U + v is in V. This axiom holds. This axiom fails. (2) U + V = V + u This axiom holds. This axiom fails. (3) U+ (v + w) = (u + v) + w This axiom holds. This axiom fails. STEP 2: Use your results from Step 1 to decide whether V is a vector space. O V is a vector space. O Vis not a vector space.

There is a $243,570 cap on the amount you may finance.

MentaI arithmetic, aIso known as "quick math," aIgebra, trigonometry, statistics, and probabiIity are some of the key mathematicaI concepts and abiIities needed in the financiaI sector. A fundamentaI knowIedge of these abiIities shouId be sufficient to quaIify you for the majority of finance occupations.

The highest home vaIue that may be financed is x.

$39,000 in avaiIabIe cash for a down payment

16% is the needed down payment percentage.

16% of the home's worth can be expressed as,

x × 16/100=$39,000

x=39,000/0.16

x=$243,570

We can then infer that the highest vaIue you couId finance is $243,570.

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Answer:

Step-by-step explanation:

Let's call the number we're looking for "x".

The problem tells us that "if the sum of a number and eight is doubled, the result is seven less than the number", which can be translated into an equation:

2(x+8) = x-7

Now let's solve for x:

2x + 16 = x - 7

2x - x = -7 - 16

x = -23

Therefore, the number we're looking for is -23.

The linear homogeneous constant-coefficient equation is y'' - 4y' + 2y = 0

A linear homogeneous constant-coefficient equation is of the form:

ay'' + by' + cy = 0

where a, b, and c are constants, and y is a function of x. The general solution of this equation can be written as:

[tex]y(x) = c_1e^{(r_1x)} + c_2e^{(r_2x)}[/tex]

where r₁ and r₂ are the roots of the characteristic equation:

ar² + br + c = 0

In our case, the general solution is given as:

y(x) = [tex]Ae^{2x}[/tex] + Bcos(2x) + Csin(2x)

To find the linear homogeneous constant-coefficient equation that has this general solution, we need to first write the general solution in the form:

[tex]y(x) = c_1e^{(r_1x)} + c_2e^{(r_2x)}[/tex]

where r₁ and r₂ are the roots of the characteristic equation. Comparing this form with the given general solution, we see that r₁ = 2, r₂ = 2i, and r3 = -2i.

Therefore, the characteristic equation is:

a(r-2)(r-2i)(r+2i) = 0

Expanding this equation, we get:

a(r³ - 2r²i - 2ri² + 4r² + 8i²r - 8i) = 0

Simplifying and grouping the real and imaginary terms, we get:

a(r³ - 4r) + 2a(r² + 4) = 0

Dividing by a, we get:

r³ - 4r + 2(r² + 4) = y'' - 4y' + 2y = 0

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Complete Question:

Find a linear homogeneous constant-coefficient equation with the given general solution

y(x)=Ae²ˣ+Bcos(2x)+Csin(2x)

Answer:

The area of a semicircle with diameter 28 cm is 98π cm², or 307.88 cm² to the nearest tenth.

Step-by-step explanation:

A semicircle is a two-dimensional shape that is exactly half of a circle.

The area of a circle is given by the formula:

[tex]\sf A=\pi r^2[/tex]

where A is the area of the circle, and r is the radius of the circle.

The diameter of a circle is twice its radius.

Given the diameter of the semicircle is 28 cm, the radius is:

[tex]\sf r = \dfrac{28}{2} = 14 \; cm[/tex]

Substituting this into the formula for the area of a circle, we get:

[tex]\sf A = \pi(14)^2[/tex]

[tex]\sf A = 196 \pi[/tex]

Finally, divide this by two to get the area of the semicircle:

[tex]\sf Area\;of\;semicircle = \dfrac{1}{2} \cdot 196 \pi[/tex]

[tex]\sf Area\;of\;semicircle = 98 \pi\; cm^2[/tex]

So the area of a semicircle with diameter 28 cm is 98π cm², or 307.88 cm² to the nearest tenth.

Initially, while maintaining the very same [tex]x-[/tex]coordinates, the negative reflect across the [tex]x-axis[/tex] modifies the signs of all locations' [tex]y-[/tex]coordinates.

A negative value is one that always has a value lower than zero and is denoted by the minus (-) symbol. In a number line, negative values are displayed just left of zero.

If a number is higher than zero, it is considered positive. Or less zero numbers are referred to as negative numbers. If a number is bigger than that or equal to zero, it is not negative. If a number falls below or equal to zero, it is non-positive.

This transforms Figure [tex]1[/tex] into a new figure with points [tex](-2, -5), (-2, -4), (-3, -4), (-3, -2), (-5, -2), (-5, -5)[/tex].

Next, the [tex]180^{0}[/tex] rotation about the origin swaps the signs of both [tex]x[/tex] and [tex]y[/tex] coordinates of each point. This results in the final figure with points [tex](2, -5), (2, -4), (3, -4), (3, -2), (5, -2), (5, -5)[/tex].

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The cube has 8 vertices. Exercise 21 states that for a polyhedron with V vertices, E edges, and F faces, the following equation holds:

V - E + F = 2

To use this equation to find the number of vertices of a cube with 12 edges and 6 faces, we first need to identify the values of E and F.

A cube has 12 edges, as given in the problem statement, and it has 6 faces since a cube has 6 square faces.

Substituting these values into the equation, we get:

V - 12 + 6 = 2

Simplifying this equation, we have:

V - 6 = 2

Adding 6 to both sides, we get:

V = 8

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