2. Consider this dilation.
Pre-image
9 cm
B
Image
3 cm B
(a) Find the scale factor. Show your work. (5 points)
(2- a.) Scale factor= Image length
Pre-image length

The statement indicates that in practical applications, it is common to extend the basic logical language with additional symbols and allow for some degree of informality in the representation of formulas, as long as it is clear that these additional symbols and formulas can be formalized in terms of the basic logical language.

The statement means that when working with formulas in a particular context or domain, it is common to introduce additional symbols beyond the basic logical symbols such as "E" (for existential quantification) and "∧" (for conjunction). These additional symbols can include defined predicates, operations, and constants.

Defined predicates refer to predicates or relations that are defined in terms of other predicates or relations. They can be used to express complex concepts or conditions in a concise manner. For example, in mathematics, one may define a predicate "Prime(x)" to represent the concept of a number being prime.

Operations refer to functions or operations that can be applied to terms or formulas. They allow for combining or manipulating expressions in a meaningful way. For instance, addition, multiplication, and exponentiation are common operations in mathematics.

Constants are symbols that represent specific, fixed values. They can be used to denote specific objects or elements within the domain of discourse. For example, in logic, "0" and "1" can be used as constants to represent the truth values "false" and "true," respectively.

Furthermore, the statement suggests that formulas may be used informally, meaning that they can be expressed or written in a less strict or precise manner. While the informal representation of formulas may be easier to understand or work with, it is understood that each of these formulas can be translated or reformulated in a way that only involves the basic logical symbols (such as "E" and "∧") and the specified nonlogical symbols (such as defined predicates, operations, and constants).

Overall, the statement acknowledges that in practical applications, it is common to extend the basic logical language with additional symbols and allow for some degree of informality in the representation of formulas, as long as it is clear that these additional symbols and formulas can be formalized in terms of the basic logical language.

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The product of the fractions is 1/[(x - 5)(x + 7)]

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

Fraction 1 = (x + 1)/(x² - 25)

Also, we have

Fraction 2 = (x + 5)/(x² + 8x + 7)

So, we have

Product = (x + 1)/(x² - 25) * (x + 5)/(x² + 8x + 7)

When factored, we have

Product = (x + 1)/(x - 5)(x + 5) * (x + 5)/(x + 1)(x + 7)

Evaluate the products

Product = 1/(x - 5) * 1/(x + 7)

This gives

Product = 1/[(x - 5)(x + 7)]

Hence, the product of the fractions is 1/[(x - 5)(x + 7)]

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Answer:

plane

point

line

Step-by-step explanation:

Based on the options provided, the kinds of symmetry in a 2D number can be:

Therefore, the possible kinds of symmetry in a 2D number can be

plane symmetry,

point symmetry, or

line symmetry.

The vertices of triangle image are L'(1, -4), M'(6, 4) and N'(7, -2).

Given that, triangle LMIN with vertices L(2, -8), M(12, 8) and N(14,-4).

Here, scale factor k=1/2

Now, by applying scale factor to the vertices, we get

L(2, -8)→1/2 (2, -8)→L'(1, -4)

M(12, 8)→1/2 (12, 8)→M'(6, 4)

N(14,-4)→1/2 (14, -4)→N'(7, -2)

Therefore, the vertices of triangle image are L'(1, -4), M'(6, 4) and N'(7, -2).

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"Your question is incomplete, probably the complete question/missing part is:"

Triangle LMIN with vertices L(2, -8), M(12, 8), and N(14,-4): k = ½.

Answer: $127.50

Step-by-step explanation: if we pull up our handy dandy calculator and do 15% of 150 it will output 22.50, 150 - 22 = 127.50 which brings the total to $127.50

Answer:

Infinite solutions (D).

Step-by-step explanation:

Here is how:

To determine the true statement about the given equation, let's simplify it step by step:

-9(x + 3) + 12 = -3(2x + 5) - 3x

Distributing the -9 and -3 on the left and right sides respectively:

-9x - 27 + 12 = -6x - 15 - 3x

Combining like terms:

-9x - 15 = -9x - 15

Now, let's analyze the equation. We have -9x on both sides, and -15 on both sides. By subtracting -9x from both sides and -15 from both sides, we obtain:

0 = 0

This equation is true regardless of the value of x. In other words, it holds for all values of x. Therefore, the equation has infinitely many solutions.

Answer:

The correct answer is: "The equation has one solution, x = 0"

Teresa's result of 95 heads is slightly higher than what she would have expected, it is still reasonably close to the expected value based on the theoretical probability of getting a head when flipping a fair coin.

Teresa's result of 95 heads in 150 coin flips is slightly higher than what she would have expected if the theoretical possibility of getting a head when flipping a coin was exactly 1/2.

When flipping a fair coin, the probability of getting a head is indeed 1/2, and the probability of getting a tail is also 1/2. Therefore, if we assume that the coin is fair and unbiased, we can use the concept of expected value to determine what Teresa would have expected.

The expected value of a single coin flip can be calculated as follows:

Expected value = (Probability of an outcome) * (Value of the outcome)

In this case, the probability of getting a head is 1/2, and the value of getting a head is 1 (since we are interested in the number of heads). Therefore, the expected value of a single coin flip is:

Expected value = (1/2) * 1 = 1/2

To determine what Teresa would have expected in 150 coin flips, we can multiply the expected value of a single coin flip by the number of flips:

Expected number of heads = (Expected value) * (Number of coin flips)

Expected number of heads = (1/2) * 150 = 75

So, if Teresa had flipped the coin 150 times and the coin was fair, she would have expected to get approximately 75 heads.

However, Teresa obtained 95 heads in her actual experiment. This result is slightly higher than the expected value of 75 heads. It's important to note that, due to the inherent randomness of coin flips, there will always be some variation between the expected and actual results. Flipping a coin 150 times is a relatively small sample size, and it is within the realm of possibility to deviate from the expected outcome.

Therefore, while Teresa's result of 95 heads is slightly higher than what she would have expected, it is still reasonably close to the expected value based on the theoretical probability of getting a head when flipping a fair coin.

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The law of sines indicates that, where the angle of elevation of the Sun is 73°, the length of the shadow of the tree is 20 ft, and the inclination of the tree is 5°, the length of the tree is 51.1 ft. The correct option is therefore;

(C) 51.1 ft.

The law of sines states that the ratio of the length of a side in a triangle to the sine of the angle facing that side is equivalent for the three sides of a triangle.

The length of the shadow = 20 feet

Angle of elevation to the Sun = 73°

The angles formed by the triangle formed by the tree are;

73°, (90 - 5) = 85°, (180 - 85 - 73) = 22°

Let l represent the length of the tree. The law of sines indicates that we get;

20/(sin(22)) = l/sin(73)

Therefore; l = sin(73°) × (20/(sin(22°))) ≈ 51.1 feet

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Answer:

First, we need to figure out how many circular rugs can fit in the living room without overlap.

One way to approach this is to find the area of each circular rug (using the formula A = πr^2, where r = 3 feet since the diameter is 6 feet).

A = π(3)^2 = 28.26 square feet

Next, we can find the area of the living room:

A = 12 x 18 = 216 square feet

To figure out how many circular rugs can fit, we can divide the living room area by the rug area:

216/28.26 ≈ 7.64

Since we can't have partial rugs, we need to round down, which means Shannon can fit 7 circular rugs in her living room without overlap.

The total area covered by the circular rugs would be:

7 x 28.26 = 197.82 square feet

Therefore, the closest answer choice to the nearest square foot is A) 170 ft².

Answer:A

Step-by-step explanation:

She can put 2 circles together width wise because 6+6 = 12

She can put 3 circles together length wise because 6+6+6 = 18

So she can put 6 circles in total.

The area of one circle is found with the equation A=πr^2

The diameter is 6ft, so the radius is 3ft, so

A=π*3ft^2

A=28.26ft²

This is the area of one circle. We need to find the area of 6 because 6 can fit in total.

28.26ft² * 6= 169.56 ft²

a. The total length of the painted white line in the javelin throwing arena is approximately 255.984 meters.

b. The total area of grass in the shaded landing area of the javelin throwing arena is approximately 1624.6 square meters.

To find the total length of the painted white line, we need to calculate the length of the two straight segments and the two circular arcs.

a) Total length of the painted white line:

Let's break it down into components:

1. The two straight segments: Each straight segment is 95 meters long, and there are two of them.

Length of straight segments = 2 * 95 = 190 meters

2. The two circular arcs:

The throwing arena is a sector of a circle with a radius of 100 meters. The angle of the sector can be calculated using trigonometry. The angle can be found by taking the inverse cosine of the ratio of the adjacent side (which is the radius minus the throwing line distance) to the hypotenuse (which is the radius).

Angle (in radians) = cos⁻¹((radius - throwing line distance) / radius)

Now, the length of each circular arc can be calculated using the formula for the length of an arc of a circle:

Length of circular arc = radius * angle

Let's calculate the angle first:

Angle (in radians) = cos⁻¹((100 - 5) / 100)

Angle = cos⁻¹(95 / 100)

Angle ≈ 0.32492 radians

Now, we can calculate the length of each circular arc:

Length of each circular arc = 100 * 0.32492 ≈ 32.492 meters

Since there are two circular arcs, the total length of the painted white line is:

Total length = 190 (straight segments) + 2 * 32.492 (circular arcs)

Total length ≈ 255.984 meters

b) Total area of grass in the shaded landing area:

The shaded landing area is the sector of the circle with a radius of 100 meters and the angle we calculated above.

The formula to calculate the area of a sector of a circle is:

Area of sector = (angle / 2π) * π * radius²

Area of the shaded landing area = (0.32492 / (2π)) * π * 100²

Area of the shaded landing area ≈ (0.32492 / 2) * 10000

Area of the shaded landing area ≈ 1624.6 square meters

So, the total area of grass in the shaded landing area is approximately 1624.6 square meters.

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The length of side b in triangle ABC is 18.7 units.

In Mathematics and Geometry, the law of sines is also referred to as sine law or sine rule and it can be defined as an equation that relates the side lengths of a triangle to the sines of its angles.

In Mathematics and Geometry, the law of sine is modeled or represented by this mathematical equation (ratio):

[tex]\frac{sinA}{a} =\frac{sinB}{b} =\frac{sinC}{c}[/tex]

In this context, the value of b can be determined as follows;

sin112/37 = sin28/b

b = 37sin28/sin112

b = 17.3705/0.9272

b = 18.7 units.

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The solution to the given simultaneous equations using Cramer's rule is x = -1, y = -4.92, z = -1.03.

To solve the given simultaneous equations using Cramer's rule, we need to find the determinants of various matrices.

Given the set of equation:

3x-3y+ 5z = 5

9x - 8y + 13z = 14

-3x+4y- 7z=-7

We start by finding the determinant of the coefficient matrix, which is denoted as D.

D = [tex]\left[\begin{array}{ccc}3&-3&5\\9&-8&13\\-3&4&-7\end{array}\right][/tex]

To calculate D, we use the formula:

D = (3 * (-8) * (-7) + (-3) * 13 * (-3) + 5 * 9 * 4) - ((-3) * (-8) * 5 + 3 * 13 * 4 + (-3) * 9 * (-7))

D = (-168 + 117 + 180) - (120 - 156 + 189)

D = 129 - 165

D = -36

Next, we need to find the determinants of the matrices obtained by replacing the columns of the coefficient matrix with the constant terms. These determinants are denoted as Dx, Dy, and Dz, respectively.

Dx = [tex]\left[\begin{array}{ccc}5&-3&5\\14&-8&13\\-7&4&-7\end{array}\right][/tex]

Dx = (-40 - 65 + 20) - (-70 - 60 + 189) = -85 - (-121) = -85 + 121

Dx= 36

Dy = [tex]\left[\begin{array}{ccc}3&5&5\\9&14&13\\-3&-7&-7\end{array}\right][/tex]

Dy = (21 + 75 + 35) - (-21 - 130 + 105) = 131 - (-46) = 131 + 46

Dy = 177

Dz = [tex]\left[\begin{array}{ccc}3&-3&5\\9&-8&14\\-3&4&-7\end{array}\right][/tex]

Dz = (39 - 39 + 0) - (-27 + 84 + 20) = 0 - (-37) = 0 + 37

Dz = 37

Finally, we can find the values of x, y, and z using the formulas:

x = Dx / D

y = Dy / D

z = Dz / D

Plugging in the values, we have:

x = 36 / -36 = -1

y = 177 / -36 ≈ -4.92

z = 37 / -36 ≈ -1.03

Therefore, the solution to the given simultaneous equations using Cramer's rule is x = -1, y ≈ -4.92, z ≈ -1.03.

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The function to model the amount remaining after t hours is:

[tex]A(t) = 500 \times e^{-0.045t}[/tex]

We have,

To model the amount remaining after t hours for the substance sodium-24, which decays at a rate of 4.5% per hour compounded continuously, we can use the formula for continuous exponential decay:

[tex]A(t) = A(0) \times e^{-rt}[/tex]

Where:

A(t) is the amount remaining after t hours,

A(0) is the initial amount,

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

r is the decay rate (expressed as a decimal).

In this case,

The initial amount A(0) is 500 grams, and the decay rate r is 4.5% per hour, which can be expressed as 0.045 (decimal form).

Thus,

The function to model the amount remaining after t hours is:

[tex]A(t) = 500 \times e^{-0.045t}[/tex]

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Based on the Triangle Inequality Theorem the possible lengths of the sides of a triangle are:

The Triangle Inequality Theorem which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side, is used to determine if three lengths can be the lengths of the sides of a triangle.

Considering each option:

A. 12cm, 5cm, 17cm:

12 + 5 > 17 (not satisfied)

5 + 17 > 12 (satisfied)

12 + 17 > 5 (satisfied)

B. 10cm, 15cm, 24cm:

10 + 15 > 24 (satisfied)

15 + 24 > 10 (satisfied)

10 + 24 > 15 (satisfied)

C. 9cm, 22cm, 11cm:

9 + 22 > 11 (satisfied)

22 + 11 > 9 (satisfied)

9 + 11 > 22 (not satisfied)

D. 21cm, 7cm, 6cm:

21 + 7 > 6 (satisfied)

7 + 6 > 21 (not satisfied)

21 + 6 > 7 (satisfied)

Hence, option B and D are correct.

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The monthly payment on the 15-year fixed rate mortgage is $515.27.

The bank requires a 10% down payment on the condo price of $80,000.

Down Payment = 10% of $80,000

Down Payment = 0.1 * $80,000

Down Payment = $8,000

Therefore, the required down payment is $8,000.

Amount of the Mortgage:

To find the amount of the mortgage, we subtract the down payment from the condo price.

Amount of Mortgage = Condo Price - Down Payment

Amount of Mortgage = $80,000 - $8,000

Amount of Mortgage = $72,000

Therefore, the amount of the mortgage is $72,000.

Now, Monthly Payment = (Loan Amount x Monthly Interest Rate) / (1 - (1 + Monthly Interest Rate) ^ (-Number of Months))

Monthly Interest Rate = Annual Interest Rate / 12

= 3.5% / 12 = 0.035 / 12

= 0.002917

and, Number of Months = 15 years x 12 months = 180 months

So, Monthly Payment = ($72,000 x 0.002917) / (1 - (1 + 0.002917) ^ (-180))

Monthly Payment ≈ $515.27

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Answer:

Step-by-step explanation: On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer.

Answer:

Step-by-step explanation:

I Think this is the  answer

On the left side find the number that is able to make that sum true for that equation. on the right side you just subtract the answer with the number to get your answer

The length of PQ in circle R is given as follows:

C) 3.14 m.

The circumference of a circle of radius r is given by the equation presented as follows:

C = 2πr.

The radius for the circle in this problem is given as follows:

r = 3 m.

Hence the circumference for the entire circle is given as follows:

C = 2 x 3.14 x 3

C = 18.84 m.

The sector has an angle measure of 60º, while the entire circumference is of 360º. hence the measure of arc PQ is given as follows:

PQ = 60/360 x 18.84

PQ = 3.14 m.

Hence option C is the correct option in the context of this problem.

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The value of the given expression:

⇒ mk = 12

⇒ m + k = 8  

The given values,

m = 6

k  = 2

We have evaluate the given expression,

mk

This is nothing but product of m and k

Since we know,

The product is the result of multiplying two or more numbers together. Assume they are two integers and, then their product is derived by multiplying both numbers together.

Therefore,

⇒ mk = 6x2

⇒ mk = 12

We have evaluate the given expression,

m + k

This is nothing but addition of m and k

Since we know that,

Addition is the process of joining two or more integers. The numbers being added are known as addends, and the result or final response obtained after the procedure is known as the total.

⇒ m + k = 6 + 2

⇒ m + k = 8          

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Formulas with free variables are said to be expressions in formal logic that is known to have variables that are not bound by quantifiers.

Variables are placeholders with changeable values in logic. A free variable in a formula has no quantifier and can take any value.

Formula meaning varies with variable values. Formulas containing variables that are not assigned specific values are frequently employed to convey  open statements or propositions that require further specification.

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In practice, we shall use in formulas other symbols, namely defined pred- icates, operations, and constants, and even use formulas informally; but it will be tacitly understood that each such formula can be written in a form that only involves and as nonlogical symbols.

Concerning formulas with free variables, we adopt the notational conven- tion that all free variables of a formula

(u1,..., Un)

are among u1, ..., un (possibly some u are not free, or even do not occur, in ). A formula without free variables is called a sentence.What is the meaning of "formulas with free variables"?

Answer: is X = 5 within the diagram.

The quadratic equation with roots 5 + i√2 and 5 - i√2 is:

x^2 - 10x + 27 = 0

To write a quadratic equation with roots 5 + i√2 and 5 - i√2, we can use the fact that complex roots occur in conjugate pairs. Therefore, the equation will have the form:

(x - root1)(x - root2) = 0

Substituting the given roots:

(x - (5 + i√2))(x - (5 - i√2)) = 0

Now, we expand the equation:

(x - 5 - i√2)(x - 5 + i√2) = 0

Using the difference of squares formula:

((x - 5)^2 - (i√2)^2) = 0

Simplifying the equation:

(x - 5)^2 + 2 = 0

Expanding the square:

x^2 - 10x + 25 + 2 = 0

Combining like terms:

x^2 - 10x + 27 = 0

Therefore, the quadratic equation with roots 5 + i√2 and 5 - i√2 is:

x^2 - 10x + 27 = 0

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No, the given two fractions/ratios do not form a proportion.

Two ratios are said to be forming proportions if and only if the reduced forms of fractions of the two ratios are the same or are equal to each other.  For example - [tex]\frac{3}{6}[/tex] and [tex]\frac{1}{2}[/tex] are in proportion because [tex]\frac{3}{6}[/tex] when reduced to the simplest form gives [tex]\frac{1}{2}[/tex] which is the same as the other given ratio.

In the given question, one ratio given is [tex]\frac{1}{3}[/tex] and the other one is [tex]\frac{28}{36}[/tex]. Reducing the second ratio to the simplest form we get [tex]\frac{7}{9}[/tex]. Clearly, the two fractions are not equal. Hence, the given two ratios are not in proportion.

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Solving a system of equations we can see that the costs are:

For a chair, 3/2 dollars.

For a table, 25/4 dollars.

Here we can define the variables:

x = cost of rending a chair.

y = cost of renting a table.

Here we can write a system of equations:

8x + 4y = 37

3x + 2y = 17

We can subtract two times the second equation from the first one:

8x + 4y - 2*(3x + 2y) = 37 - 2*17

2x = 3

x = 3/2

And the value of y is:

3*(3/2) + 2*y = 17

2y = 17 - 9/2 = 25/2

y = 25/2/2

y = 25/4

These are the costs in dolllars.

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[tex] \boxed{\rm{Similarity \: shape}}[/tex]

[tex]\begin{aligned} \frac{AB}{DE}&= \frac{BC}{EF}\\ \frac{36}{24}&=\frac{15}{x} \\ x &= \frac{\cancel{^{ \green{2}}24} \times 15}{\cancel{36_{ \green{3}}}} \\ x&= \frac{2 \times 15}{3} \\ x &= \bold{10} \\ \\\small{\blue{\mathfrak{That's \: it \: :)}}} \end{aligned}[/tex]

Answer:

Part A: The actual measurement of the piece of wood could be between 9.65 and 9.75 inches.

Part B: The measurement has three significant digits.

Step-by-step explanation:

Significant digits are the digits in a number that carry meaning in terms of the precision or accuracy of the measurement. In this case, the number 9.7 has three significant digits, which means that the measurement is accurate to the tenths place.

A true statement based on the given information is that ΔCAT and ΔDOG are similar triangles, meaning their corresponding angles are congruent and their corresponding sides are in proportion.

If all pairs of corresponding angles and sides in triangles ΔCAT and ΔDOG are congruent, it implies that the two triangles are similar. In similar triangles, the corresponding angles are congruent, and the corresponding sides are in proportion.

Based on this information, the following true statement can be made:

The ratio of the lengths of the corresponding sides in ΔCAT and ΔDOG is equal.

For example, if the corresponding sides are CA and DO, the ratio CA/DO will be equal to the ratio of the lengths of the other corresponding sides.

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Answer:

C. y + 4 = 4(x+3)

Step-by-step explanation:

Point slope form: y-y1 = m(x-x1)

Substitute the given slope 4 and point (-3, -4)

y-(-4) = 4(x - (-3))

Simplify

y + 4 = 4 (x+3)

Area of a polygon formula: = 2(1+[tex]\sqrt{2}[/tex]) side²

To find one side: 96 ÷ 8 = 12

implement formula

≈695.29351