Two sides of a triangle have lengths 43 and 67. The angle included between these sides measures 27degrees°. To the nearest hundreth, what is the length of the third side?

We can express the number 98 in exponential form as 10 raised to the power of 2. This means that by multiplying the base, which is 10, by itself twice, we obtain the value of 98.

To express 98 in exponential form, we need to determine the base and exponent that can represent the number 98.

Exponential form represents a number as a base raised to an exponent. Let's find the base and exponent for 98:

We can express 98 as 10 raised to a certain power since the base 10 is commonly used in exponential notation.

To find the exponent, we need to determine how many times we can divide 98 by 10 until we reach 1. This will give us the power to which 10 needs to be raised.

98 ÷ 10 = 9.8

Since 9.8 is still greater than 1, we need to continue dividing by 10.

9.8 ÷ 10 = 0.98

Now, we have reached a value less than 1, so we stop dividing.

From these calculations, we can see that 98 can be expressed as 10 raised to the power of 1 plus the number of times we divided by 10:

98 =[tex]10^1[/tex] + 2

Therefore, we can write 98 in exponential form as:

98 = [tex]10^3[/tex]

In summary, 98 can be expressed in exponential form as 10^2. The base is 10, and the exponent is 2, indicating that we multiply 10 by itself two times to obtain 98.

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Indianapolis: Ratio as decimal: 0.006939, Owners per 100: 0.7

New York: Ratio as decimal: 0.049018, Owners per 100: 4.9

Cairo: Ratio as decimal: 0.000554, Owners per 100: 0.1

Beijing: Ratio as decimal: 0.045336, Owners per 100: 4.5

Tokyo: Ratio as decimal: 0.064198, Owners per 100: 6.4

To calculate the ratio as a decimal and the owners per 100 for each city, we need to divide the number of owners by the total population and perform some additional calculations.

Let's calculate the ratio as a decimal for each city:

For Indianapolis:

Number of Owners: 6,245

Total Population: 0.90 million

Ratio as decimal: 6,245 / 0.90 million

= 6,245 / 900,000

= 0.006939

For New York:

Number of Owners: 911,216

Total Population: 18.6 million

Ratio as decimal: 911,216 / 18.6 million

= 911,216 / 18,600,000

= 0.049018

For Cairo:

Number of Owners: 10,598

Total Population: 19.1 million

Ratio as decimal: 10,598 / 19.1 million

= 10,598 / 19,100,000

= 0.000554

For Beijing:

Number of Owners: 959,611

Total Population: 21.2 million

Ratio as decimal: 959,611 / 21.2 million

= 959,611 / 21,200,000

= 0.045336

For Tokyo:

Number of Owners: 1,700,510

Total Population: 26.5 million

Ratio as decimal: 1,700,510 / 26.5 million

= 1,700,510 / 26,500,000

= 0.064198

Now, let's calculate the owners per 100 for each city by multiplying the ratio as a decimal by 100:

Owners per 100 for Indianapolis: 0.006939 * 100 = 0.6939

Owners per 100 for New York: 0.049018 * 100 = 4.9018

Owners per 100 for Cairo: 0.000554 * 100 = 0.0554

Owners per 100 for Beijing: 0.045336 * 100 = 4.5336

Owners per 100 for Tokyo: 0.064198 * 100 = 6.4198

Therefore, rounding the ratio as a decimal to 6 places, and the owners per 100 to one decimal, we have:

Indianapolis: Ratio as decimal: 0.006939, Owners per 100: 0.7

New York: Ratio as decimal: 0.049018, Owners per 100: 4.9

Cairo: Ratio as decimal: 0.000554, Owners per 100: 0.1

Beijing: Ratio as decimal: 0.045336, Owners per 100: 4.5

Tokyo: Ratio as decimal: 0.064198, Owners per 100: 6.4

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The transformation of g(x) to f(x) is g(x) is shifted up 15 units and shifted left by 16 units

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

The functions f(x) and g(x)

Where, we have

f(x) = 1/(x + 16) + 15

g(x) = 1/x

From the above, we have

Horizontal difference = 16 - 0 = 16

Vertical difference = 15 - 0 = 15

This means that the transformation of g(x) to f(x) is g(x) is shifted up 15 units and shifted left by 16 units

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Answer:

16 deliveries

Step-by-step explanation:

238 ≤ 10d + 78

160 ≤ 10d

16 ≤ d

Thus, he needs to make at least 16 deliveries to make at least $238 in one day.  Any less than 16 deliveries will cause him to fall short of his goal and any more than 16 deliveries will cause him to exceed his goal.

The 9th arrangement will have 36 dots, the 10th arrangement will have 45 dots, and when adding the number of dots in any two consecutive arrangements, the result will be the number of dots in the next arrangement.

To determine the number of dots in the 9th and 10th arrangements, we need to understand the pattern in the given sequence: 1, 3, 6, 10, 15, ...

To find the pattern, we can observe the differences between consecutive terms: 2, 3, 4, 5, ...

We notice that these differences increase by 1 each time. This indicates that the sequence is formed by adding consecutive positive integers to the previous term.

Using this pattern, we can determine the 9th and 10th arrangements as follows:

9th arrangement:

Starting with the 1st arrangement, we add the positive integers consecutively: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36.

Therefore, the 9th arrangement will have 36 dots.

10th arrangement:

Starting with the 1st arrangement, we add the positive integers consecutively: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45.

Therefore, the 10th arrangement will have 45 dots.

Now, let's address the observation when we take any two consecutive arrangements and add the number of dots.

If we take the nth arrangement and the (n+1)th arrangement, we can observe that the number of dots in the (n+1)th arrangement will be the sum of the number of dots in the nth arrangement and the number of dots added in the (n+1)th step.

For example, let's consider the 2nd and 3rd arrangements:

2nd arrangement: 3 dots

3rd arrangement: 3 + 3 (adding 2, the next positive integer) = 6 dots

This pattern holds true for any two consecutive arrangements in the sequence.

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Answer:

(A) AA similarity

Step-by-step explanation:

In ΔABC,

∠A + ∠B + ∠C = 180

∠A + 27 + 90 = 180

∠A = 180 - 90 - 27

∠A = 63

Comparing ΔABC and ΔMNP,

∠A = ∠M = 63

∠C = ∠P = 90

Therfore, by AA property, the two triangles are similar

The instantaneous rate of change of f(x)=4-2 [tex]x^{2}[/tex] at x=0.5 is −2. This means that at that specific point, the function is changing at a rate of -2 units per unit change in x.

The instantaneous rate of change of a function at a specific point can be found by taking the derivative of the function and evaluating it at that point. In this case, we have the function f(x)=4-2 [tex]x^{2}[/tex] and we want to find the instantaneous rate of change at x=0.5.

To find the derivative of f(x), we apply the power rule for differentiation. Taking the derivative of each term, the derivative of 4 is 0, and the derivative of −2[tex]x^{2}[/tex] is −4x. Therefore, the derivative of f(x) is  (x)=−4x.

Now, to find the instantaneous rate of change at x=0.5, we substitute

x=0.5 into the derivative function. f′(0.5)=−4(0.5)=−2.

This means that the function is changing at a rate of -2 units per unit change in x.

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If the total height between two floors is 9'0 1/2" and 14 risers are to be used in a set of stairs, the height of each riser would be approximately 7.75 inches.

To find the height of each riser in a set of stairs given the total height between two floors, we need to divide the total height by the number of risers.

First, let's convert the total height to a consistent unit. The distance between two floors is given as 9'0 1/2". We can convert this to inches by multiplying the feet by 12 and adding the remaining inches:

9 feet = 9 * 12 = 108 inches

0 1/2 inch = 0.5 inch

Total height = 108 + 0.5 = 108.5 inches

Next, we divide the total height by the number of risers (14) to find the height of each riser:

Height of each riser = Total height / Number of risers

Height of each riser = 108.5 inches / 14

Using a calculator, we can compute:

Height of each riser ≈ 7.75 inches

Therefore, the height of each riser in the set of stairs would be approximately 7.75 inches.

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The transformation T maps the unit square to a figure with an area of 24 and a vertex at (5,3) which gives: T(x, y) = (6x + 5, 8y + 3).

To find a transformation T that maps the unit square to a figure with an area of 24 and a vertex at (5,3), we can use a combination of scaling and translation.

Let's denote the unit square as S, and the transformed figure as F.

To obtain a rectangle with an area of 48 and a vertex at (5,3), we can scale the unit square by multiplying the x-coordinates by 6 and the y-coordinates by 8, and then translate the scaled rectangle by adding (5,3) to the coordinates of each point.

Therefore, the transformation T can be expressed as:

T(x, y) = (6x + 5, 8y + 3)

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The amount of inches that the plant grows in 14 days is given as follows:

420 inches.

The proportional relationship that models the situation is given as follows:

i = 30d.

This means that the plant grows by 30 inches every day.

After 14 days, we have that d = 14, hence the size of the plant after 14 days is given as follows:

i = 30 x 14

i = 420 inches.

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The number of subsets in the given set is as follows:

128.

Considering a set with n elements, the number of subsets in the set is the nth power of 2, that is:

[tex]2^n[/tex]

The set in this problem is composed by integers between 2 and 8, hence it has these following elements:

{2, 3, 4, 5, 6, 7, 8}.

The set has four elements, meaning that n = 7, hence the number of subsets is given as follows:

[tex]2^7 = 128[/tex]

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Answer:

D. √3

Step-by-step explanation:

The tangent ratio is given by:

tan (θ) = opposite/adjacent, where

Thus, we plug in 8√3 for the opposite side and 8 for the adjacent side, which gives us:

tan (60) = (8√3) / 8

Thus reduces down to √3 so D. is the correct answer.

Answer:

y= -x²-4x+2

Step-by-step explanation:

write in vertex form

a(x-h)²+k

in our case h = -2 and k= 6

y=a(x+2)²+6

now we just need to solve for a. we know that when x= 1 y = -3. plug these values in and solve for a

-3= a(1+2)²+6

-9=9a

a= -1

thus the formula is -(x+2)²+6

generally, teachers want things in standard form, so expand the exponent and simplify.

-(x²+4x+4)+6

y= -x²-4x+2

Answer:

[tex]y = -x^2 - 4x + 2[/tex]

Step-by-step explanation:

The equation of a parabola in vertex form is:

[tex]y = a(x - h)^2 + k[/tex]

where (h, k) is the vertex of the parabola.

In this case, the vertex is (-2, 6), so h = -2 and k = 6.

We also know that the parabola passes through the point (1, -3).

Plugging these values into the equation, we get:

[tex]-3 = a(1 - (-2))^2 + 6[/tex]

[tex]-3 = a(3)^2 + 6[/tex]

-9 = 9a

a = -1

Substituting a = -1 into the equation for a parabola in vertex form, we get the equation of the parabola:

[tex]y = -1(x + 2)^2 + 6[/tex]

This equation can also be written as:

[tex]y = -x^2 - 4x -4+6\\y=x^2-4x+2[/tex]

The equation of the line in the graph that passes through (0,1) and (4,0) is  [tex]y = -\frac{1}{4}x + 1[/tex].

The equation of line is expressed as:

y = mx + b

Where m is slope and b is the y-intercept.

From the image, the graph passes through points (0,1) and (4,0).

First, we calculate the slope (m) using the formula:

[tex]m = \frac{y_2 - y_1}{x_2 - x_1} \\\\m = \frac{0 - 1 }{4 - 0} \\\\m = -\frac{1 }{4 }[/tex]

Next, plug the slope m = -1/4 and one point (0,1) into the point-slope form and simplify:

( y - y₁ ) = m( x - x₁ )

[tex]y - 1 = -\frac{1}{4}( x - 0 ) \\\\y - 1 = -\frac{1}{4}x\\\\y = -\frac{1}{4}x + 1[/tex]

Therefore, the equation of the line is [tex]y = -\frac{1}{4}x + 1[/tex].

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The average speed in miles per minute during the interval of 30 and 40 is -1.5 miles per minute

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

The graph

During the interval of 30 and 40, we have

Distances = 15 and 0

So, we have

Average speed = (15 - 0)/(30 - 40)

Evaluate

Average speed = -1.5

Hence, the average speed in miles per minute is -1.5

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To calculate the ratio as a decimal, we divide the number of owners by the total population for each city.

For Indianapolis: Ratio = 6,245 / 0.9 million = 0.006938

For New York: Ratio = 911,216 / 18.6 million = 0.049019

For Cairo: Ratio = 10,598 / 19.1 million = 0.000554

For Beijing: Ratio = 959,611 / 21.2 million = 0.045270

For Tokyo: Ratio = 1,700,510 / 26.5 million = 0.064234

To calculate the owners per 100, we multiply the ratio by 100.

For Indianapolis: Owners per 100 = 0.006938 * 100 = 0.7 (rounded to one decimal place)

For New York: Owners per 100 = 0.049019 * 100 = 4.9 (rounded to one decimal place)

For Cairo: Owners per 100 = 0.000554 * 100 = 0.1 (rounded to one decimal place)

For Beijing: Owners per 100 = 0.045270 * 100 = 4.5 (rounded to one decimal place)

For Tokyo: Owners per 100 = 0.064234 * 100 = 6.4 (rounded to one decimal place)

Therefore, the ratio as a decimal and the owners per 100 for each city are as follows:

Indianapolis: Ratio = 0.006938, Owners per 100 = 0.7

New York: Ratio = 0.049019, Owners per 100 = 4.9

Cairo: Ratio = 0.000554, Owners per 100 = 0.1

Beijing: Ratio = 0.045270, Owners per 100 = 4.5

Tokyo: Ratio = 0.064234, Owners per 100 = 6.4

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Q1- The annual depreciation for the equipment is $2,000.

Q2- The book value of the equipment at the end of Year 3 is $6,500.

Q1: What is the annual depreciation for the equipment?

To calculate the annual depreciation, we need to determine the difference between the initial value and the salvage value, and divide it by the number of years.

Initial value = $12,500

Salvage value = $2,500

Number of years = 5

Annual depreciation = (Initial value - Salvage value) / Number of years

= ($12,500 - $2,500) / 5

= $10,000 / 5

= $2,000

Therefore, the annual depreciation for the equipment is $2,000.

Q2: What is the book value of the equipment at the end of Year 3?

The book value of the equipment at the end of a specific year can be calculated by subtracting the accumulated depreciation from the initial value.

Initial value = $12,500

Annual depreciation = $2,000

Number of years = 3

Accumulated depreciation = Annual depreciation * Number of years

= $2,000 * 3

= $6,000

Book value at the end of Year 3 = Initial value - Accumulated depreciation

= $12,500 - $6,000

= $6,500

Therefore, the book value of the equipment at the end of Year 3 is $6,500.

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The height of the upper end of the handrail above the floor is approximately 3.467726 meters.

To calculate the height of the upper end of the handrail above the floor, we can use trigonometry and the given information about the length of the handrail and the angle of inclination.

Let's break down the problem into two right triangles:

The first right triangle is formed by the handrail, the floor, and a vertical line connecting the lower end of the handrail to the floor. The vertical line represents the height of the lower end above the floor, which is given as 0.7m.

The second right triangle is formed by the handrail, the floor, and a horizontal line parallel to the floor, connecting the upper end of the handrail to the floor. This horizontal line represents the height we need to calculate.

Now, let's apply trigonometric ratios to find the height of the upper end of the handrail above the floor.

In the first right triangle:

Opposite side = height of the lower end = 0.7m

Hypotenuse = length of the handrail = 5.4m

Using the sine function:

sin(angle) = Opposite / Hypotenuse

sin(40°) = 0.7 / 5.4

Now, let's solve for the sin(40°):

sin(40°) ≈ 0.64279

Multiplying both sides of the equation by 5.4:

0.64279 * 5.4 ≈ 3.467726

So, the length of the vertical line in the second right triangle (representing the height of the upper end of the handrail above the floor) is approximately 3.467726 meters.

Therefore, the height of the upper end of the handrail above the floor is approximately 3.467726 meters.

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Answer: x=2, y=4

Step-by-step explanation:

Since the lines are parallel, we can say that the larger triangle and the smaller triangle are similar. Thus. x+3/(2y-1) = (3/2x+2)/(3y-5). Also, note that there are hash lines, telling us that 2y-1 and 3y-5 are equal. Thus 2y-1=3y-5, and y=4. Plugging y=4 into the first equation yields: (x+3)/(7)=(3/2x+2)/(7), or x+3=3/2x+2. Then 1/2x=1, x=2.

Thus: x=2, and y=4

Answer:

x = 2

y = 4

Step-by-step explanation:

If a line parallel to one side of a triangle intersects the other two sides, then this line divides those two sides proportionally.

As the line bisects the side of the triangle with the y-variables, then the line is the midsegment of the triangle. This means that the line also bisects the side of the triangle with the x-variables.

Therefore, the two expressions with the x-variable are equal.

Similarly, the two expressions with the y-variable are equal.

Solving for x:

[tex]\boxed{\begin{aligned}\dfrac{3}{2}x+2&=x+3\\\\\dfrac{3}{2}x+2-x&=x+3-x\\\\\dfrac{1}{2}x+2&=3\\\\\dfrac{1}{2}x+2-2&=3-2\\\\\dfrac{1}{2}x&=1\\\\2 \cdot \dfrac{1}{2}x&=2 \cdot 1\\\\x&=2\end{aligned}}[/tex]

Solving for y:

[tex]\boxed{\begin{aligned}3y-5&=2y-1\\\\3y-5-2y&=2y-1-2y\\\\y-5&=-1\\\\y-5+5&=-1+5\\\\y&=4\end{aligned}}[/tex]

x = 8.33 and the solution to the equation 12x = 100 is x = 8.33 (rounded to the nearest hundredth).

To solve the equation 12x = 100, we need to isolate the variable x. We can do this by dividing both sides of the equation by 12.

12x = 100

Dividing both sides by 12:

(12x)/12 = 100/12

Simplifying:

x = 100/12

Now, let's calculate the value of x.

x = 8.33 (rounded to the nearest hundredth)

Therefore, the solution to the equation 12x = 100 is x = 8.33 (rounded to the nearest hundredth).

Here's the step-by-step solution:

12x = 100 (given equation)

Divide both sides by 12: (12x)/12 = 100/12

Simplify: x = 8.33

Please note that the answer is rounded to the nearest hundredth as requested.

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The members of the given set in this problem are given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

The union and intersection sets of multiple sets are defined as follows:

The intersection of the sets Y and Z for this problem is given as follows:

Y ∩ Z = {0, 6, 23, 26}

(which are the elements that belong to both of the sets).

The union of the above set with the set X is given as follows:

X U (Y ∩ Z) = {p, q, r, 0, 6, 21, 22, 23, 26}

(elements that belong to at least one of the sets).

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When 'x' is 5, the Algebraic  expression 3x - 4 evaluates to 11.

To find the algebraic expression for "4 less than the product of 3 and some number," let's break it down step by step.

First, let's define the unknown number as 'x.' Since the problem states "the product of 3 and some number," we can express this as 3 * x or simply 3x.

Next, we want to subtract 4 from this product. The phrase "4 less than" indicates subtraction, so we subtract 4 from 3x. This can be represented as 3x - 4.

In summary, the algebraic expression for "4 less than the product of 3 and some number" is 3x - 4, where 'x' represents the unknown number.

To calculate the value of this expression for a specific value of 'x,' you substitute that value into the expression and simplify. For example, if 'x' is 5, you would substitute 5 into the expression:

3(5) - 4

This simplifies to:

15 - 4 = 11

So, when 'x' is 5, the expression 3x - 4 evaluates to 11.

In general, the algebraic expression allows us to represent a mathematical relationship between quantities using symbols and operations. By substituting specific values into the expression, we can calculate the corresponding results.

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Note the complete question:

Algebraic expression for 4 less than the product of 3 and some number

What is the algebraic expression for "4 less than the product of 3 and some number?

Please help me understand the meaning of how to calculate.

Answer:

polynomial; left 3 right 1

Step-by-step explanation:

The type of function is a polynomial.

The parent function is translated by moving -2x^3 to the left 3 and right 1.

Answer:

The function is a quadratic function.

Series of transformations:

Step-by-step explanation:

A quadratic function is a polynomial function of degree 2 (the highest exponent of the variable is 2).

Therefore, the given equation, y = -2(x + 3)² + 1, represents a quadratic function as the term (x + 3)² indicates that the highest power of the x-variable is 2.

[tex]\hrulefill[/tex]

The parent function of a quadratic function is y = x².

To translate the parent function to produce the given equation, we need to apply this series of transformations:

1. Horizontal translation

When "a" is added to the x-variable, the graph is shifted "a" units to the left. Therefore, the addition of 3 inside the parentheses shifts the graph horizontally to the left by 3 units.

2. Vertical Stretch

The coefficient in front of the squared term (x + 3)² indicates a vertical stretch or compression. In this case, since the coefficient greater than 1, it stretches the graph vertically, making it narrower compared to the parent function.

3. Reflection

As the stretch coefficient is negative, the graph is reflected across the x-axis, meaning the parabola opens downwards.

4. Vertical translation

When "a" is added to the function, the graph is shifted "a" units up. Therefore, the addition of 1 to the function shifts the graph vertically up by 1 unit.

In summary, the series of transformations that maps the parent function to the given function is:

Answer:

The rates are not equivalent since Maria saves $0.50 more per month than Ralph.

Step-by-step explanation:

We can determine if two rates are equivalent by comparing the rates at which they save per month.

Maria's savings per month:

Both 50 and 4 can be divided by 2, which gives us 25/2.  As a regular number, this becomes 12.5/1 which means Maria saves $12.5 per month.

Ralph's savings per month:

Both 60 and 5 can be divided by 5, which gives us 12.  Thus, Ralph saves $12 per month.

Thus, the rates are not equivalent as Maria saves $0.50 more per month than Ralph.

The proportional sides/corresponding sides of the triangles are ST/XY = RT/YZ = SR/XZ.

The value of a and b are 5 and 20 respectively.

Corresponding sides refers to a pair of matching sides that are in the same spot in two different shapes.

ST/XY = RT/YZ = SR/XZ

ST/XY = 3/15

= 1/5

RT/YZ = 4 ÷ 1/5

= 4 × 5/1

YZ, b = 20

Hypotenuse² = opposite² + adjacent²

5² = 3² + 4²

25 = 9 + 16

25 = 25

Therefore,

SR = a = 5

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To prepare a July 31 Balance Sheet for Academic Learning Services, you would need access to the financial statements and records provided in the class, as well as the information from Practical Exercise

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Answer: The first one ⊄

Step-by-step explanation: A is not a subset of E.

However, E is a subset of A

To factor the expression 4x^2 - 25, we can use the difference of squares formula, which states that a^2 - b^2 can be factored as (a + b)(a - b).

In this case, we have 4x^2 - 25, which can be written as (2x)^2 - 5^2. Comparing it with the difference of squares formula, we can identify that a = 2x and b = 5. Therefore, the correct option is:

b. (2x)^2 - (5)^2

Using the difference of squares formula, we can factor it as follows:

(2x + 5)(2x - 5)

Hence, the correct factorization of 4x^2 - 25 is (2x + 5)(2x - 5), which is equivalent to option b.

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Identify the matrix A: C. A is a 3 × 3 square matrix.

The negative (additive inverse) of A is: -A =  [tex]\left[\begin{array}{ccc}1&0&-5\\-3&-6&1\\4&7&-1\end{array}\right][/tex]

In Mathematics and Geometry, a square matrix can be defined as a type of matrix that is composed of an equal number of both rows and columns. Generally speaking, m × m matrix is typically referred to as a square matrix of order m.

In this context, we can reasonably infer and logically deduce that the number of rows in matrix A is the same as the number of columns in matrix A;

3 rows = 3 columns;

m × m = 3 × 3

From additive inverse postulate, we have the following:

A + (-A) = 0

[tex]\left[\begin{array}{ccc}-1&0&5\\3&6&-1\\-4&-7&1\end{array}\right]+\left[\begin{array}{ccc}1&0&-5\\-3&-6&1\\4&7&-1\end{array}\right]=0[/tex]

-A =  [tex]\left[\begin{array}{ccc}1&0&-5\\-3&-6&1\\4&7&-1\end{array}\right][/tex]

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Answer:

The 3.907 is approximate.

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

Explanation:

x = number of hours that elapse

y = f(x) = number of tokens

If we use a graphing tool like a TI84 or GeoGebra, then the approximate solution to -3(2)^(x+1) + 90 = 0 is roughly x = 3.907

At around 3.907 hours is when the number of tokens is y = 0. Therefore, this is the approximate upper limit for the domain. The lower limit is x = 0.

The domain spans from x = 0 to roughly x = 3.907, and we shorten that down to 0 ≤ x ≤ 3.907

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Plug in x = 0 to find y = 84. This is the largest value in the range.

The smallest value is y = 0.

The range spans from y = 0 to y = 84, so we get 0 ≤ y ≤ 84

Keep in mind y is the number of tokens. A fractional amount of tokens does not make sense, so we must have y be a whole number 1,2,3,...,83,84.

The x value can be fractional because 3.907 hours for instance is valid.

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Extra info: