Construct a difference table to predict the next term of the sequence.
6, 2, 5, 24, 68, 146, ...?

The value of x is 63

To determine the value of the variable, we have that;

From the information given, we have that;

x and x - 36 are complementary angles

Now, equate the angles to 90 degrees, we have;

x + x - 36= 90

collect the like terms, we get;

2x = 90 + 36

add the terms

2x =1 26

Divide by the coefficient

x = 63

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Complete question:

The angles X,(x-36)  are complementary. What is the value of x

The number of triangles in the diagram that can be mapped to one another by similarity transformations include the following: D. 3.

In Mathematics and Geometry, two triangles are said to be similar when the ratio of their corresponding side lengths are equal and their corresponding angles are congruent.

Additionally, the lengths of three (3) pairs of corresponding sides or corresponding side lengths are proportional to the lengths of corresponding altitudes when two (2) triangles are similar.

Based on the side, side, side (SSS) similarity theorem, we can logically deduce the following congruent and similar triangles:

ΔABC

ΔDEF

ΔGHI

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Answer:

375/100

Reduce to lowest term

15/4 or 3 3/4

1. The x - intercepts of the parabola are

2. The meaning of the x-intercepts are the plane takes of at x = 2.5 s and lands at x = 7.5 s

3. The vertex of the parabola is at (5, 80).

A parabola is a curved shape

1. Given the parabola above, to find the x - intercepts, we proceed as follows.

The x-intercepts are the points at which the graph cuts the x-axis.

They are

2. The meaning of the x-intercepts in this problem are the points where the plane takes off and lands on the ground.

The plane takes of at x = 2.5 s and lands at x = 7.5 s

3. The vertex is the maximum point on the graph.

So, we see that the vertex is at x = 5 s and y = 80 ft

So, the vertex is at (5, 80).

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The sprinter will complete the race in approximately 17.07 seconds.

To calculate the time for the race, we need to consider two parts: the acceleration phase and the constant velocity phase.

Acceleration Phase:

The acceleration of the sprinter is 1.64 m/s², and the distance covered during this phase is 25 m. We can use the equation of motion to calculate the time taken during acceleration:

v = u + at

Here:

v = final velocity (which is the velocity at the end of the acceleration phase)

u = initial velocity (which is 0 since the sprinter starts from rest)

a = acceleration

t = time

Rearranging the equation, we have:

t = (v - u) / a

Since the sprinter starts from rest, the initial velocity (u) is 0. Therefore:

t = v / a

Plugging in the values, we get:

t = 25 m / 1.64 m/s²

Constant Velocity Phase:

Once the sprinter reaches the end of the acceleration phase, the velocity remains constant. The remaining distance to be covered is 100 m - 25 m = 75 m. We can calculate the time taken during this phase using the formula:

t = d / v

Here:

d = distance

v = velocity

Plugging in the values, we get:

t = 75 m / (v)

Since the velocity remains constant, we can use the final velocity from the acceleration phase.

Now, let's calculate the time for each phase and sum them up to get the total race time:

Acceleration Phase:

t1 = 25 m / 1.64 m/s²

Constant Velocity Phase:

t2 = 75 m / v

Total race time:

Total time = t1 + t2

Let's calculate the values:

t1 = 25 m / 1.64 m/s² = 15.24 s (rounded to two decimal places)

Now, we need to calculate the final velocity (v) at the end of the acceleration phase. We can use the formula:

v = u + at

Here:

u = initial velocity (0 m/s)

a = acceleration (1.64 m/s²)

t = time (25 m)

Plugging in the values, we get:

v = 0 m/s + (1.64 m/s²)(25 m) = 41 m/s

Now, let's calculate the time for the constant velocity phase:

t2 = 75 m / 41 m/s ≈ 1.83 s (rounded to two decimal places)

Finally, let's calculate the total race time:

Total time = t1 + t2 = 15.24 s + 1.83 s ≈ 17.07 s (rounded to two decimal places)

Therefore, the sprinter will complete the race in approximately 17.07 seconds.

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i. The first two terms of the series are 50/3 and 100/9, and the fifth term is 800/243.

ii. The first two terms of the series are 0 and 3, and the fifth term is 48.

The first term of the series is can be obtained by substituting n=1 into the formula

Thus;

first term = 25(2/3[tex])^1[/tex]= 25(2/3) = 50/3

The second term of the series is obtained by substituting n=2 into the formula

second term = 25(2/3[tex])^2[/tex] = 25(4/9) = 100/9

The fifth term of the series is obtained by substituting n=5 into the formula

fifth term = 25(2/3[tex])^5[/tex] = 25(32/243) = 800/243

Therefore, the first two terms of the series are 50/3 and 100/9, and the fifth term is 800/243.

(ii) The first term of the series is obtained by substituting n=1 into the formula

first term = [tex]3(1-1)^2 = 3(0)^2 = 0[/tex]

The second term of the series is obtained by substituting n=2 into the formula

second term = [tex]3(2-1)^2 = 3(1)^2 = 3[/tex]

The fifth term of the series is obtained by substituting n=5 into the formula

fifth term = [tex]3(5-1)^2 = 3(16) = 48[/tex]

Therefore, the first two terms of the series are 0 and 3, and the fifth term is 48.

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The angles are matches as;

a. 65 degrees

b. 62 degrees

c. 33 degrees

d. 38 degrees

First, we need to know the properties of a triangle.

These properties are given as;

Then, we have from the information given that;

1. x + 28 + 87 = 180

collect the like terms, we have;

x = 180 - 115

x = 65 degrees

2. x + 16 + 102 = 180

collect like terms

x = 180 - 118

x = 62 degrees

3.  x + 96 + 51 = 180

collect like terms

x = 180 - 147 = 33 degrees

4. x + 119 + 23 = 180

x = 38 degrees

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Answer:

x>-3

Step-by-step explanation:

x-7<-10

x<-10+7

x<-3

x>-3

Ayaan drank 11/4 bottles of water during the soccer game, which can be expressed as a mixed numeral 2 and 3/4.

To convert the fraction 11/4 to a mixed numeral, we need to determine the whole number part and the fractional part.

Divide the numerator (11) by the denominator (4).

11 ÷ 4 = 2 remainder 3

The quotient 2 represents the whole number part, and the remainder 3 represents the fractional part.

Write the mixed numeral using the whole number part and the fractional part.

The mixed numeral for 11/4 is:

2 and 3/4

Therefore, Ayaan drank 11/4 bottles of water during the soccer game, which can be expressed as a mixed numeral 2 and 3/4.

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The correct rule for reflection across the y-axis is "Ty-axis (x, y) → (-x, y)."

The rule for reflection depends on the axis of reflection. In this case, we are given the options of Ty-axis (reflection across the y-axis) and Tx-axis (reflection across the x-axis). Let's analyze each option:

Ty-axis (x, y) → (-x, y):

This rule states that for a reflection across the y-axis, the x-coordinate of a point is negated while the y-coordinate remains the same. For example, if we have a point (2, 3) and reflect it across the y-axis, it becomes (-2, 3). The x-coordinate is negated, but the y-coordinate remains the same.

Ty-axis (x, y) → (x, y):

This rule states that for a reflection across the y-axis, both the x-coordinate and y-coordinate remain the same. This means that there is no change in the coordinates of the point after the reflection.

Tx-axis (x, y) → (-x, y):

This rule states that for a reflection across the x-axis, the y-coordinate of a point remains the same while the x-coordinate is negated. For example, if we have a point (2, 3) and reflect it across the x-axis, it becomes (2, -3). The y-coordinate remains the same, but the x-coordinate is negated.

Tx-axis (x, y) → (x, y):

This rule states that for a reflection across the x-axis, both the x-coordinate and y-coordinate remain the same. This means that there is no change in the coordinates of the point after the reflection.

Based on the given options, the correct rule for the reflection is Ty-axis (x, y) → (-x, y). This means that when reflecting a point across the y-axis, the x-coordinate is negated while the y-coordinate remains the same.

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So the probability of getting one head and one tail is 50%.since there are no other possible outcomes.

When we flip a coin, there are two possible outcomes: heads (H) or tails (T). Tossing a coin is a random experiment that follows a binomial distribution since each toss is independent, and there are only two possible outcomes.

We can use the binomial probability formula to find the probability of a specific event happening.Here are the answers to each part of the question:b) TT (two tails)

To calculate the probability of getting two tails in a row, we multiply the probability of getting tails on the first toss by the probability of getting tails on the second toss.

P(TT) = P(T) x P(T) = 0.5 x 0.5 = 0.25 = 25%So the probability of getting two tails in a row is 25%.c) HT and TH (one head and one tail)To calculate the probability of getting one head and one tail,

we multiply the probability of getting a head on the first toss by the probability of getting a tail on the second toss, and then add it to the probability of getting a tail on the first toss and a head on the second toss.

P(HT or TH) = P(H) x P(T) + P(T) x P(H) = 0.5 x 0.5 + 0.5 x 0.5 = 0.5 = 50%

Note that the probability of getting two heads is the same as the probability of getting two tails (25%), and the probability of getting any combination of heads and tails is 100% (1),

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Answer: 1.5mg of medicine is to be added to 500ml of water to gain a solution of 3mg per liter

Step-by-step explanation:

3mg per liter is needed. That is 3mg of medicine in 1000ml of water.

As we have only 1.5 mg of medicine that is half of 3mg. We have to add a half of one liter of water. that is 500 ml of water.

The function [tex] \sf f(x) [/tex] is defined as:

[tex] \sf f(x) = x^3 + \frac{3}{2}x^2 - 6x + 10 [/tex]

To find the stationary points of [tex] \sf f [/tex], we need to find the values of [tex] \sf x [/tex] where the derivative of [tex] \sf f(x) [/tex] is equal to zero.

First, let's find the derivative of [tex] \sf f(x) [/tex]:

[tex] \sf f'(x) = 3x^2 + 3x - 6 [/tex]

To find the stationary points, we set [tex] \sf f'(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 3x^2 + 3x - 6 = 0 [/tex]

We can factor the quadratic equation as follows:

[tex] \sf 3(x^2 + x - 2) = 0 [/tex]

Now, we solve for [tex] \sf x [/tex] by factoring further:

[tex] \sf 3(x + 2)(x - 1) = 0 [/tex]

This gives us two solutions: [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

So, the stationary points of [tex] \sf f(x) [/tex] are [tex] \sf x = -2 [/tex] and [tex] \sf x = 1 [/tex].

To determine the intervals where [tex] \sf f(x) [/tex] is increasing, we need to analyze the sign of the derivative [tex] \sf f'(x) [/tex] in different intervals. We can use the values of [tex] \sf x = -2 [/tex], [tex] \sf 1 [/tex], and any other value between them.

For [tex] \sf x < -2 [/tex], we choose [tex] \sf x = -3 [/tex] as a test point:

[tex] \sf f'(-3) = 3(-3)^2 + 3(-3) - 6 = 12 > 0 [/tex]

For [tex] \sf -2 < x < 1 [/tex], we choose [tex] \sf x = 0 [/tex] as a test point:

[tex] \sf f'(0) = 3(0)^2 + 3(0) - 6 = -6 < 0 [/tex]

For [tex] \sf x > 1 [/tex], we choose [tex] \sf x = 2 [/tex] as a test point:

[tex] \sf f'(2) = 3(2)^2 + 3(2) - 6 = 18 > 0 [/tex]

From the above analysis, we can conclude that [tex] \sf f(x) [/tex] is increasing in the intervals [tex] \sf (-\infty, -2) [/tex] and [tex] \sf (1, \infty) [/tex].

To find the inflection point of [tex] \sf f [/tex], we need to determine where the concavity changes. This occurs when the second derivative of [tex] \sf f(x) [/tex] changes sign.

The second derivative of [tex] \sf f(x) [/tex] is:

[tex] \sf f''(x) = 6x + 3 [/tex]

To find the inflection point, we set [tex] \sf f''(x) = 0 [/tex] and solve for [tex] \sf x [/tex]:

[tex] \sf 6x + 3 = 0 [/tex]

[tex] \sf 6x = -3 [/tex]

[tex] \sf x = -\frac{1}{2} [/tex]

Therefore, the inflection point of [tex] \sf f(x) [/tex] is [tex] \sf x = -\frac{1}{2} [/tex].

For an input of 2, the output value of the function include the following: D. after 2 hours, the cost is $6.

In Mathematics and Geometry, a linear function refers to a type of function whose equation is graphically represented by a straight line on the xy-plane or cartesian coordinate.

This ultimately implies that, a linear function has the same (constant) slope and it is typically used for uniquely mapping an input variable to an output variable, which both increases simultaneously.

In this context, we can logically deduce that the total cost of parking after 2 hours is equal to 6 dollars based on the graph shown above.

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Answer:

To find the expression for (s+t)(x), we simply add the two functions s(x) and t(x): (s + t)(x) = s(x) + t(x) = 4x - 5 + x + 6 = 5x + 1

To find the expression for (s•t)(x), we multiply the two functions s(x) and t(x): (s•t)(x) = s(x) * t(x) = (4x - 5) * (x + 6) = 4x^2 + 19x - 30

To evaluate (s-t)(2), we substitute 2 for x in the expression for (s-t)(x): (s-t)(2) = s(2) - t(2) = (4*2 - 5) - (2 + 6) = 3 - 8 = -5

Therefore, the expression for (s+t)(x) is 5x+1, the expression for (s•t)(x) is 4x^2+19x-30, and (s-t)(2) is -5.

Step-by-step explanation:

Answer:

D

Step-by-step explanation:

A and D are both equidistant from the x- axis, that is

A is 3 units above the x- axis and D is 3 units below the x- axis

then the x- axis is the line of symmetry

The solution to the system of equations, y = 3x + 15 and 3x + 3y = 9 is one unique solution, which is (-3, 6) [see image attached below].

To graph the system of equations and determine the solution, let's start with each equation individually:

y = 3x + 15

3x + 3y = 9

To graph the first equation, y = 3x + 15, we can plot a line with a slope of 3 and a y-intercept of 15. This line will have a positive slope, meaning it goes upward from left to right.

To graph the second equation, 3x + 3y = 9, we can rearrange it to the slope-intercept form y = (-1)x + 3, or simplified as y = -x + 3. This equation represents a line with a slope of -1 and a y-intercept of 3. This line will have a negative slope, meaning it goes downward from left to right.

Now, let's analyze the graphed lines and their intersection to determine the solution:

From the graphs as shown in the image attached below, we can see that the lines intersect at the point (-3, 6). This means that the solution to the system of equations is a single unique solution.

Therefore, the correct answer is option b. There is one unique solution (-3, 6).

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Answer:

third option

Step-by-step explanation:

under a counterclockwise rotation of 90° about the origin

a point (x, y ) → (y, - x )

Then

A (- 1, - 2 ) → (- 2, - (- 1) ) → (- 2, 1 )

B (2, - 2 ) → (- 2, - 2 )

C (1, - 4 ) → (- 4, - 1 )

The new coordinates are (d) A = (2, -1) B = (2, 2) and C = (4, 1)

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

The figure,

Where, we have

A = (-1, -2)

B = (2, -2)

C = (1, -4)

The rule of 90 degrees counterclockwise is

(x, y) = (-y, x)

Using the above as a guide, we have the following:

A = (2, -1)

B = (2, 2)

C = (4, 1)

Hence, the new coordinates are (d) A = (2, -1) B = (2, 2) and C = (4, 1)

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The annual rates of return for Asman and Salinas are as follows: Asman - 22.22%, 9.09%, 16.67%; Salinas - 60.00%%.

1. To calculate the annual rates of return, we need to determine the percentage change in stock prices from one time period to another.

2. For Asman stock, the price data is as follows:

  - Time 1: $30

  - Time 2: $11

  - Time 3: $29

  - Time 4: $12

3. The rate of return you would have earned on Asman stock from time 1 to time 2 can be calculated using the formula:

  [(Ending Price - Beginning Price) / Beginning Price] * 100

  Substituting the values, we get:

  [(11 - 30) / 30] * 100 = -63.33% (rounded to two decimal places)

4. The rate of return you would have earned on Asman stock from time 2 to time 3 can be calculated similarly:

  [(29 - 11) / 11] * 100 = 163.64% (rounded to two decimal places)

5. The rate of return you would have earned on Asman stock from time 3 to time 4 can be calculated:

  [(12 - 29) / 29] * 100 = -58.62% (rounded to two decimal places)

6. For Salinas stock, the price data is as follows:

  - Time 1: $30

  - Time 2: $12

  - Time 3: $31

  - Time 4: $34

7. The rate of return you would have earned on Salinas stock from time 1 to time 2 can be calculated:

  [(12 - 30) / 30] * 100 = -60.00% (rounded to two decimal places)

8. Therefore, the annual rates of return for Asman and Salinas are as follows:

  - Asman: -63.33%, 163.64%, -58.62% (rounded to two decimal places)

  - Salinas: -60.00% (rounded to two decimal places)

9. The annual rates of return indicate the percentage change in the value of the stock over a one-year period. A positive rate of return indicates a gain, while a negative rate of return indicates a loss. These figures help investors assess the performance of their investments and make informed decisions.

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Alex and Juana canoed 75% of the total distance on the first day of their trip.

To find the percentage of the total distance they canoed on the first day, we need to divide the distance they traveled on the first day by the total distance and then multiply by 100.

Total distance of the canoe trip = 32 miles

Distance traveled on the first day = 24 miles

Calculate the percentage.

Percentage = (Distance traveled on the first day / Total distance) [tex]\times[/tex] 100

Percentage = (24 miles / 32 miles) [tex]\times[/tex] 100

Percentage = (3/4) [tex]\times[/tex] 100

Percentage = 75%

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Answer:

Step-by-step explanation:

The simplified values of each given expressions involving factorials are:

1) 3!6! = 4,320       2) 5!7! = 604,800

Factorials are the product of a given number and all positive integers smaller than it, denoted by an exclamation mark (!).

To evaluate and simplify the expressions involving factorials, calculate the factorials and then simplify the resulting values you get:

1. 3!6!

To calculate 3! (3 factorial), we multiply all the integers from 1 to 3:

3! = 3 × 2 × 1 = 6

Next, we calculate 6! (6 factorial) by multiplying all the integers from 1 to 6:

6! = 6 × 5 × 4 × 3 × 2 × 1 = 720

Therefore, 3!6! = 6 × 720 = 4,320.

2. 5!7!

5! = 5 × 4 × 3 × 2 × 1 = 120

7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Therefore, 5!7! = 120 × 5,040 = 604,800.

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Labeling variables and expressions clearly is important because it helps ensure clarity, precision, and understanding in mathematical or scientific contexts.

Here are a few reasons why it is important to label variables and expressions:

Avoid confusion: Clear labeling prevents confusion, especially when dealing with complex equations or multiple variables. It helps distinguish between different quantities and ensures that each variable represents a specific aspect of the problem or situation.

Enhance communication: Precise labeling allows for effective communication of mathematical ideas and concepts. It helps convey information accurately to others, whether it's through written work, presentations, or discussions.

Promote understanding: By labeling variables and expressions clearly, it becomes easier for both the reader and the writer to understand the meaning and purpose behind each term. It aids in interpreting the mathematical relationships and allows for a better grasp of the overall problem or equation.

Enable error detection: When variables and expressions are properly labeled, it becomes easier to identify any errors or inconsistencies in calculations or equations. It allows for a systematic review and helps catch mistakes in formulas, units, or overall logic.

Facilitate problem-solving: Clear labeling enables effective problem-solving by providing a clear framework for approaching mathematical or scientific challenges. It helps organize information, track relevant variables, and establish logical connections between different components of a problem.

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Step-by-step explanation:

The ONE  solution to this system of two lines is the point where the two lines cross   at   (1,4)

The ladder will reach a Height of approximately 43.46 meters on the building.

To find out how high on the building the ladder will reach, we can use the Pythagorean theorem. The Pythagorean theorem 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 ladder forms a right triangle with the ground and the building. The base of the ladder is 7 meters, the height of the building is what we need to find, and the length of the ladder is given as 44 meters.

Using the Pythagorean theorem, we can set up the equation:

(Height of the building)^2 + 7^2 = 44^2

Simplifying the equation, we have:

(Height of the building)^2 + 49 = 1936

Subtracting 49 from both sides, we get:

(Height of the building)^2 = 1887

To find the height of the building, we take the square root of both sides:

Height of the building = √1887

Calculating the square root of 1887, we find that the height of the building is approximately 43.46 meters.

Therefore, the ladder will reach a height of approximately 43.46 meters on the building.

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Adult admission costs $18, and child admission costs $5.To check:2(5) + 18 = 28, which is true3(5) + 18 = 48, which is also true.

Let's begin by letting x be the cost of adult admission, and y be the cost of a child admission, so we have two equations:2y + x = 28
3y + x = 48

Subtracting the first equation from the second one, we get:3y + x - (2y + x) = 48 - 28y = 20Therefore, we can solve for x by plugging in y=5 in either equation.

Using the first equation, we have:2(5) + x = 28x = 18

Thus, adult admission costs $18, and child admission costs $5.To check:2(5) + 18 = 28, which is true3(5) + 18 = 48, which is also true.

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Answer:

Step-by-step explanation:

Step 1:First Make 3/4 and 5/6 into like fractions.Find the L.C.M of 4 and  

            6,which is 12.

            3*3/4*3=9/12

             5*2/6*2=10/12

Step 2:Subtract 9/12 from 10/12,which is 1/12.

The equation of the parabola is:

y = (x + 4)² - 3

We can see that we have the graph of a parabola.

Remember that the vertex form of a parabola whose leading coefficient is a and whose vertex is (h, k) is:

y = a*(x - h)² + k

Here we can see that the vertex is at (-4, -3)

So we can write:

y = a*(x + 4)² - 3

We also can see that the function passes through (-3, -2), replacing that we will get:

-2 = a*(-3 + 4)² - 3

-2 = a - 3

-2 + 3 = a

1 = a

The equation is:

y = (x + 4)² - 3

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Answer:

(5, 4)

Step-by-step explanation:

(x + 2, y + 3)

(3  + 2, 1 + 3)

(5, 4)

Answer:

B' (5, 4 )

Step-by-step explanation:

the translation rule (x, y ) → (x + 2, y + 3 )

means add 2 to the original x- coordinate and add 3 to the original y- coordinate , then

B (3, 1 ) → B' (3 + 2, 1 + 3 ) → B' (5, 4 )