X
45°
X =
Find x.
17
X
45°
AVC

The equation is true when x = 9,x = 9 is a solution to this equation.

To determine which equations have x = 9 as a possible solution, we need to check each equation individually. Here are the equations to consider:

3x - 18 = 15

Substituting x = 9, we have:

3(9) - 18 = 15

27 - 18 = 15

9 = 15

The equation is not true when x = 9. Therefore, x = 9 is not a solution to this equation.

2(x + 4) = 26

Substituting x = 9, we have:

2(9 + 4) = 26

2(13) = 26

26 = 26

The equation is true when x = 9. Therefore, x = 9 is a solution to this equation.

5x + 3 = 2x + 30

Substituting x = 9, we have:

5(9) + 3 = 2(9) + 30

45 + 3 = 18 + 30

48 = 48

The equation is true when x = 9

Based on the analysis, x = 9 is a possible solution for equations 2 and 3.

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The x variable represents the number of tickets sold.

The y variable represents the total amount of money raised from ticket sales.

The equation for the scenario is y = 2.50x.

In the given scenario, the number of tickets sold is represented by the variable x, and the total amount of money raised from ticket sales is represented by the variable y. Since each ticket is priced at $2.50, the equation relating the number of tickets sold (x) and the total amount of money raised (y) is y = 2.50x.

Here's an explanation of each element in the response:

1. The x variable represents the number of tickets sold: This statement correctly identifies the variable x as representing the number of tickets sold. In the equation y = 2.50x, x represents the independent variable, which is the quantity we want to determine.

2. The y variable represents the total amount of money raised from ticket sales: This statement correctly identifies the variable y as representing the total amount of money raised from ticket sales. In the equation y = 2.50x, y represents the dependent variable, which is determined based on the value of x.

3. The equation for the scenario is y = 2.50x: This equation is derived from the given information that each ticket is priced at $2.50. Multiplying the price per ticket by the number of tickets sold gives the total amount of money raised, which is represented by y in the equation.

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A median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side.

The term that describes a line segment connecting a vertex of a triangle to the midpoint of the opposite side is the "median." In triangle geometry, a median is a line segment that joins a vertex of a triangle to the midpoint of the opposite side.

To understand the concept of a median, let's consider a triangle ABC. The midpoint of side BC is denoted as M, and vertex A is connected to M by a line segment. This line segment AM is referred to as the median from vertex A.

Medians have some interesting properties and play a significant role in triangle geometry. Here are a few key characteristics of medians:

1. Medians Divide the Triangle into Two Equal Areas:

Each median of a triangle divides the triangle into two regions with equal areas. The point where all three medians intersect is called the centroid, which is also the center of mass of the triangle.

2. Medians are Concurrent:

The three medians of a triangle are always concurrent, meaning they intersect at a single point called the centroid. This centroid divides each median in a 2:1 ratio, with the longer segment adjacent to the vertex.

3. Medians Divide the Triangle into Six Congruent Triangles:

The medians of a triangle divide the triangle into six smaller congruent triangles. Each of these triangles shares a common vertex with the original triangle.

4. Medians Determine the Centroid:

The centroid of a triangle is the point of intersection of the three medians. It is the balance point of the triangle, where the triangle would perfectly balance on a needle.

In summary, a median is a line segment connecting a vertex of a triangle to the midpoint of the opposite side. Medians have unique properties, including dividing the triangle into equal areas, being concurrent at the centroid, dividing the triangle into congruent triangles, and determining the balance point of the triangle.

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Answer:

[tex]\textsf{8.} \quad \textsf{(A)}\;\;\overline{XB}[/tex]

[tex]\textsf{9.} \quad \textsf{(D)}\;\;\overleftrightarrow{BD}[/tex]

Step-by-step explanation:

The radius is the distance from the center of a circle to any point on its circumference.

The center of the given circle is point X.

Therefore, the radii in the given circle are line segments XB, XA and XC.

[tex]\hrulefill[/tex]

A tangent is a straight line that touches a circle at only one point.

The line BD touches the circle at point B.

Therefore, the tangent of the given circle is line BD.

Answer:

8. A

9. D

Step-by-step explanation:

The radius is a straight line from the midpoint to the circle's circumference.

A Tangent is a line going through the circumference of the circle.

To determine the type of correlation between two variables on a graph, we can examine the pattern or relationship exhibited by the data points.

If the data points on the graph form a roughly linear pattern with a positive slope, it indicates a positive correlation. This means that as one variable increases, the other variable also tends to increase.

Conversely, if the data points on the graph form a roughly linear pattern with a negative slope, it indicates a negative correlation. In this case, as one variable increases, the other variable tends to decrease.

Additionally, the strength of the correlation can be assessed by how closely the data points align with the overall trendline. If the points are tightly clustered around the trendline, it suggests a strong correlation, while scattered points indicate a weaker correlation.

By visually inspecting the graph and observing the direction and pattern of the data points in relation to the trendline, we can determine the type of correlation exhibited by the variables.

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Rhe point-slope equation of a line with the labeled point (3, -4) is y + 4 = m(x - 3), where 'm' represents the slope of the line.

To find the point-slope equation of a line using the coordinates of a labeled point, you can use the following formula:
y - y₁ = m(x - x₁)
In this formula, (x₁, y₁) represents the coordinates of the labeled point, and m represents the slope of the line.
Given the coordinates (3, -4) of the labeled point, we can substitute these values into the formula:
y - (-4) = m(x - 3)
Simplifying this equation, we get:
y + 4 = m(x - 3)
This is the point-slope equation of the line.
Now, it's important to note that the problem does not provide information about the slope of the line. Therefore, we cannot determine the exact point-slope equation without knowing the slope. The point-slope equation requires the slope value to be defined.
If you have the slope of the line, let's say it is represented by the variable 'm', you can substitute that value into the equation to get the specific point-slope equation. For example, if the slope is 2, the equation becomes:
y + 4 = 2(x - 3)

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Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).

Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).

Statement 3: DC = DC (reflexive property of equality).

Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).

Statement 5: AC = BD (by CPCTC).

The statement that completes Jimmy's proof is:

DC = DC (reflexive property of equality)

The reflexive property of equality states that any quantity is equal to itself. In this case, statement 3 is stating that the diagonal DC is equal to itself, which is true by the reflexive property of equality.

Therefore, the completed proof is:

Statement 1: In triangle ADC and BCD, AD = BC (opposite sides of a rectangle are congruent).

Statement 2: Angle ADC = Angle BCD (angles of a rectangle are 90°).

Statement 3: DC = DC (reflexive property of equality).

Statement 4: Triangle ADC and BCD are congruent (by SAS postulate).

Statement 5: AC = BD (by CPCTC).

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An example of a data set where the mean is equal to one of the numbers in the set is 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, with a mean of 11.

Here's an example of a data set where the mean is equal to one of the numbers in the set:

Data set: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20

In this data set, the mean (average) value is calculated by summing up all the numbers in the set and dividing by the total number of values. In this case, the sum of the numbers is 110, and since there are 10 numbers in the set, the mean is 110/10 = 11.

As we can see, the number 11 is present in the data set itself and coincidentally, it is also the mean value of the set. This happens because the other numbers are symmetrically distributed around the mean, balancing out to yield the same value.

It's important to note that this is just one example, and there can be various data sets where the mean matches one of the numbers. The occurrence of such a scenario depends on the values within the data set and their distribution.

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Question: Data set [2, 4, 6, 8, 10, 12, 14, 16, 18, 20]

Consider the data set provided above. Is there any number in the data set that is equal to the mean of the data set?

Answer:

-2,1,6,13,22

Step-by-step explanation:

cn = n^2 -3

Let n=1

c1 = 1^2 -3 = 1-3 = -2

Let n=2

c2 = 2^2 -3 = 4-3 = 1

Let n=3

c3 = 3^2 -3 = 9-3 = 6

Let n=4

c4 = 4^2 -3 = 16-3 = 13

Let n=5

c5 = 5^2 -3 = 25-3 = 22

The answer is:

45÷3[90÷2{6+3(19+16)}]

= 45÷3[90÷2*12] (Simplifying inside brackets first)

= 45÷3[90÷24]

=45÷3 *3

= 135

Therefore, the correct answer is: 135

The sum diverges to negative infinity.

Here we want to find the value of the sum:

[tex]\sum_{m=1}^{ \infty}} (-11/2)*(3/2)^{m + 1}[/tex]

So, that sum goes for infinite values of m, that is bad because you can see that the term with an exponent is larger than 1.

So when m is a really large value, then the term will also be a really large value, which means that the fuction eventually diverges to negative inifnity.

The usual rule that we need to check is that, for large values of m, as m increases, the absolute value of each term decreases.

Here this cleraly does not happen, so the sum diverges.

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Rounding to the nearest whole number, the work required to pump the oil out of the spout is approximately 5,068,032π J.

To find the work required to pump the oil out of the spout, we need to consider the potential energy of the oil. The work done is equal to the change in potential energy.The potential energy of an object is given by the formula: PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.

Given that the density of the oil is 900 kg/m^3 and the tank is half full, we can determine the mass of the oil. The volume of the tank is calculated using the formula for the volume of a cylinder: V = πr^2h, where r is the radius and h is the height.

The volume of the tank is (1/2)π(12^2)(4) = 288π m^3.

Since the oil is half full, the volume of the oil is (1/2)(288π) m^3.

The mass of the oil is the density multiplied by the volume:

m = (900 kg/m^3)(1/2)(288π m^3) = 129,600π kg.

The height of the oil is 4 m.

Now, we can calculate the potential energy:

PE = mgh = (129,600π kg)(9.8 m/s^2)(4 m) = 5,068,032π J.

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The whole number that has no predecessor is 1.

499 is to the left of 500 on a number line.

The successor of the greatest 5-digit number is 100,000.

The additive identity is 0.

The number of whole numbers is infinite.

1 is called the multiplicative identity.

The result of (77) × 99 is 7,623.

The smallest whole number is 0.

The greatest two-digit number exactly divisible by 18 is 90.

The greatest 7-digit number using the digits 4, 6, and 9 with repetition is 999,9999.

Rearranging the numbers using the property of addition:

7326 + 139 + 674 + 861 = (7326 + 861) + 674 + 139 = 8187 + 674 + 139 = 9,000 + 674 + 139 = 9,813.

Finding the product using the distributive property:

798 x 998 = (700 + 90 + 8) x (900 + 90 + 8) = 700 x 900 + 700 x 90 + 700 x 8 + 90 x 900 + 90 x 90 + 90 x 8 + 8 x 900 + 8 x 90 + 8 x 8 = 630,000 + 63,000 + 5,600 + 81,000 + 8,100 + 720 + 7,200 + 720 + 64

= 1,448,504.

The largest 6-digit number exactly divisible by 45 is 999,990.

Simplifying the expression:

75 - [30 + (3 x (18 ÷ 6))] = 75 - [30 + (3 x 3)] = 75 - [30 + 9] = 75 - 39 = 36.

Ramesh buys 15 computers and 15 printers.

Cost of one computer = Rs. 75,326

Cost of one printer = Rs. 8,265

Using the distributive property of multiplication:

Total cost = (Cost of one computer x Number of computers) + (Cost of one printer x Number of printers)

Total cost = (Rs. 75,326 x 15) + (Rs. 8,265 x 15)

Total cost = Rs. 1,129,890 + Rs. 123,975

Total cost = Rs. 1,253,865.

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The relationship shown in the diagram can be described as follows: For each input value, there is a corresponding output value. The output value is obtained by performing certain operations on the input value according to the rules specified in the diagram.

2.2.1: The relationship shown in the diagram represents a function where each input value corresponds to a specific output value. The diagram may include various operations or rules to transform the input values into their respective output values.

2.2.2: Using the information provided in the flow diagram, we can complete the table as follows:

  - For input 0, the output is 1.

  - For input 1, the output is 2.

  - For input 2, the output is 4.

  - For input 4, the output is LO.

  - For input 5, the output is 182.

  - For input 182, the output is 2.

  - For input 2, the output is 4.

  - For input -1, the output is 12-10.

  - For input 12-10, the output is 2.

2.2.3: To calculate the input value for the given output value of -21, we follow these steps:

  - Start with the output value -21.

  - Reverse the operations or rules specified in the diagram to transform the output back into the input.

  - Apply the reverse operations in the opposite order to obtain the input value.

Please note that without a specific diagram or additional information, it is challenging to provide precise steps for reversing the operations or rules. The steps may vary depending on the complexity and specifics of the diagram.

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The width of the cooling tower at the base is approximately 20 meters

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

[tex]\frac{x^2}{400} - \frac{(y - 110)^2}{2304} = 1[/tex]

The above equation is an equation of a hyperbols

The general equation for a hyperbola that has a center at (h, k) is

[tex]\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1[/tex]

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

a² = 400

So, we have

a = 20

Hence, the width is 20 meters

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Question

Cooling towers are used to remove or expel heat from a process. A cooling towers walls are modeled by. x^2/400 - (y - 110)^2/2304 = 1 where the measurements are in meters

What is the width of the cooling tower at the base of the structure? Round your answer to the nearest whole number

The density of Liquid A is equal to the density of Liquid C when they are mixed in the specified ratio.

To determine the density of Liquid C, we need to find the mass and volume of Liquid C and then calculate the density by dividing the mass by the volume.

Given that the density of Liquid A is in a 4:3 ratio with the density of Liquid B, and the mass of Liquid A is in a 5:2 ratio with the mass of Liquid B, we can assume that the ratio of their volumes is also 5:2. This is because density is the ratio of mass to volume.

When 3 cans of Liquid B are mixed with 1 can of Liquid A to make Liquid C, the volume ratio remains the same. So, the volume of Liquid C would be 5 + 3 = 8 units.

Since the density is the mass divided by the volume, the density of Liquid A would remain the same. Therefore, the density of Liquid A is equal to the density of Liquid C.

In conclusion, the density of Liquid A is equal to the density of Liquid C.

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The conclusions that can be drawn from the data shown in the table are:

- Twelve tickets cost $30.00.

- Each additional ticket costs $2.50.

- The table is a partial representation.

- (27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table.

The statement "(27.50, 11), (30, 12), and (32.50, 13) are the ordered pairs represented in the table" is correct. From the data shown in the table, we can draw the following conclusions:

1. Twelve tickets cost $30.00: Looking at the "Tickets" column, we can see that the entry "12" corresponds to the "Total Cost" entry of $30.00. Therefore, we can conclude that purchasing twelve tickets would cost $30.00.

2. Each additional ticket costs $2.50: By examining the "Tickets" column, we can observe that for each increase of one ticket (from 11 to 12, and from 12 to 13), the "Total Cost" in the second column increases by $2.50. This consistent pattern suggests that each additional ticket costs $2.50.

3. The table is a partial representation: The table only displays three rows of data, showing the "Tickets" and "Total Cost" for x values of 11, 12, and 13. Since the table does not provide information for all possible values of x, it is a partial representation of the relationship between the number of tickets and the total cost.

The statement "Thirty tickets cost $12.00" is not supported by the given data. The table does not include an entry for 30 tickets, and none of the given entries correspond to that value.

These ordered pairs match the values shown in the table, with the first element representing the number of tickets (x) and the second element representing the total cost (y) in dollars.

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Although the chef used fresh ingredients, Karen knew the pasta dish was tasty but not healthy.Type of Parallel Structure: Parallelism using correlative conjunctions

Every year, I go on a long hiking trip where I like to take a break away from the hustle of the city and enjoy the peacefulness within nature.

Type of Parallel Structure: Parallelism in a series of items

Derek enjoys playing baseball with his friends, going on camping trips with his dad, and traveling to different cities throughout the year.

Type of Parallel Structure: Parallelism in a series of items

I like playing hockey more than I like to play soccer.

Type of Parallel Structure: Parallelism in a comparative sentence

Jonathan enjoys watching comedy at the movie theater more than he likes watching horror films at the movie theater.

Type of Parallel Structure: Parallelism in a comparative sentence

When I go to the park, I like bringing a blanket and to pack a picnic basket full of sandwiches and fruit.

Type of Parallel Structure: Parallelism using the same verb tense

In these sentences, the type of parallel structure used in each sentence has been matched correctly.

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Based on the line of best fit in the provided image, it appears that for y = 14, the estimated x-value is approximately 6.

By examining the line of best fit, we can estimate the x-value corresponding to y = 14. In the image provided, the line of best fit appears to be a straight line passing through several data points. Let's assume this line can be approximated by the equation y = mx + b, where m represents the slope and b represents the y-intercept.

To find the x-value when y = 14, we can substitute y = 14 into the equation and solve for x. However, since we don't have the equation explicitly, we will have to estimate the x-value based on the visual representation of the line of best fit.

Looking at the image, we can observe that the line of best fit intersects the y = 14 mark at approximately x ≈ 6. This is an estimation based on the position of the line relative to the given point.

Please note that this estimation is subject to the accuracy of the plotted points and the line of best fit in the image. For a more precise answer, the actual equation of the line of best fit or additional data would be required.

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The 98% confidence interval for the true mean statistics anxiety score (μ) is approximately (39.037, 39.763).

To calculate the 98% confidence interval for the true mean statistics anxiety score (μ) for all undergraduate students in the U.S., we can use the formula:

Confidence interval = sample mean ± (critical value * standard error)

First, we need to find the critical value associated with a 98% confidence level. Since we are assuming a normal distribution, we can use the Z-table or a statistical software to find this value. For a 98% confidence level, the critical value is approximately 2.33.

Next, we calculate the standard error (SE) using the formula:

SE = standard deviation / √sample size

In this case, the standard deviation is 0.9 and the sample size is 33. Plugging these values into the formula, we get: SE = 0.9 / √33 ≈ 0.156

Now, we can calculate the confidence interval:Confidence interval = 39.4 ± (2.33 * 0.156)

Simplifying the expression:Confidence interval ≈ 39.4 ± 0.363

This gives us the interval (39.037, 39.763). This means we are 98% confident that the true mean statistics anxiety score for all undergraduate students in the U.S. falls within this interval.

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The correct step that comes after constructing a circle when constructing an inscribed square is to use a straightedge to draw a diameter of the circle (option C).

When constructing an inscribed square by hand, the step that comes after constructing a circle is to use a straightedge to draw a diameter of the circle. Therefore, the correct answer is C.

To understand why drawing a diameter comes after constructing a circle, let's review the steps involved in constructing an inscribed square:

1. Start by constructing a circle: To do this, you would use a compass and a fixed point as the center to draw a circle.

2. Draw a diameter of the circle: A diameter is a line segment that passes through the center of the circle and divides it into two equal parts. Using a straightedge, you can draw a straight line that passes through the center of the circle.

3. Find the midpoint of the diameter: The midpoint is the point on the diameter that divides it into two equal parts. You can use a compass or measure the distance from each end of the diameter to find the midpoint.

4. Draw a perpendicular bisector: With the midpoint as the center, use a compass to draw an arc that intersects the diameter on both sides. This arc will create two points on the diameter.

5. Connect the points: Use a straightedge to connect the two points on the diameter. This line segment will be one side of the inscribed square.

6. Repeat the process: Repeat steps 2 to 5 to draw the other three sides of the square, using the circle as a guide.

By drawing a diameter of the circle, you establish a reference line that will be the base for constructing the sides of the inscribed square. It allows you to accurately position the square within the circle and ensure that its vertices lie on the circumference.

Therefore, the correct step that comes after constructing a circle when constructing an inscribed square is to use a straightedge to draw a diameter of the circle (option C).

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The probability is given as follows:

0.278 = 27.8%.

The event is not mutually exclusive, as the probability is different of zero.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

The total number of outcomes when two dice are rolled is given as follows:

6² = 36.

The desired outcomes are given as follows:

Hence the probability is given as follows:

(6 + 4)/36 = 5/18 = 0.278 = 27.8%.

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Answer:

B

Step-by-step explanation:

the secant- secant angle LMN is half the difference of the measures of the intercepted arcs , that is

∠ LMN = [tex]\frac{1}{2}[/tex] ( KP - LN)

20° = [tex]\frac{1}{2}[/tex] (96 - LN) ← multiply both sides by 2 to clear the fraction

40° = 96° - LN ( subtract 96° from both sides )

- 56° = - LN ( multiply both sides by - 1 )

56° = LN

Average speed of the humpback whale: 27 kilometers per hour.

To calculate the average speed of the humpback whale, we can use the formula:

Average speed = Total distance / Total time

Given that the humpback whale traveled a distance of 13.5 kilometers in 30 minutes, we need to convert the time to hours. There are 60 minutes in an hour, so 30 minutes is equal to 0.5 hours.

Now, we can substitute the values into the formula:

Average speed = 13.5 kilometers / 0.5 hours

Average speed = 27 kilometers per hour

Therefore, the average speed of the humpback whale is 27 kilometers per hour.

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Tom's percentage score is above 75%, which is the passing threshold, we can conclude that Tom has indeed passed the assessment.

To determine if Tom has passed the assessment, we need to calculate his percentage score out of the total marks.

Percentage Score = (Tom's Score / Total Marks) * 100

Given that Tom's score is 98 marks and the total marks are 128:

Percentage Score = (98 / 128) * 100 ≈ 76.5625%

Since Tom's percentage score is above 75%, which is the passing threshold, we can conclude that Tom has indeed passed the assessment.

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Answer: OC. The cost of packaging per weight unit is given by x / 3.

To find the cost of packaging per weight unit, we need to calculate the cost of packaging (given by x^2) divided by the weight of the box (given by x + 2).

Let's substitute x + 2 for the weight in the cost function:

Cost of packaging per weight unit = (Cost of packaging) / (Weight of the box)

= (x^2) / (x + 2)

Now, let's simplify this expression:

Cost of packaging per weight unit = x^2 / (x + 2)

To further simplify, we can divide both the numerator and denominator by x:

Cost of packaging per weight unit = (x * x) / (x * (1 + 2))

= x / (1 + 2)

= x / 3

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Answer:

The 2nd option is correct, a 90 degree counterclockwise rotation about the origin and then a reflection across the y-axis

Step-by-step explanation:

The explicit formula for the sequence 12, 112, 212, 312, 412 is a_n = 100n + 12.

The explicit formula for the given sequence is:

a_n = 100n + 12

In the given sequence, each term is obtained by adding 100 to the previous term. The first term is 12, and each subsequent term is obtained by adding 100 to the previous term.

Using the formula, we can calculate any term in the sequence by substituting the corresponding value of n. For example:

a_1 = 100(1) + 12 = 112

a_2 = 100(2) + 12 = 212

a_3 = 100(3) + 12 = 312

a_4 = 100(4) + 12 = 412

Therefore, the explicit formula for the sequence 12, 112, 212, 312, 412 is a_n = 100n + 12.

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Answer:

Refer to the step-by-step explanation, please follow along very carefully. Answers are encased in two boxes.

Step-by-step explanation:

Given the following function, find it's derivative using the definition of derivatives. Evaluate the function when θ=1, 11, and 3/11

[tex]p(\theta)=\sqrt{11\theta}[/tex]

[tex]\hrulefill[/tex]

The definition of derivatives states that the derivative of a function at a specific point measures the rate of change of the function at that point. It is defined as the limit of the difference quotient as the change in the input variable approaches zero.

[tex]f'(x) = \lim_{{h \to 0}} \dfrac{{f(x+h) - f(x)}}{{h}}[/tex][tex]\hrulefill[/tex]

To apply the definition of derivatives to this problem, follow these step-by-step instructions:

Step 1: Identify the function: Determine the function for which you want to find the derivative. In out case the function is denoted as p(θ).

[tex]p(\theta)=\sqrt{11\theta}[/tex]

Step 2: Write the difference quotient: Using the definition of derivatives, write down the difference quotient. The general form of the difference quotient is (f(x+h) - f(x))/h, where "x" is the point at which you want to find the derivative, and "h" represents a small change in the input variable. In our case:

[tex]p'(\theta) = \lim_{{h \to 0}} \dfrac{{p(\theta+h) - p(\theta)}}{{h}}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11(\theta + h)} - \sqrt{11\theta} }{h}[/tex]

Step 3: Take the limit:

We need to rationalize the numerator. Rewriting using radical rules.

[tex]p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11(\theta + h)} - \sqrt{11\theta} }{h} \\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11\theta + 11h} - \sqrt{11\theta} }{h}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} }{h}[/tex]

Now multiply by the conjugate.

[tex]p'(\theta)= \lim_{h \to 0} \dfrac{\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} }{h} \cdot \dfrac{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} } \\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{(\sqrt{11}\sqrt{\theta+h} - \sqrt{11}\sqrt{\theta} )(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )}{h(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )} \\\\\\[/tex]

[tex]\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{11h}{h(\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} )}\\\\\\\Longrightarrow p'(\theta)= \lim_{h \to 0} \dfrac{11}{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }[/tex]

Step 4: Simplify the expression: Evaluate the limit by substituting the value of h=0 into the difference quotient. Simplify the expression as much as possible.

[tex]p'(\theta)= \lim_{h \to 0} \dfrac{11}{\sqrt{11}\sqrt{\theta+h} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{\sqrt{11}\sqrt{\theta+(0)} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{\sqrt{11}\sqrt{\theta} + \sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11}\sqrt{\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11\theta} }\\\\\\\Longrightarrow p'(\theta)= \dfrac{11}{2\sqrt{11\theta} }[/tex]

[tex]\therefore \boxed{\boxed{p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta} }}}[/tex]

Thus, we have found the derivative on the function using the definition.

It's important to note that in practice, finding derivatives using the definition can be a tedious process, especially for more complex functions. However, the definition lays the foundation for understanding the concept of derivatives and its applications. In practice, there are various rules and techniques, such as the power rule, product rule, and chain rule, that can be applied to find derivatives more efficiently.[tex]\hrulefill[/tex]

Now evaluating the function at the given points.  

[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}; \ p'(1)=??, \ p'(11)=??, \ p'(\frac{3}{11} )=??[/tex]

When θ=1:

[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(1)= \dfrac{\sqrt{11} }{2\sqrt{1}}\\\\\\\therefore \boxed{\boxed{p'(1)= \dfrac{\sqrt{11} }{2}}}[/tex]

When θ=11:

[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(11)= \dfrac{\sqrt{11} }{2\sqrt{11}}\\\\\\\therefore \boxed{\boxed{p'(11)= \dfrac{1}{2}}}[/tex]

When θ=3/11:

[tex]p'(\theta)= \dfrac{\sqrt{11} }{2\sqrt{\theta}}\\\\\\\Longrightarrow p'(\frac{3}{11} )= \dfrac{\sqrt{11} }{2\sqrt{\frac{3}{11} }}\\\\\\\therefore \boxed{\boxed{p'(\frac{3}{11} )= \dfrac{11\sqrt{3} }{6}}}[/tex]

Thus, all parts are solved.

The two cars must be approximately 177.34 feet apart from each other for Spiderman to have different depression angles to each car.

To find the distance between the two cars, we can use trigonometry and the concept of similar triangles. Let's denote the distance between Spiderman and the nearest car as d1 and the distance between Spiderman and the farther car as d2.

In a right triangle formed by Spiderman, the height of the hotel, and the line of sight to the nearest car, the tangent of the depression angle (52 degrees) can be used:

tan(52) = 600 / d1

Rearranging the equation to solve for d1:

d1 = 600 / tan(52)

Similarly, in the right triangle formed by Spiderman, the height of the hotel, and the line of sight to the farther car, the tangent of the depression angle (46 degrees) can be used:

tan(46) = 600 / d2

Rearranging the equation to solve for d2:

d2 = 600 / tan(46)

Using a calculator, we can compute:

d1 ≈ 504.61 feet

d2 ≈ 681.95 feet

The distance between the two cars is the difference between d2 and d1:

Distance = d2 - d1

Plugging in the values, we have:

Distance ≈ 681.95 - 504.61

Distance ≈ 177.34 feet

For more such question on angles. visit :

https://brainly.com/question/25770607

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