Compute
lim n→[infinity] √3n⁴+5n−n²

Each step of constructing a rectangle is described below and justification is also described.

To construct a rectangle using the construction for congruent segments and the construction for a line perpendicular to another line through a point on the line, follow these steps:

Draw a line segment AB as the base of the rectangle.

Using the construction for congruent segments, construct a line segment CD that is congruent to AB.

Draw a perpendicular line to AB at point A using the construction for a line perpendicular to another line through a point on the line. Let this line intersect CD at point E.

Draw a perpendicular line to AB at point B using the same construction. Let this line intersect CD at point F.

Connect points C and E with a straight line.

Connect points D and F with a straight line.

Finally, connect points C and D, as well as points E and F, with straight lines to complete the rectangle.

Justification for each step:

This step is the starting point of the construction.

We use the construction for congruent segments to create segment CD, which will be parallel to AB and form the other side of the rectangle.

We use the construction for a line perpendicular to another line through a point on the line to draw a perpendicular line to AB through point A, which will be one of the corners of the rectangle.

We use the same construction to draw a perpendicular line to AB through point B, which will be another corner of the rectangle.

We connect point C and point E with a straight line to form one side of the rectangle.

We connect point D and point F with a straight line to form the other side of the rectangle.

Finally, we connect points C and D, as well as points E and F, with straight lines to complete the rectangle and create the last two sides of the rectangle.

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Answer:

30°

Step-by-step explanation:

complementary angles sum to 90°

then the complement of 60° = 90° - 60° = 30°

For a tapered cylinder with a taper per foot (T) of 0.75, 0.85, and 0.95, and given that the initial radius (R) is 4 in.

To find the length (L) for each taper per foot (T), we can rearrange the formula T= 24(R-r)/L to solve for L. Substituting the given values of R=4 in. and r=3 in., we have:

0.75 = 24(4-3)/L

0.85 = 24(4-3)/L

0.95 = 24(4-3)/L

Simplifying these equations, we get:

0.75L = 24

0.85L = 24

0.95L = 24

Dividing both sides of each equation by the corresponding coefficient, we find:

L = 24/0.75

L = 24/0.85

L = 24/0.95

Evaluating these expressions, we get:

L ≈ 32 in.

L ≈ 28.24 in.

L ≈ 25.26 in.

Therefore, the length (L) for taper per foot values of 0.75, 0.85, and 0.95, with R=4 in. and r=3 in., are approximately 32 in., 28.24 in., and 25.26 in., respectively.

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To find the volume of a rectangular prism with a square base of side length x and a surface area of 100 square feet, we can use the formula for the surface area of a rectangular prism.

The surface area of a rectangular prism is given by the formula SA = 2lw + 2lh + 2wh, where l, w, and h represent the length, width, and height of the prism, respectively. Since we have a rectangular prism with a square base, the length and width are both equal to x, and the height can be denoted as h. Given that the surface area is 100 square feet, we can set up the equation as follows:

100 = 2x^2 + 2xh + 2xh. Simplifying the equation, we have:

100 = 2x^2 + 4xh. Now, to express the volume of the box as a function of x, denoted as v(x), we need to use the formula for the volume of a rectangular prism, which is V = lwh. Since the base is a square, the length and width are both equal to x. Therefore, the volume can be expressed as:

v(x) = x^2h.

So, the volume of the rectangular prism, as a function of x, is v(x) = x^2h.

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Answer:

a = 8

Step-by-step explanation:

The question has given us the following equation of a line:

[tex]y-2= \frac{2}{7}(x-1)[/tex],

and told us that it contains the point (a, 4). It then asks us to find the value of a.

To do this, we have to understand the following: since point (a, 4) is on the given line, these coordinates satisfy the equation of the line. In other words, if we substitute the given values into the equation, the equality will still be valid.

Therefore, we can simply substitute (a, 4) into the equation and then solve for a:

[tex]y-2= \frac{2}{7}(x-1)[/tex]

⇒ [tex]4 - 2 = \frac{2}{7}(a-1)[/tex]

⇒ [tex]2 = \frac{2}{7}(a - 1)[/tex]

⇒ [tex]2 \div \frac{2}{7} = a - 1[/tex]      [Dividing both sides of the equation by [tex]\frac{2}{7}[/tex]]

⇒ [tex]2 \times \frac{7}{2} = a-1[/tex]      [Dividing by a fraction is the same as multiplying by its reciprocal]

⇒  [tex]\frac{14}{2} = a - 1[/tex]

⇒ [tex]7= a - 1[/tex]

⇒ [tex]a = 7 + 1[/tex]            [Adding 1 to both sides]

⇒ [tex]a = \bf 8[/tex]

Therefore, the value of a is 8.

The value of x is 6.

The value of x can be found with the help of Pythagoras theorem and Perpendicular Bisector theorem. As we can see in the diagram, the radius of the circle is 8 and point T and U are perpendicular. The value of PQ is 3x-4 and RS is 14. So, we will draw two lines from point H to point Q and point R making two perpendicular angled triangles.

So, according to the diagram:

In ΔHRU, [tex]HR^{2}[/tex] = [tex]HU^{2} + UR^{2}[/tex]

As we know, a perpendicular radius bisects the chord,

UR = 1/2 * RS

UR = 1/2 * 14

UR = 7

So, [tex]HR^{2}[/tex] = [tex]8^{2} + 7^{2}[/tex]

[tex]HR^{2}[/tex] = 113

As we know that the radii of the circles are equal, so:

[tex]HR^{2} = HQ^{2}[/tex] [tex]= 113[/tex]

Similarly, in ΔHQT

[tex]HQ^{2} = HT^{2} + TQ^{2}[/tex]

[tex]TQ^{2} = 113 - 64[/tex]

[tex]TQ^{2} = \sqrt[]{49}[/tex]

[tex]TQ^{2} = 7[/tex]

As we know, a perpendicular radius bisects the chord,

TQ = 1/2 * PQ

PQ = 2 * TQ

PQ = 2 * 7

PQ = 14

But, in the question we have been given PQ as 3x - 4, so:

3x - 4 = 14

3x = 18

x = 6

Therefore, value of x when PQ=3x-4 and RS=14 is 6.

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The levels of factor is 4 when the degree of freedom is 3 .

Given,

Degree of freedom = 3

Now,

If factor A has a levels, then numerator degree of freedom for factor A is

df  = A - 1

Here,

In the given case degree of freedom is 3 so levels of factors,

A = 3+1 = 4.

Therefore given statement is false.

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Complete question :

A two-factor analysis of variance produces an f-ratio for factor a that has df = 3, 36. this analysis is comparing three different levels of factor A. True False.

The solution of number after multiplication is,

⇒ [tex]2^{29/12}[/tex]

We have to give that,

An expression to simplify,

⇒ ⁴√8 × ∛32

Now, Multiplying numbers is not possible without simplifying because the nth power of numbers is not the same.

Hence, We can simplify the numbers as,

⇒ ⁴√8 × ∛32

⇒ ⁴√(2×2×2) × ∛(2×2×2×2×2)

⇒ ⁴√(2)³ × ∛2⁵

⇒ [tex]2^{3/4} * 2^{5/3}[/tex]

⇒ [tex]2^{\frac{3}{4} + \frac{5}{3} }[/tex]

⇒ [tex]2^{29/12}[/tex]

Therefore, The solution of number after multiplication is,

⇒ [tex]2^{29/12}[/tex]

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The answers in part (b) represent the fines for violating the speed limits on the freeway. In part (b), we are given three values for x: 25, 60, and 80. Let's evaluate each of them in the piecewise function F(x).

For F(25), we look at the first condition, which states that if the speed is less than 40 mi/h, the output is 0. Since 25 is less than 40, F(25) would be 0.

For F(60), we consider the second condition. If the speed is between 40 and 65 mi/h (inclusive), the output is the square root of the speed minus 40. So, F(60) would be √(60 - 40) = √20 = 2√5.

For F(80), we focus on the third condition. If the speed is greater than 65 mi/h, the output is 70. Since 80 is greater than 65, F(80) would be 70.

Based on these evaluations, it is clear that the answers in part (b) represent the fines for violating the speed limits on the freeway. F(25) and F(80) both result in non-zero values, indicating that fines would be incurred for driving at those speeds. On the other hand, F(60) yields a non-zero value as well, representing a fine for exceeding the speed limit between 40 and 65 mi/h. Therefore, the fines for violating the speed limits on the freeway are represented by the answers in part (b).

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The result of the operation (3x/2) + (5x/2) is 4x.

A common denominator is a shared multiple of the denominators of two or more fractions. It is used to simplify the process of adding or subtracting fractions.

To explain with an example, let's consider the fractions 1/4 and 2/5. These fractions have different denominators, which means they cannot be added or subtracted directly. To add or subtract them, we need to find a common denominator.

To perform the indicated operation (3x/2) + (5x/2), we combine like terms by adding the coefficients of x.

The common denominator for the fractions is 2, so we can write the expression as:

(3x + 5x) / 2

Adding the coefficients of x gives:

8x / 2

Simplifying the expression further:

8x / 2 = 4x

Therefore, the result of the operation (3x/2) + (5x/2) is 4x.

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The frequency distribution table might look like this:

Class Interval | Frequency

35,000 - 40,000 | 8

40,000 - 45,000 | 12

45,000 - 50,000 | 10

... | ...

To construct a frequency distribution with a lower class limit of 35,000 and a class width of 5,000, we would need the actual income data and the frequency (number of occurrences) for each income value. Without the specific data, I cannot provide the frequency distribution table. However, I can guide you on how to create one using your data.

Assuming you have a dataset of income values and their frequencies, follow these steps to construct the frequency distribution:

Determine the class boundaries:

The lower class limit is 35,000.

The upper class limit can be found by adding the class width to the lower class limit: 35,000 + 5,000 = 40,000.

Class boundaries define the range of values for each class interval. In this case, the class boundaries for the first class are 35,000 (inclusive) and 40,000 (exclusive).

Create the class intervals:

Based on the class boundaries, you can create class intervals. For example, the first class interval would be "35,000 - 40,000".

Collect the data values and count the frequency of each value falling within the given class boundaries:

Go through your dataset and count the number of income values that fall within each class interval. Sum up the frequencies for each class interval.

Create a frequency distribution table:

Create a table with two columns: "Class Interval" and "Frequency".

List the class intervals in the "Class Interval" column.

Enter the frequencies in the "Frequency" column corresponding to each class interval.

For example, the frequency distribution table might look like this:

Class Interval | Frequency

35,000 - 40,000 | 8

40,000 - 45,000 | 12

45,000 - 50,000 | 10

... | ...

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complete question:

(a) Construct a frequency distribution. Use a first class having a lower class limit of 35,000 and a class width of 5000.

Income

Frequency

The value of x that makes ABCD an isosceles quadrilateral is x = 11.

To determine the value of x that makes ABCD an isosceles quadrilateral, we need to find the condition where AB is congruent to CD and BC is congruent to AD.

In an isosceles quadrilateral, opposite angles are congruent.

So, we can set up an equation based on the given angle measures:

m∠ABC = m∠CDA   (opposite angles)

4x + 11 = 2x + 33   (substituting the given angle measures)

4x - 2x = 33 - 11

2x = 22

x = 11

Hence, the value of x that makes ABCD an isosceles quadrilateral is x = 11.

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Assumption: Assume that 4y + 17 = 41.

an indirect proof of the statement "If 4y + 17 = 41, then y = 6," we make the assumption that 4y + 17 = 41.

In an indirect proof, we assume the opposite of what we want to prove and then demonstrate that it leads to a contradiction. By assuming that 4y + 17 = 41, we can proceed with logical steps to show that this assumption leads to a contradiction, which would then validate the original statement that y = 6.

By assuming the opposite and working through the logical steps, we aim to demonstrate that no other solution is possible except y = 6, reinforcing the validity of the statement.

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The point(s) of intersection between the parabola [tex]y = 2x^2[/tex] and the line y = 4x - 2 can be found by setting the equations equal to each other and solving for x is (1, 2)

The point(s) of intersection between the parabola  [tex]y = 2x^2[/tex] and the line y = 4x - 2 can be found by setting the equations equal to each other and solving for x.

To find the point(s) of intersection, we equate the two equations:

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

This is a quadratic equation, which can be solved by setting it equal to zero:

[tex]2x^2 - 4x + 2 = 0[/tex]

To solve the quadratic equation, we can use the quadratic formula:

[tex]x = (-b \pm \sqrt{b^2 - 4ac}) / (2a)[/tex]

In this case, a = 2, b = -4, and c = 2. Plugging in these values into the quadratic formula, we get:

[tex]x = (-(-4)\pm \sqrt{(-4)^2 - 4 * 2 * 2}) / (2 * 2)\\x = (4 \pm\sqrt{16 - 16}) / 4\\x = (4\pm \sqrt{0}) / 4\\x = (4 \pm0) / 4\\x = 1[/tex]

Therefore, the line y = 4x - 2 intersects the parabola [tex]y = 2x^2[/tex]at the point (1, 2).

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The population median commute time of the employees of Asons is 40 minutes.

The median is a measure of central tendency that represents the middle value in a dataset when the data is arranged in ascending or descending order. In this case, the commute times provided are 47.4, 45, 35, and 40 minutes.

To determine the median, we arrange the commute times in ascending order: 35, 40, 45, 47.4. Since there are four values in the dataset, the middle value is the average of the two central values, which are 40 and 45. Taking the average of these two values gives us the median of 40 minutes.

Therefore, the population median commute time of the employees of Asons is 40 minutes, indicating that half of the employees have a commute time of 40 minutes or less, while the other half have a commute time of 40 minutes or more.

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The next time the red line and blue line trains will arrive at the station at the same time will be in 40 minutes.

To determine the next time the red line and blue line trains will arrive at the station simultaneously, we need to find the least common multiple (LCM) of the time intervals between their arrivals. The red line arrives every 8 minutes, and the blue line arrives every 10 minutes.

The LCM of 8 and 10 is 40. Therefore, it will take 40 minutes for both trains to arrive at the station at the same time again.

As for the blue line and yellow line trains, their time intervals between arrivals are 10 minutes and 12 minutes, respectively. To find the next time they will arrive at the station simultaneously, we again need to find the LCM of these time intervals.

The LCM of 10 and 12 is 60. Therefore, it will take 60 minutes for both the blue line and yellow line trains to arrive at the station at the same time again.

Finally, to determine when all three trains (red line, blue line, and yellow line) will arrive at the station simultaneously, we need to find the LCM of their respective time intervals. The time intervals are 8 minutes, 10 minutes, and 12 minutes.

The LCM of 8, 10, and 12 is 120. Thus, it will take 120 minutes for all three trains to arrive at the station at the same time.

In summary, the next time the red line and blue line trains will arrive at the station at the same time is in 40 minutes, the next time the blue line and yellow line trains will arrive simultaneously is in 60 minutes, and the next time all three trains will arrive together is in 120 minutes.

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The system will be consistent and will have unique solution.

Given,

Coefficient matrix of 5 equation in 5 variables.

Here,

Let A be a 5x5 coefficient matrix such that its each column has a pivot element, then

Rank A = 5, rank of augmented matrix [A|b] = 5 and number of unknowns = 5

Rank A = rank of augmented matrix [A|b] = number of unknowns = 5

Hence , System is consistent

There is a unique intersection point of all three lines, so values of variables is unique.

Hence, solution of system if unique.

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The type of variable that represents the names of the automobile manufacturers would be the categorical variable.

A variable is defined as the quantity that may change within the context of a mathematical problem, research work or an experiment.

There are various types of variables that include the following:

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The total production after specialization would be 3F + 6C = 3 + 6 = 9 units.

Betty: F = 3 - 3C

Ann: F = 6 - 1.5C

Let's calculate the opportunity costs for both Betty and Ann for Fish and Coconuts:

The opportunity cost of Fish for Betty is the slope of her PPF, which is the negative of the coefficient of C in her equation. So, the opportunity cost of Fish for Betty is -(-3) = 3 Coconuts.

The opportunity cost of Coconuts for Betty is the inverse of the slope, which is 1/3 Fish per Coconut.

Similarly, the opportunity cost of Fish for Ann is the negative of the coefficient of C in her equation, which is -(-1.5) = 1.5 Coconuts.

The opportunity cost of coconut for Ann is the inverse of the slope, which is 2/3 Fish per Coconut.

Comparative advantage in Fish:

Betty has a lower opportunity cost of Fish (3 Coconuts) compared to Ann (1.5 Coconuts). Therefore, Betty has a comparative advantage in Fish production.

Comparative advantage in Coconuts:

Ann has a lower opportunity cost of Coconuts (2/3 Fish) compared to Betty (1/3 Fish). Therefore, Ann has a comparative advantage in Coconut production.

Absolute advantage:

Betty has an absolute advantage in Fish production since she can produce 3 Fish per day compared to Ann's 1 Fish per day. Ann has an absolute advantage in Coconut production since she can produce 2 Coconuts per day compared to Betty's 1 Coconut per day.

If in Autarky (without trade) Betty produces 1F and 1C, and Ann produces 1F and 2C, the total production would be:

Betty: 1F + 1C = 2 units

Ann: 1F + 2C = 3 units

If each specializes in the production of the good in which she has a comparative advantage, Betty would produce only Fish and Ann would produce only Coconuts. The total production would be:

Betty: 3F (since she has a comparative advantage in Fish)

Ann: 6C (since she has a comparative advantage in Coconuts)

The total production after specialization would be 3F + 6C = 3 + 6 = 9 units.

The gain from specialization is the increase in total production compared to Autarky, which is 9 units - 5 units (total production in Autarky) = 4 units.

The range for the terms-of-trade (TOT) represents the rate at which Fish and Coconuts can be exchanged between Betty and Ann. Since we have only the information about their individual production, we cannot determine the exact range of the TOT.

The social PPF for this society, considering only Betty and Ann, would be the sum of their individual PPFs.

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The volume of the cone is,

V = 28.1 millimeters

We have to give that,

An oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters.

Since the Volume of the cone is,

V = πr²h/3

Here, r = 1.6 millimeters

h = 10.5 millimeters

Substitute all the values,

V = 3.14 × 1.6² × 10.5 / 3

V = 80.4/3

V = 28.1 millimeters

Therefore, The volume is,

V = 28.1 millimeters

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The X-bar and R chart method is suitable for monitoring continuous data, such as measurements or weights. The p chart is used to monitor the proportion of nonconforming items in a sample, while the c chart is used when the count of nonconformities is being monitored.

The X-bar and R chart method is utilized when the focus is on continuous data. It consists of two charts: the X-bar chart, which tracks the average or mean of a process, and the R chart, which monitors the range or variation within subgroups. This method is commonly used in industries where measurements or weights are critical, such as manufacturing processes, chemical production, or quality control in laboratories. The X-bar chart helps identify shifts or trends in the process mean, while the R chart detects variations within subgroups.

The p chart is employed when the objective is to monitor the proportion of nonconforming items in a sample. It is useful when dealing with attribute data, where each item is classified as either conforming or nonconforming. The p chart is commonly used in industries where the focus is on the quality of a product or service, such as in pharmaceutical manufacturing, food processing, or customer satisfaction surveys. This chart helps identify any significant changes in the proportion of nonconforming items, enabling timely corrective actions.

The c chart is employed when the count of nonconformities is being monitored. It is particularly suitable for situations where the number of nonconformities can vary within a sample, such as defects in a product or errors in a process. The c chart is frequently used in industries where the occurrence of defects needs to be controlled, like automotive manufacturing, electronics production, or healthcare facilities. By monitoring the count of nonconformities, the c chart helps detect any unusual variations and facilitates the implementation of corrective measures to maintain process stability and quality standards.

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The triangle with side lengths 11, 12 and 24 is a obtuse triangle.

Let us assume a, b and c are three sides of a triangle in such a way that c is the longer side.

if c²=a²+b² then the triangle is a right triangle.

if c²>a²+b² then the triangle is a obtuse triangle.

if c²<a²+b² then the triangle is a acute triangle.

Let us take c=24, a=11 and b=12.

24²=11²+12²

576=121+144

576>265

Since, 576>265 the triangle is a obtuse triangle.

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In the Bertrand Duopoly Competition, choosing a price of $0 is a dominated strategy for each firm because they can always earn a higher profit by setting a positive price. Similarly, setting a price of $6 is also a dominated strategy for both firms. Choosing a price higher than the monopoly price is also a dominated strategy, as it results in lower profits. The rationalizable strategies in this game are setting a positive price between $0 and the monopoly price.

(a) Setting a price of $0 is a dominated strategy for each firm because their rival firm can undercut them by setting a slightly positive price, resulting in zero profits for the firm setting $0. Similarly, setting a price of $6 is dominated because the rival firm can set a slightly lower price and capture the entire market demand, leaving the firm setting $6 with zero profits.

(b) The monopoly price represents the highest price that a firm can set while still maximizing its profits. Any price higher than the monopoly price will result in a decrease in demand and lower profits. Thus, choosing a price higher than the monopoly price is a dominated strategy.

(c) The rationalizable strategies in this game are the set of prices between $0 and the monopoly price. The Nash Equilibrium occurs when both firms set the monopoly price, as neither firm has an incentive to deviate from this strategy.

(d) In the Nash Equilibrium, both firms set the monopoly price, resulting in a quantity sold in the market that corresponds to the demand at the monopoly price. Each firm earns a profit equal to half of the total industry profits, as they split the resulting demand equally between them.

(e) The market outcome in Bertrand competition with the Nash Equilibrium price and quantity is different from the monopoly outcome. In the monopoly outcome, the single firm sets the monopoly price and quantity, earning higher profits compared to the Nash Equilibrium in Bertrand competition. The presence of competition in the Bertrand model drives prices down towards marginal costs, resulting in lower profits for both firms compared to the monopoly outcome.

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1) The sum is 4.9.

2) The sum is 7.4.

To add numbers while maintaining the correct number of significant digits, follow these steps:

Determine the number with the fewest decimal places (or significant figures) among the numbers being added.

Add the numbers together.

Round the result to match the number of decimal places (or significant figures) of the least precise number.

Let's apply these steps to the given examples:

1) 2.1 + 2.824:

The number 2.1 has one decimal place, and 2.824 has three decimal places. So, we need to round the result to one decimal place.

2.1 + 2.824 = 4.924

Since the number 2.1 has one decimal place, the final result should also have one decimal place. Therefore, the sum is 4.9.

2) 3.2 + 4.19:

The number 3.2 has one decimal place, and 4.19 has two decimal places. So, we need to round the result to one decimal place.

3.2 + 4.19 = 7.39

Since the number 3.2 has one decimal place, the final result should also have one decimal place. Therefore, the sum is 7.4.

In both cases, the sum is rounded to match the least precise number in terms of decimal places (or significant figures). This helps maintain the appropriate level of precision in the final result, consistent with the original approximations.

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Complete question =

Add the following numbers, leaving the results with the correct number of significant digits if each number is assumed to be approximate

1) 2.1 + 2.824

2) 3.2 + 4.19

The calculated depth of the well in feet is 402.5 feet

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

Time, t = 5 seconds

The depth of the well in feet is calculated as

d = 1/2gt²

substitute the known values in the above equation, so, we have the following representation

d = 1/2 * 32.2 * 5²

Evaluate

d = 402.5

Hence, the depth is 402.5 feet

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Answer:

Step-by-step explanation:

To apply the Rational Root Theorem to the equation 2x³ - 5x + 4 = 0, we need to determine the possible rational roots. The Rational Root Theorem states that any rational root of a polynomial equation with integer coefficients must be of the form p/q, where p is a factor of the constant term (in this case, 4) and q is a factor of the leading coefficient (in this case, 2).

The factors of 4 are ±1, ±2, and ±4.

The factors of 2 are ±1 and ±2.

Therefore, the possible rational roots can be expressed as:

±1/1, ±1/2, ±2/1, ±2/2, ±4/1, ±4/2.

Simplifying these fractions:

±1, ±1/2, ±2, ±1, ±4, ±2.

Now, we need to check if any of these possible rational roots are actual roots of the equation. We can do this by substituting each value into the equation and checking if the equation equals zero.

Checking ±1:

For x = 1:

2(1)³ - 5(1) + 4 = 2 - 5 + 4 = 1 ≠ 0

For x = -1:

2(-1)³ - 5(-1) + 4 = -2 + 5 + 4 = 7 ≠ 0

Checking ±1/2:

For x = 1/2:

2(1/2)³ - 5(1/2) + 4 = 1/4 - 5/2 + 4 = -3/4 ≠ 0

For x = -1/2:

2(-1/2)³ - 5(-1/2) + 4 = -1/4 + 5/2 + 4 = 15/4 ≠ 0

Checking ±2:

For x = 2:

2(2)³ - 5(2) + 4 = 16 - 10 + 4 = 10 ≠ 0

For x = -2:

2(-2)³ - 5(-2) + 4 = -16 + 10 + 4 = -2 ≠ 0

Checking ±4:

For x = 4:

2(4)³ - 5(4) + 4 = 128 - 20 + 4 = 112 ≠ 0

For x = -4:

2(-4)³ - 5(-4) + 4 = -128 + 20 + 4 = -104 ≠ 0

None of the possible rational roots ±1, ±1/2, ±2, ±4 are actual roots of the equation 2x³ - 5x + 4 = 0.

Therefore, this equation does not have any rational roots.

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Answer:

2πr = 4, so r = 2/π

Curved surface area = 2π(2/π)(9) = 36 cm²

The volume of a cylinder with a radius of 3 cm and height of 8 cm is approximately 226.1 cubic centimeters. This is calculated using the formula V = πr²h, where π is approximately 3.14.

The volume of a cylinder is given by the formula V = πr²h, where r is the radius and h is the height of the cylinder.

In this case, the radius is 3 centimeters and the height is 8 centimeters. Substituting these values into the formula, we get V = π(3)²(8) = π(9)(8) = 72π.

To find the approximate value, we can use the approximation π ≈ 3.14. So, the volume is approximately V ≈ 72(3.14) = 226.08 cubic centimeters. Rounding to the nearest tenth, the volume is approximately 226.1 cubic centimeters.

To find the volume of a cylinder, we multiply the area of the base (which is a circle) by the height of the cylinder. In this case, the radius of the cylinder is given as 3 centimeters. Therefore, the area of the base (circle) is calculated as πr² = π(3)² = 9π square centimeters.

The height of the cylinder is given as 8 centimeters. Multiplying the area of the base by the height gives us the volume: V = 9π * 8 = 72π cubic centimeters.

To find the approximate value, we can use the approximation π ≈ 3.14. So, the volume is approximately 72 * 3.14 = 226.08 cubic centimeters. Rounding to the nearest tenth, we get approximately 226.1 cubic centimeters.

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The explicit formula for the given sequence is aₙ = 1 * 3^(n-1), and the tenth term is 19683. This formula can be used to find any term in the sequence by plugging in the corresponding term number.

The given sequence seems to be a geometric sequence, where each term is obtained by multiplying the previous term by a common ratio. In this case, the common ratio appears to be 3 because each term is three times the previous term.

To find the explicit formula for the sequence, we can use the general formula for a geometric sequence: aₙ = a₁ * r^(n-1), where aₙ represents the nth term, a₁ is the first term, r is the common ratio, and n is the term number.

For the given sequence, the first term (a₁) is 1, and the common ratio (r) is 3. Plugging these values into the formula, we have:

aₙ = 1 * 3^(n-1)

Now, to find the tenth term (a₁₀), we substitute n = 10 into the formula:

a₁₀ = 1 * 3^(10-1)

    = 1 * 3^9

    = 1 * 19683

    = 19683

Therefore, the tenth term of the given sequence is 19683.

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The value of the product (-4+5 i)(-4-5 i)is 41.

To find the product of the complex numbers (-4+5i) and (-4-5i).

we can use the FOIL method, which stands for First, Outer, Inner, Last.

First, we multiply the first terms of each binomial:

(-4) × (-4) = 16

Outer, we multiply the outer terms:

(-4) ×  (-5i) = 20i

Inner, we multiply the inner terms:

(5i) ×  (-4) = -20i

Last, we multiply the last terms:

(5i) ×  (-5i) = -25i²

Now, we simplify the result:

Remember that i×  is defined as -1, so we have:

-25i² = -25 ×  (-1) = 25

Putting it all together, the product of (-4+5i) and (-4-5i) is:

16 + 20i - 20i + 25

= 16 + 25

= 41

Therefore, (-4+5i)(-4-5i) equals 41.

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