Determine which function has the greatest rate of change as x approaches infinity. f(x) = 2x − 10 g(x) = 16x − 4 h(x) = 3x2 − 7x 8 there is not enough information to determine the answer.

Answer: Therefore, the team was producing 10 parts per hour prior to switching to the new technology.

Step-by-step explanation:Let's denote the number of parts produced per hour before the technology upgrade by x.

After the upgrade, the team produces 6 more parts per hour than before, so their new production rate is x + 6 parts per hour.

In 8 hours, the team produces 8x parts in total.

In 6 hours with the new technology, the team produces 120% of what they previously produced in 8 hours, or 1.2(8x) = 9.6x parts in total.

We can set up an equation based on the information above:

6(x + 6) = 9.6x

Simplifying the equation:

6x + 36 = 9.6x

Subtracting 6x from both sides:

36 = 3.6x

Dividing both sides by 3.6:

x = 10

The second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.

Newton's method to approximate a root of the equation 5sin(x) = x.

We are given the initial approximation x1 = 1. To find the second approximation x2 and the third approximation x3, we need to follow these steps:

Step 1: Write down the given function and its derivative. f(x) = 5sin(x) - x f'(x) = 5cos(x) - 1

Step 2: Apply Newton's method formula to find the next approximation. x_{n+1} = x_n - f(x_n) / f'(x_n)

Step 3: Calculate the second approximation x2 using x1 = 1. x2 = x1 - f(x1) / f'(x1) x2 = 1 - (5sin(1) - 1) / (5cos(1) - 1) x2 ≈ 1.112141637097

Step 4: Calculate the third approximation x3 using x2. x3 = x2 - f(x2) / f'(x2) x3 ≈ 1.112141637097 - (5sin(1.112141637097) - 1.112141637097) / (5cos(1.112141637097) - 1) x3 ≈ 1.130884826739

So, the second approximation x2 is approximately 1.112141637097, and the third approximation x3 is approximately 1.130884826739.

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The function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet is y = 12x - 4

Let's start by considering the initial amount of printer paper in the closet before any boxes are used. Since each box contains 12 reams of printer paper, if there are x boxes in the closet, then the total number of reams of paper is given by 12x.

Now, if a worker uses 4 reams from one of the boxes, then the total number of reams of paper remaining in the closet is (12x - 4). If we define y as the total number of reams of paper remaining in the closet, then we have:

y = 12x - 4

This function shows the relationship between y, the total number of reams of printer paper remaining in the closet, and x, the number of boxes in the closet.

As x increases, the total number of reams of paper in the closet increases as well. However, each time a worker uses 4 reams of paper from a box, the total number of reams of paper in the closet decreases by 4.

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Curved surface area of cone: 255π or approx. 801.41 sq units with radius 15 and height 8.

The curved surface area of a right circular cone can be calculated using the formula:

A = πrℓ

where A is the area of the curved surface,

r is the radius of the base of the cone, and

ℓ is the slant height of the cone.

To find the slant height, we can use the Pythagorean theorem:

ℓ² = r² + h²

where h is the height of the cone.

Substituting the given values, we get:

ℓ² = 15² + 8²

ℓ² = 225 + 64

ℓ² = 289

ℓ = √289

ℓ = 17

Now, substituting the values of r and ℓ in the formula for curved surface area, we get:

A = πrℓ

A = π(15)(17)

A = 255π

Therefore, the area of the curved surface of the cone is 255π square units, or approximately 801.41 square units.

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The budgeted ending raw material inventory for May is 2,560 pounds, calculated by taking 25% of the next month's budgeted production (12,800 units) multiplied by 2 pounds per unit.

To solve this problem, we need to calculate the budgeted production and raw material inventory for June, July, and August.

Budgeted production = 12,800 units + 30% * 16,800 units = 17,440 units

Raw material inventory = 25% * 17,440 units * 2 pounds = 8,720 pounds

Budgeted production = 16,800 units + 30% * 14,800 units = 20,840 units

Raw material inventory = 25% * 20,840 units * 2 pounds = 10,420 pounds

Budgeted production = 14,800 units + 30% * 20,840 units = 20,632 units

Raw material inventory = 25% * 20,632 units * 2 pounds = 10,316 pounds

To find the budgeted ending finished goods inventory for June, we need to subtract the budgeted sales for June from the budgeted production for June and add the budgeted ending finished goods inventory for May:

Budgeted ending finished goods inventory for June = 17,440 units - 12,800 units + 3,840 units = 8,480 units

Similarly, we can find the budgeted ending finished goods inventory for July and August:

Budgeted ending finished goods inventory for July = 20,840 units - 16,800 units + 8,480 units = 12,520 units

Budgeted ending finished goods inventory for August = 20,632 units - 14,800 units + 12,520 units = 18,352 units

Therefore, the budgeted ending finished goods inventory for June, July, and August are 8,480 units, 12,520 units, and 18,352 units, respectively. The budgeted raw material inventory for June, July, and August are 8,720 pounds, 10,420 pounds, and 10,316 pounds, respectively.

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The correct equation to determine the number of people (p) who can go to the amusement park is: 100.25 = 13.75p + 17.75.

Here's the step-by-step explanation:

1. The total amount they have to spend is $100.25.
2. The cost of parking is $17.75, which is a one-time expense.
3. The cost of admission per person is $13.75.

To find out how many people can go, you need to account for both the parking cost and the cost of tickets for each person. Therefore, the equation is:

100.25 (total amount) = 13.75p (cost per person times the number of people) + 17.75 (cost of parking)

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To find the length of the ladder needed to reach the roof of the shed that is 35 ft from the ground with the base of the ladder 12 ft from the shed, you can use the Pythagorean theorem. The theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the ladder, in this case) is equal to the sum of the squares of the other two sides (the height and the distance from the shed).

Step 1: Identify the sides of the triangle.
- Height (a): 35 ft (vertical side)
- Distance from the shed (b): 12 ft (horizontal side)
- Ladder length (c): Hypotenuse

Step 2: Apply the Pythagorean theorem.
- a² + b² = c²
- 35² + 12² = c²

Step 3: Calculate the squares and sum them.
- (35 * 35) + (12 * 12) = c²
- 1225 + 144 = c²
- 1369 = c²

Step 4: Find the length of the ladder (c).
- c = √1369
- c = 37

The ladder needs to be 37 ft long to reach the roof of the shed.

Shortening the distance between the ladder and the shed will affect the height of the ladder by making it steeper. This will cause the ladder to be higher above the ground, but it may also make it less stable and more difficult to climb.

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the new coordinates of the rhombus W'X'Y'Z' after a dilation with scale factor k=3 are: [tex]W'(3,0), X'(12,-3), Y'(15,-12), Z'(6,-9)[/tex]

To find the new coordinates of the image after dilation, we need to multiply the coordinates of each vertex by the scale factor k = 3.

Let's start with vertex W(1,0):

Multiply the x-coordinate by  [tex]3: 1 *\times 3 = 3[/tex]

Multiply the y-coordinate by [tex]3: 0 \times 3 = 0[/tex]

So the new coordinates of W' are [tex](3,0).[/tex]

Next, let's look at vertex X(4,-1):

Multiply the x-coordinate by [tex]3: 4 \times 3 = 12[/tex]

Multiply the y-coordinate by [tex]3: -1 \times 3 = -3[/tex]

So the new coordinates of X' are [tex](12,-3).[/tex]

Now for vertex Y(5,-4):

Multiply the x-coordinate by [tex]3: 5 \times 3 = 15[/tex]

Multiply the y-coordinate by [tex]3: -4 \times3 = -12[/tex]

So the new coordinates of Y' are  [tex](15,-12).[/tex]

Finally, let's consider vertex Z(2,-3):

Multiply the x-coordinate by  [tex]3: 2 \times 3 = 6[/tex]

Multiply the y-coordinate by  [tex]3: -3 \times3 = -9[/tex]

So the new coordinates of Z' are [tex](6,-9)[/tex]  .

Therefore, the new coordinates of the rhombus  [tex]W'X'Y'Z'[/tex] after a dilation with scale factor k=3 are:

[tex]W'(3,0)[/tex]

[tex]X'(12,-3)[/tex]

[tex]Y'(15,-12)[/tex]

[tex]Z'(6,-9)[/tex]

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Answer: 20 kg

Step-by-step explanation:

You follow the line of best fit until 50cm

Then you trace across and look at the x-axis.

There you will find that the dog will be 20kg at 50cm using the line of best fit.

Therefore, (70+72)/2 = **71 minutes** is the median time. B)42.5 minutes as a result. C,D)The range is determined by deducting the dataset's smallest value from highest value.

A)When the data are organized in order of magnitude, the median time is the middle value. The median in this situation is the average of the 14th and 15th values, which are 70 and 72, respectively. There are 28 data points in this situation. Therefore, (70+72)/2 = **71 minutes** is the median time.

b) The median of the lowest half of the data constitutes the lower quartile (Q1). We must arrange the data in descending order of magnitude before determining the median of the first half of the data in order to determine Q1.  is the average of the seventh and eighth values, which are 40 and 45, respectively, in the first half of the data, which consists of 14 values. Q1 = (40+45)/2 = **42.5 minutes as a result.

b) The median of the upper half of the data constitutes the upper quartile (C). We must first organise the data in descending order of magnitude before determining the median of the remaining data in order to determine Q3. Q3 is the average of the seventh and eighth values from the last, which are 90 and 105, respectively, in the second half of the data, which consists of 14 values. Q3 = (90+105)/2 = **97.5 minutes**3 as a result.

d) The range is determined by deducting the dataset's smallest value from highest value.

In this instance, Ben's practise time can be anywhere from **10 minutes** to **180 minutes**. Range then equals maximum value - minimum value, which in this case is 180 - 10 = **170 minutes**

e) Extreme values that are beyond the typical range of a dataset's values are known as outliers. We can use a criterion that states that any value that sits more than 1.5 times the interquartile range (IQR) below Q1 or above Q3 is regarded as an outlier to ascertain whether there are outliers in this dataset. When Q1 is subtracted from Q3, the result is the IQR: Q3 - Q1 = 97.5 - 42.5 = **55 minutes**3. By using this rule, we can see that the dataset contains the outliers **180** and **120** minutes.

The term "interquartile range" is IQR. It is a measure of variability that is based on quartilizing a dataset. The first quartile (Q1) is subtracted from the third quartile to determine the IQR. (Q3). It is a representation of the middle 50% of the data's range.

f) A box-and-whisker plot illustrates a dataset's quartiles, outliers, and range1. For Ben's practice, here's how to create a box-and-whisker plot:

- Create a number line with all the values Ben practised with.

- Draw a box spanning Q1 through Q3.

- Inside the box, at the location of Q2, draw a vertical line. (the median).

Draw whiskers from the box's two ends to all values that are not outliers.

- Place every outlier on the graph as a separate point, outside of any whiskers.

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Answer:

  b. g(t) = 1.5 sin (π t) +3.5

Step-by-step explanation:

You want to choose the function that has the given graph.

At t = 0, the graph shows a value of 3.5. The sine of 0 is 0, so this eliminates choice A.

At t = 1/2, the graph shows a value of 5. The values given by the different formulas are ...

  b. g(1/2) = 1.5·sin(π/2) +3.5 = 5 . . . . . matches the graph

  c. h(1/2) = 1.5·sin(π) + 3.5 = 3.5 . . . . no match

  d. h(1/2) = 1.5·sin(π/4) +3.5 = 0.75√2 +3.5 . . . . no match

__

Additional comment

The horizontal distance for one period of the graph (from peak to peak, for example) is T = 2 seconds. If the sine function is sin(ωt), then the value of ω is ...

  ω = 2π/T = 2π/2 = π

This tells you the function g(t) = 1.5·sin(πt)+3.5 is the correct choice.

The correct option regarding the data is B. Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the left.

A histogram has score on the x-axis, and frequency on the y-axis. A score of 4 has a frequency of 1; 5, 1; 6, 2; 7, 3; 8, 5; 9, 7; 10, 6.

It shtbe noted that Mean = 8.2, median = 9, mode = 9, midrange = 7; skewed to the left.

This statement describes a distribution with a mean equal to the median and a mode that is likely less than the mean and the median. The fact that the distribution is skewed to the left indicates that the tail of the distribution is longer on the left side, and that there may be some low outliers that are pulling the mean towards the left.

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To determine whether the function f(2) converges or diverges, we need to evaluate the limit of the function as x approaches 2. We can rewrite the function as:

f(2) = 1 / (x² + √x) = 1 / (x² + x^(1/2))

As x approaches 2, both x² and x^(1/2) approach 2, so we can substitute 2 for both of these terms:

f(2) = 1 / (2² + 2^(1/2)) = 1 / (4 + 1.414) ≈ 0.176

Therefore, f(2) converges to a finite value of approximately 0.176, and does not diverge.

Based on the given information, let's analyze the function f(x) = 1 / (x² + √x). To determine if the function converges or diverges, we can examine its behavior as x approaches infinity.

As x gets larger, both x² and √x increase, but x² increases at a much faster rate. Therefore, the denominator (x² + √x) will become larger and larger as x approaches infinity. Consequently, the value of the function f(x) = 1 / (x² + √x) will approach 0.

Since the function approaches 0 as x goes to infinity, we can conclude that the function f(x) = 1 / (x² + √x) converges.

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Points a,b and c satisfied the Pythagoras theorem. Thus, the triangle abc is a right angled triangle.

Every triangle has inner angles that add up to 180 degrees. A right angle and a right triangle are both formed when one of their internal angles is 90 degrees.

Given coordinates :

a(5,2) , b(2,-3) and c(-8,3).

Find the distance between the points using the distance formula:

d = √[(x2 - x1)² + (y2 - y1)²]

ab = √[(2 - 5)² + (- 3 - 2)²]

ab = √[(-3)² + (- 5)²]

ab = √[9 + 25]

ab = √34

ab² = 34

bc = √[(2 + 8)² + (- 3 - 3)²]

bc  = √[(10)² + (- 6)²]

bc  = √[100 + 36]

bc  = √136

bc²  = 136

ac = √[(-8 - 5)² + (3 - 2)²]

ac = √[(-13)² + (1)²]

ac = √[169 + 1]

ac = √170

ac² = 170

Now,

(ac)² = (bc)² + (ab)²

170 = 136 + 24

170 = 170

This, points a,b and c satisfied the Pythagoras theorem. Thus, the triangle abc is a right angled triangle.

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Answer:

34/73

Step-by-step explanation:

37 + 34 + 2 = number of customers = 73

73 is our denominator.

34 is the number of people who used a credit card.

34 is our numerator.

Put the two together, and you get 73! Enjoy!

In this scenario, the sampling units are four households, the variable of interest is the average food cost, and auxiliary information associated with the units is the number of people in each household and total food cost.

Sampling Units: The sampling units are the four households selected from the 25,000 households in the population.

They are as follows:

1. Household with 4 people that spent $150 on food

2. Household with 2 people that spent $100 on food

3. Household with 4 people that spent $200 on food

4. Household with 3 people that spent $140 on food

Variable of Interest: The variable of interest is the average cost on food per household for a week.

Auxiliary Information: The auxiliary information associated with the units includes the number of people in each household and the total amount spent on food for that week.

To estimate the average cost on food per household for a week, follow these steps:

1. Calculate the total cost on food for all four households: $150 + $100 + $200 + $140 = $590

2. Divide the total cost by the number of households: $590 / 4 = $147.50

So, the estimated average cost on food per household for a week is $147.50.

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Answer: 135 and 45

Step-by-step explanation:

We can read off from these equations the gradients of the two lines: (3) and (-2).

Then we quote the trigonometric identity tan(A-B) = [tan(A)-tan(B)] / [1+tan(A)tan(B)]

Substituting tan(A)=3 and tan(B)=-2 gives tan(A-B) = [(3)-(-2)] / [1+(3)(-2)] = 5/-5 = -1

So A-B = 135°.

That is the obtuse angle between the two lines, so the acute angle is 45°.

The marginal profit in dollars for the sale of expensive watches when approximately $1912.61.

Find the marginal profit in dollars from the sale?

We need to find the marginal profit in dollars from the sale of x expensive watches for the given profit function P(x) = 0.08x² - 5x + 6x^0.2 - 5200 when (a) x = 300, (b) x = 2000, (c) x = 5000, and (d) x = 12,000.

Find the derivative of the profit function P(x), which represents the marginal profit.
P'(x) = dP(x)/dx = 0.16x - 5 + (6 * 0.2 * x^(-0.8))

Calculate the marginal profit for each specified value of x:

x = 300:
P'(300) = 0.16(300) - 5 + (6 * 0.2 * 300^(-0.8)) ≈ 42.57

x = 2000:
P'(2000) = 0.16(2000) - 5 + (6 * 0.2 * 2000^(-0.8)) ≈ 317.52

x = 5000:
P'(5000) = 0.16(5000) - 5 + (6 * 0.2 * 5000^(-0.8)) ≈ 794.57

x = 12,000:
P'(12,000) = 0.16(12,000) - 5 + (6 * 0.2 * 12,000^(-0.8)) ≈ 1912.61

So, the marginal profit in dollars for the sale of expensive watches when (a) x = 300 is approximately $42.57, (b) x = 2000 is approximately $317.52, (c) x = 5000 is approximately $794.57, and (d) x = 12,000 is approximately $1912.61.

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Answer:

The first box and whisker plot

Step-by-step explanation:

A box and whisker plot gives you the five number summary for a set of data.  The five number summary is

Maximum and minimum:

We know from the data that the minimum value is 100 and the maximum value is 200.  However, because both boxes available as answer choices have the correct minimum and maximum, we'll need to find more data.

Median:

We can start finding the median first by arranging the data from the least to greatest.  Then, we find the middle of the data.  Because there are 9 points and 9 is odd, we know that there will be 4 points to the left of the median and 4 points to the right of the median:

100, 100, 120, 120, 150, 165, 180, 180, 200

150 has 4 numbers both on its left and right sides so its the median.

Because both of the plots available as answer choices have the correct median, we we'll need to find more data.

First Quartile/Q1:

In order to find Q1, we must find the middle number of the four numbers to the left of the median.

Because we have an even number of points, we will get two middle numbers, 100 and 120.  To find the middle of all four points, we average these two numbers:

(100 + 120) / 2 = 220 / 2 = 110

Only the first box has the accurate Q1 value, so it's our answer.

We don't have to find Q3, since both boxes have the correct Q3, but only the first box has the correct minimum, correct Q1, correct median, correct Q3, correct maximum.

The statement "The harmonic series: 1+1/2+1/3+1/4+... diverges, but when its terms are squared the resulting series converges." is True.

The harmonic series is defined as the sum of the reciprocals of the natural numbers: Σ(1/n) for n = 1 to ∞. This series is known to diverge, meaning that its sum tends to infinity as more terms are added.

However, when the terms of the harmonic series are squared, we get a new series called the p-series, with p=2: Σ(1/n^2) for n = 1 to ∞. The p-series converges if p > 1, which is true for p=2. Thus, the series Σ(1/n^2) converges to a finite sum.

In conclusion, the given statement is true, as the harmonic series diverges, but its squared terms result in a convergent series.

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Answer:

x = 16

Step-by-step explanation:

(2x + 16) = 48

Subtract 16 with the positive 16 to cancel the numbers.

Subtract 16 with 48.

2x = 32

divide 32 by 2 to isolate the x.

32/2 = 16

x = 16

a. The total length of time, T, it will take for the dog to reach the frisbee is  143.22

b. A natural closed interval that limits reasonable values of x is  [0, 220] is a reasonable closed interval for x.

c. The value of x that will minimize the time, T, that it takes for the dog to retrieve the frisbee is 143.22

Let's start by breaking down the problem into two parts: the time it takes for the dog to run along the shore, and the time it takes for the dog to swim in the water. Let's call the distance the dog runs along the shore "d1" and the distance the dog swims in the water "d2".

To find d1, we can use the Pythagorean theorem:

d1 = sqrt(x^2 + 60^2)

To find d2, we can use the fact that the total distance the dog travels is equal to 220 feet:

d2 = 220 - x

Now we can use the formulas for distance, rate, and time to find the total time it takes for the dog to retrieve the frisbee:

T = d1/13 + d2/4.3

Substituting our expressions for d1 and d2, we get:

T = [sqrt(x^2 + 3600)]/13 + (220 - x)/4.3

To find the value of x that minimizes T, we can take the derivative of T with respect to x, set it equal to zero, and solve for x:

dT/dx = x/13sqrt(x^2 + 3600) - 1/4.3 = 0

Multiplying both sides by 13sqrt(x^2 + 3600), we get:

x = (13/4.3)sqrt(x^2 + 3600)

Squaring both sides and solving for x, we get:

x ≈ 143.22

So the dog should enter the water after running about 143.22 feet down the shoreline to minimize the total time it takes to retrieve the frisbee.

To check that this is a minimum, we can take the second derivative of T with respect to x:

d^2T/dx^2 = (13x^2 - 46800)/(169(x^2 + 3600)^(3/2))

Since x^2 and 3600 are both positive, the numerator is positive when x is not equal to zero, and the denominator is always positive. Therefore, d^2T/dx^2 is always positive, which means that x = 143.22 is indeed the value that minimizes T.

As for the natural closed interval that limits reasonable values of x, we know that x has to be greater than zero (since the dog needs to run at least some distance along the shoreline before entering the water), and it has to be less than or equal to 220 (since the frisbee is 220 feet down the shoreline from the dog). So the interval [0, 220] is a reasonable closed interval for x.

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The line y=10x will in this instance pass through most of the data points, demonstrating that it is a good fit for the data.

A good line of fit should travel across the greatest number of data points and exhibit a positive connection.

A relationship between two variables in which rising values of one cause rising values of the other. On a scatter plot, it is shown as a positive slope.

The line y=10x will in this instance pass through most of the data points, demonstrating that it is a good fit for the data.

The line will be favourably sloped, so as the duration of an accessible bike rental increases, so does the total cost charged.

The scatterplot confirms this, proving that the line y=10x is a good match for the data.  

This indicates that the data points are nearly aligned with the line but not exactly so.

A good line of fit should travel across the greatest number of data points and exhibit a positive connection.

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Answer: C

Step-by-step explanation: after rotating, if you split it in half horizontally, you have two cones

The three-dimensional figure formed when the triangle is rotated around the dashed line through B and C is a cone.

A cone is a three-dimensional geometric form with a flat base and a smooth, tapering apex or vertex. A cone is made up of a collection of line segments, half-lines, or lines that link the base's points to the apex, which is a common point on a plane that does not include the base.

When we rotate a two-dimensional shape around an axis, we create a three-dimensional solid. This process is known as "revolution" or "rotational symmetry".

In this particular case, we have a triangle that can be rotated around the line segment that connects points B and C. If we were to rotate the triangle around this axis, we would create a three-dimensional solid. To figure out what kind of solid this is, we can think about the cross-sections that would be created if we were to slice through the solid perpendicular to the axis of rotation.

If we were to slice through the solid perpendicular to the axis of rotation, we would get a circle. This means that the solid created by rotating the triangle is a cylinder.

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(a) WAT is not true for the open interval (0,1) with function f(x) = 1/x.

(b) WAT holds for the closed set [a,∞) with any continuous function f(x).

(a) The Weierstrass Approximation Theorem (WAT) is not true if we replace the closed interval [a,b] with the open interval (a,b). A counterexample is the function f(x) = 1/x on the open interval (0,1). This function is continuous on (0,1) but it is not uniformly continuous, so it cannot be uniformly approximated by a polynomial.

(b) The Weierstrass Approximation Theorem holds if we replace [a,b] with the closed set [a,∞). That is, if f(x) is a continuous function on [a,∞), then for any ε > 0, there exists a polynomial p(x) such that |f(x) - p(x)| < ε for all x in [a,∞). The proof is similar to the proof of the original theorem using the Bernstein polynomials.

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The smallest side of the triangle makes an angle of approximately 20.6 degrees with the ground.

To determine the angle the smallest side will make with the ground, we can use the law of sines. The law of sines states that for any triangle ABC:

a/sin(A) = b/sin(B) = c/sin(C)

where a, b, and c are the lengths of the sides opposite the angles A, B, and C, respectively.

Let's label the sides of our triangle as follows:

The longest side (sitting on the ground) is side c

The second longest side is side b

The smallest side is side a

We know that side b is 2'3" long, which is equivalent to 27 inches. We also know that side a is 10 inches long. We can use the law of sines to solve for the angle opposite side a:

sin(A) = (a/c) * sin(C)

We can solve for sin(C) by using the fact that the sum of the angles in any triangle is 180 degrees:

C = 180 - A - B

We know that angle B is 18 degrees, so we can substitute that into our equation for C:

C = 180 - A - 18

C = 162 - A

Substituting this expression for C into our equation for sin(A), we get:

sin(A) = (a/c) * sin(162 - A)

We know that c is the longest side of the triangle and therefore opposite the largest angle. Since we are interested in the angle opposite side a, we can assume that angle A is the smallest angle in the triangle. We can use this assumption to simplify our equation for sin(A):

sin(A) = (a/c) * sin(162)

Plugging in the values for a, c, and sin(162), we get:

sin(A) = (10/27) * 0.951

sin(A) = 0.352

Taking the inverse sine of both sides, we get:

A = sin^-1(0.352)

A ≈ 20.6 degrees

Therefore, the smallest side of the triangle makes an angle of approximately 20.6 degrees with the ground.

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The arc length (s) for a circle with radius 4.27 meters and central angle 2.16 radians is approximately 9.22 meters.

To find the arc length (s) for a circle with radius (r) and central angle (θ), you can use the formula:

s = r * θ

In this case, the radius (r) is 4.27 meters, and the central angle (θ) is 2.16 radians. Plug these values into the formula:

s = 4.27 * 2.16

Now, multiply the values:

s ≈ 9.2232

Round the answer to two decimal places:

s ≈ 9.22 meters

So, the arc length (s) for a circle with radius 4.27 meters and central angle 2.16 radians is approximately 9.22 meters.

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Answer:

31.33 degrees

Step-by-step explanation:

The question is asking to find angle x.  You can use sine to find out x because sine is opposite/hypotenuse but since you are finding an angle measurement, it would be to the power of -1.  So:

sine^-1=13/25

31.33

Definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).

To evaluate the definite integral ∫(9x^2 - 4x - 1)dx, you need to first find the indefinite integral (also known as the antiderivative) of the function 9x^2 - 4x - 1. The antiderivative is found by applying the power rule of integration to each term separately:
∫(9x^2)dx = 9∫(x^2)dx = 9(x^3)/3 = 3x^3
∫(-4x)dx = -4∫(x)dx = -4(x^2)/2 = -2x^2
∫(-1)dx = -∫(1)dx = -x
Now, sum these results to obtain the antiderivative:
F(x) = 3x^3 - 2x^2 - x
∫(9x^2 - 4x - 1)dx from a to b = F(b) - F(a)

To evaluate the definite integral ∫(9x^2 - 4x - 1)dx =, we need to use the formula for integrating polynomials. Specifically, we use the power rule of integration, which states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration.
Using this formula, we integrate each term in the given expression separately. Thus, we have:
∫(9x^2 - 4x - 1)dx = (9∫x^2 dx) - (4∫x dx) - ∫1 dx
                  = 9(x^3/3) - 4(x^2/2) - x + C
                  = 3x^3 - 2x^2 - x + C
Next, we need to evaluate this definite integral. A definite integral is an integral with limits of integration, which means we need to substitute the limits into the expression we just found and subtract the result at the lower limit from the result at the upper limit. Let's say our limits are a and b, with a being the lower limit and b being the upper limit. Then, we have:
∫(9x^2 - 4x - 1)dx from a to b = [3b^3 - 2b^2 - b] - [3a^3 - 2a^2 - a]
                                              = 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a)
Therefore, the definite integral of ∫(9x^2 - 4x - 1)dx from a to b is 3(b^3 - a^3) - 2(b^2 - a^2) - (b - a).

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