Full Boat Manufacturing has projected sales of $115. 5 million next year. Costs are expected to be $67. 4 million and net investment is expected to be $12. 3 million. Each of these values is expected to grow at 9 percent the following year, with the growth rate declining by 1 percent per year until the growth rate reaches 5 percent, where it is expected to remain indefinitely. There are 4. 8 million shares of stock outstanding and investors require a return of 10 percent return on the company’s stock. The corporate tax rate is 21 percent

The factors that help to determine sample size are the desired confidence level and the margin of error. Therefore, the correct option is D) both b and c.

The desired confidence level and margin of error are two important factors that help to determine the sample size. The confidence level represents the level of certainty that the sample mean is close to the true population mean, while the margin of error is the range of error that is acceptable in the estimation of the population mean.

Both of these factors are interdependent, and an increase in either of them would require a larger sample size to achieve a certain level of accuracy. Therefore, carefully considering these factors and determining an appropriate sample size is essential for obtaining valid and reliable results.

The population size can also have an impact on the sample size calculation, but it is not a direct factor. So, the correct answer is D).

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Answer:

the probability of selecting a blue sock from the laundry basket is 2/7 or approximately 0.2857.

Step-by-step explanation:

The probability of selecting a blue sock can be found by dividing the number of blue socks by the total number of socks in the basket:

Probability of selecting a blue sock = Number of blue socks / Total number of socks

Probability of selecting a blue sock = 4 / 14

Simplifying the fraction by dividing both the numerator and denominator by 2 gives:

Probability of selecting a blue sock = 2 / 7

Answer:

The distance between these parallel lines is 2 - (-7) = 9 units.

A. Approximately 9.85% of New York City commutes are less than 30 minutes

B. Approximately 16.91% of New York City commutes are between 30 and 35 minutes.

A. To find the percent of New York City commutes that are less than 30 minutes, we need to calculate the z-score using the formula:

z = (x - μ) / σ

where x is the value we are interested in (30 minutes), μ is the mean commute time (39.7 minutes), and σ is the standard deviation (7.5 minutes).

z = (30 - 39.7) / 7.5 = -1.29

We can use a standard normal distribution table or calculator to find the area to the left of z = -1.29, which gives us:

P(z < -1.29) = 0.0985

Therefore, approximately 9.85% of New York City commutes are less than 30 minutes.

B. To find the percent of New York City commutes that are between 30 and 35 minutes, we need to calculate the z-scores for both values using the same formula:

z1 = (30 - 39.7) / 7.5 = -1.29

z2 = (35 - 39.7) / 7.5 = -0.62

We can then find the area between these two z-scores using a standard normal distribution table or calculator, which gives us:

P(-1.29 < z < -0.62) = P(z < -0.62) - P(z < -1.29) = 0.2676 - 0.0985 = 0.1691

Therefore, approximately 16.91% of New York City commutes are between 30 and 35 minutes.

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The first and fourth sequences show a geometric series.

Consequently, the sequences 2, 3, 4.5, 6.75, … and 4, 8, 16, 32, … are the geometric series.

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Complete Question:-

Which of the following show a geometric series? select all that apply.

1) 2 3 4.5 6.75 …

2) 2 2.5 3 3.5 …

3) 6 7 9 12 …

4) 4 8 16 20 …

5) 8 4 2 1 …

Answer:

Step-by-step explanation:

The value of k is 11 at which a and 6 will be orthogonal (form a 90 degree angle).

To find the value of k that makes vectors a and 6 orthogonal, we need to use the dot product formula:

a · 6 = 2(-1) + 3(5) + (-1)k = 0

Simplifying the above equation, we get:

-2 + 15 - k = 0

Combining like terms, we get:

13 - k = 0

Therefore, k = 13.

However, we need to check if this value of k makes vectors a and 6 orthogonal.

a · 6 = 2(-1) + 3(5) + (-1)(13) = 0

The dot product is zero, which means vectors a and 6 are orthogonal.

Thus, the final answer is k = 11.

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1 hour =60 minutes

Step-by-step explanation:

Let M be the time in minutes . T be temperature in Farhenheit. From 45th min to end of the hour there remains a steady temperature. after that gyms starts to cools down . For time 45≤M≤60, Temperature T=77oF.To find a) Is M a function of T ? we know that Temperature changes with respect to time . So M is independent variable and T is dependent variable . so M cannot be a function of T .

The sum of the series under the interval (3, 6) will be negative 4.

Given that:

Series, ∑ (2k - 10)

A series is a sum of sequence terms. That is, it is a list of numbers with adding operations between them.

The sum of the series under the interval (3, 6) is calculated as,

∑₃⁶ (2k - 10) = (2 x 3 - 10) + (2 x 4 - 10) + (2 x 5 - 10) + (2 x 6 - 10)

∑₃⁶ (2k - 10) = - 4 - 2 + 0 + 2

∑₃⁶ (2k - 10) = -4

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The possible dimensions of the rectangular prism having volume  =  12 in³,  are  Length = 2 in, width = 3 in, height = 2/3 in, and Length = 1 in, width = 12 in, height = 1/12 in.

To find the possible dimensions of the prism, we need to consider that the volume of a rectangular prism is given by the formula V = lwh, where l, w, and h are the length, width, and height of the prism, respectively.

Since the volume of the prism is given as 12 in³, we can write: 12 = lwh
Now, we need to find two sets of dimensions that satisfy this equation, where one of the dimensions is a fraction.

Let's try the first set of dimensions:
l = 2 in
w = 3 in
h = 2/3 in

Plugging these values into the formula for the volume, we get:
V = lwh
V = 2 in × 3 in × 2/3 in
V = 4 in³

This confirms that the volume of the prism is indeed 12 in³, and that one of the dimensions (height) is a fraction.

Now, let's try another set of dimensions:
l = 1 in
w = 12 in
h = 1/12 in

Again, plugging these values into the formula for the volume, we get:
V = lwh
V = 1 in × 12 in × 1/12 in
V = 1 in³

This set of dimensions also satisfies the condition that the volume of the prism is 12 in³, with one of the dimensions (height) being a fraction.

Therefore, the possible dimensions of the prism are:
- Length = 2 in, width = 3 in, height = 2/3 in
- Length = 1 in, width = 12 in, height = 1/12 in.

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With regard to the similar triangles,

In ΔACD

BE ∥ CD

In ΔACD and ΔABE

BE ∥ CD

∠ACD =∠ABE           (corresponding angles)

∠ADC = ∠AEB          (corresponding angles)

∠A = ∠A                    (common angle)

ACD ∼ ΔABE

So,  The corresponding sides are in proportion.

Now, find ED

AB/BC = AE/ED
ED = AE (BC/AB)
ED = 26(5/20)
ED = 6.5cm


For BE

AB/AC = BE/CD
BE = CD (AB/AC)
BE = 18 (20/25)
BE  = 14.4cm

Now, find BE

Therefore, the length of ED is 6.5cm and the length of BE is 14.4 cm.

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To find the arc length between each car on the Ferris Wheel, we need to first find the measure of the central angle formed by each car.

The Ferris Wheel has a diameter of 150 ft, which means its radius is half that of 75 ft. We can use the formula s = rθ, where s is the arc length, r is the radius, and θ is the central angle in radians.

Since we have 25 cars on the Ferris Wheel, we can divide the circle into 25 equal parts, each representing the central angle formed by each car.

The total central angle of the circle is 2π radians (or 360 degrees), so each central angle formed by each car is:

(2π radians) / 25 = 0.2513 radians (rounded to four decimal places)

Now we can use this central angle and the radius of the Ferris Wheel to find the arc length between each car:

s = rθ
s = 75 ft * 0.2513
s = 18.8475 ft (rounded to four decimal places)

Therefore, the approximate arc length between each car on the Ferris Wheel is approximately 18.85 ft.

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To find the length of a'b', we first need to know the scale factor of the dilation. The scale factor is given by the ratio of the corresponding side lengths in the original and diluted figures.

In this case, we are given that the original figure Aabc has been diluted by a factor of √2. So the length of each side in the dilated figure aa'b'c is √2 times the length of the corresponding side in Aabc.

To find the length of a'b, we can use the Pythagorean theorem in the right triangle aa'b'. Since we know that ab is one of the legs of this triangle, we can find its length as follows:

ab = (a'b' / √2) * sin(28°)

We are not given the length of ab or a in the original figure, so we cannot find their exact values. However, we can find the measure of angle A using the Law of Sines in triangle Aab:

sin(A) / ab = sin(62°) / b

where b is the length of side bc in Aabc. Solving for sin(A) and substituting the expression for ab that we found earlier, we get:

sin(A) = (sin(62°) / b) * [(a'b' / √2) * sin(28°)]

Since we know the values of sin(62°) and sin(28°), we can simplify this expression and use a value for b (if it is given in the problem) to find sin(A) and then A.

At point (-43,0), f has a maximum. At point (0,4), f has a minimum. At point (463,0), f has a maximum. At point (0,0), f may have a critical point. none of the given points are critical points of the function f (x, y) = xy + 33 - 48y.

Hi! To analyze the critical points of the function f(x, y) = xy + 33 - 48y, we first need to find the partial derivatives with respect to x and y:

fx = ∂f/∂x = y
fy = ∂f/∂y = x - 48

Now, we can analyze the given points:

1. (-43, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this point is not a critical point.

2. (0, 4)
At this point, fx = 4 and fy = 0. Again, both partial derivatives are not equal to 0, so this is not a critical point.

3. (463, 0)
At this point, fx = 0 and fy = 463 - 48 = 415. Since both partial derivatives are not equal to 0, this is not a critical point.

4. (0, 0)
At this point, fx = 0 and fy = -48. Since both partial derivatives are not equal to 0, this is not a critical point.

5. (0, -4)
At this point, fx = -4 and fy = -48. Again, both partial derivatives are not equal to 0, so this is not a critical point.

In summary, none of the given points are critical points of the function f(x, y) = xy + 33 - 48y.

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To find the maximums and minimums of the function f(x,y)=x^2+y^2+xy in the region {(x,y): x^2+y^2<=1}, we need to use the method of Lagrange multipliers.

First, we need to find the gradient of the function and set it equal to the gradient of the constraint (which is the equation of the circle x^2+y^2=1).

∇f(x,y) = <2x+y, 2y+x>
∇g(x,y) = <2x, 2y>

So, we have the equations:
2x+y = 2λx
2y+x = 2λy
x^2+y^2 = 1

Simplifying the first two equations, we get:
y = (2λ-2)x
x = (2λ-2)y

Substituting these into the equation of the circle, we get:
x^2+y^2 = 1
(2λ-2)^2 x^2 + (2λ-2)^2 y^2 = 1
(2λ-2)^2 (x^2+y^2) = 1
(2λ-2)^2 = 1/(x^2+y^2)

Solving for λ, we get:
λ = 1/2 or λ = 3/2

If λ = 1/2, then we get x = -y and x^2+y^2=1, which gives us the critical points (-1/√2, 1/√2) and (1/√2, -1/√2). We can plug these into the function to find that f(-1/√2, 1/√2) = f(1/√2, -1/√2) = -1/4.

If λ = 3/2, then we get x = 2y and x^2+y^2=1, which gives us the critical point (2/√5, 1/√5). We can plug this into the function to find that f(2/√5, 1/√5) = 3/5.

Therefore, the local maximum is (2/√5, 1/√5) with a value of 3/5, the local minimum is (-1/√2, 1/√2) and (1/√2, -1/√2) with a value of -1/4, and the absolute maximum is also (2/√5, 1/√5) with a value of 3/5, and the absolute minimum is on the border, which occurs at (0,1) and (0,-1) with a value of 0.

There are no critical points in the interior of the disk (not the border) that are not extremes or saddle points.
(i) Local extrema:
To find the local extrema, we first find the partial derivatives of f(x, y) with respect to x and y:

f_x = 2x + y
f_y = 2y + x

Set both partial derivatives equal to zero to find critical points:

2x + y = 0
2y + x = 0

Solving this system of equations, we find that the only critical point is (0, 0).

(ii) Absolute extrema:
To determine whether the critical point is an absolute maximum, minimum, or saddle point, we must examine the second partial derivatives:

f_xx = 2
f_yy = 2
f_xy = f_yx = 1

Compute the discriminant: D = f_xx * f_yy - (f_xy)^2 = 2 * 2 - 1^2 = 3

Since D > 0 and f_xx > 0, the point (0, 0) is an absolute minimum of the function.

(iii) Critical points and their classification:
The only critical point in the interior of the disk is (0, 0). As determined earlier, this point is an absolute minimum. No saddle points or other extrema are present within the interior of the disk.

To find any extrema on the boundary of the disk (x^2 + y^2 = 1), we use the method of Lagrange multipliers. However, as the boundary is not part of the domain specified in the question, we will not delve into that here.

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The total inventory costs consist of two parts: the cost of storing the shoes and the cost of reordering the shoes. The cost of storing the shoes is given by the formula:

Cost of storing = $6 per shoe pair per year x number of shoe pairs

Since the shoes are stored for a year, this cost is incurred annually. The cost of reordering the shoes is given by the formula:

Cost of reordering = $14 per order + $7 per shoe pair x number of shoe pairs

This cost is incurred each time the store places an order with the manufacturer.

Let x be the number of shoe pairs in each order from the manufacturer. The number of orders needed to sell 1750 shoe pairs each year is given by:

Number of orders = 1750 shoe pairs / x shoe pairs per order

The total inventory costs can be expressed as:

Total cost = Cost of storing + Cost of reordering

Substituting the formulas for the two costs and the expression for the number of orders, we get:

Total cost = $6 per shoe pair per year x 1750 shoe pairs + ($14 per order + $7 per shoe pair x x shoe pairs) x (1750 shoe pairs / x shoe pairs per order)

Simplifying this expression, we get:

Total cost = $10,500 + $14(1750/x) + $7(1750)

Total cost = $10,500 + $24,500/x

Therefore, the function that models the total inventory costs as a function of x is:

Total cost(x) = $10,500 + $24,500/x

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Answer: 8

Step-by-step explanation:

Declan swims faster than Madison, with a speed of 7/2 miles per hour compared to Madison's speed of 2/5 miles per hour.

Declan swims faster than Madison. While Madison swims 2/5 mile in an hour, which means her speed is 2/5 miles per hour, Declan swims 7/10 mile in 1/5 hour, which means his speed is 7/2 miles per hour. This indicates that Declan's speed is greater than Madison's speed.

In fact, we can see that Declan's speed is almost 4 times faster than Madison's speed. This means that Declan can cover a greater distance in the same amount of time compared to Madison.

Therefore, based on the given information, we can confidently say that Declan swims faster than Madison.

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Answer:

21.5

Step-by-step explanation:

First we can solve for c using the pythagoreom theorem. (probably didn't spell that right)

A squared + B squared =  C squared

9 squared + 3 squared = c squared

81+9= c squared

90=c squared

90 square root is (rounded to the nearest tenth) 9.5

c=9.5

Then we can add 9.5+9+3= 21.5

The derivative of the function f(x) = (x³ - 5x)(2x-1) after simplification is 6x⁴ - 10x³ - 10x².

We apply the product rule and simplify to determine the derivative of,

f(x) = (x³ - 5x)(2x-1).

The product rule is used to determine the derivative of the given function f(x),

h(x) = a.b, then after applying product rule,

h'(x) = (a)(d/dx)(b) + (b)(d/dx)(a).

Applying this for function f,

f'(x) = 6x⁴ - 25x² - 10x³ + 15x²

f'(x) = 6x⁴ - 10x³ - 10x².

Therefore, f'(x) = 6x⁴ - 10x³ - 10x² is the derivative of f(x) after simplifying the function.

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Answer: 8

Step-by-step explanation:

CF looks like the radius of the circle.

ED looks like the diameter.

.: ED = 2 x CF = 2 x 4 = 8

.: ED = 8

Answer:

The length of ED, or the diameter, is 8

Step-by-step explanation:

As the other person explained, CF is the radius, as the radius is from the centermost point to the edge.  I also see that ED is the diameter, as the diameter is from edge to edge, going through the centermost point.  Therefore, since the diameter is double the radius, we can solve this with the following equation:

2 * r = d, where r is radius and d is diameter.

2 * 4 = d

8 = d

Answer is original amount paid to the transportation company was $800.

Let's work backwards from the final amount of $400 to find the original amount paid to the transportation company.

First, we know that percentage given is 80% of what was left after wages went towards supplies. So, if $400 was left after wages were paid, then:

0.8(400) = $320 went towards supplies.

Next, we know that 60% of the original amount went towards wages. So, if $320 went towards supplies, then the remaining amount that went towards wages was:

0.4(original amount) = $320

Solving for the original amount:

Original amount = $320 / 0.4 = $800

Therefore, the original amount paid to the transportation company was $800.

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Answer:

  (D)  7/10

Step-by-step explanation:

You want the rate of change of y with respect to x for the relation ...

  (2/5)x -(4/7)y = 3/2

Solving for y, we have ...

  2/5x -3/2 = 4/7y . . . . . . . . . . add 4/7y -3/2

  (7/4)(2/5)x -(7/4)(3/2) = y . . . . multiply by 7/4

  7/10x -21/8 = y . . . . . . . . . . simplify

The rate of change is the coefficient of x: 7/10.

Answer:

1 hour to build a website

5 hours to update websites

Step-by-step explanation:

x is the hours to build a website

y is the hours to update websites

x + y = 6 ------> 25x + 25y = 150

25x + 15y = 100

10y = 50

y = 5

25x + 15 (5) = 100

25x + 75 = 100

25x = 25

x = 1

So, LaShawn needs 1 hour to build a website and 5 hours to update websites to allow him to reach his goals.

The 21st digit in the decimal expansion of 1/7 is 2 and the 5280th digit in the decimal expansion of 5/17 is 5.

a. To find the 21st digit in the decimal expansion of 1/7 we need to find the decimal expansion. The decimal expansion of 1/7 is a repeating decimal

= 1/7 = 0.142857142857142857…

The sequences 142857 repeat indefinitely. To find the 21st digit, we can divide 21 by the length of the repeating sequence,

= 21 / 6 = 3

Therefore, the third digit in the repeating sequence is 2

b.To find the 5280th digit in the decimal expansion of 5/17 we need to find the decimal expansion. The decimal expansion of 5/17 is a repeating decimal is

=  5/17 = 0.2941176470588235294117647…

The repeating sequences are 2941176470588235

The 5280th digit = 5280 / length of the repeating sequence,

5280 / 16 = 0

Therefore, the 5280th digit is the last digit in the repeating sequence, which is 5.

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Answer:

25

Step-by-step explanation:

let x = the amount of money that shelly has.

let y = the amount of money that john has.

if shelly give john 5 dollars, then they both have the same amount of money.

this leads to the equation:

x-5 = y+5

if john give shelly 5 dollars, then shelly has twice as much money as john has.

this leads to the equation:

x+5 = 2(y-5)

solve for x in each equation to get:

x-5 = y+5 leads to:

x = y+10

x+5 = 2(y-5) leads to:

x+5 = 2y-10 which becomes:

x = 2y-15

you have 2 expressions that are equal to x.

they are:

x = y+10

x = 2y-15

you can set these expressions equal to each other to get:

y+10 = 2y-15

subtract y from both sides of this equation and add 15 to both sides of this equation to get:

y = 25

since x = 2y-15, this leads to:

x = 2(25)-15 which becomes:

x = 35

the equation x = y + 10 leads to the same answer of:

y =35

you have:

x = 25

y = 35

The surface area of the given regular pyramid is 84 cm^2.

To find the surface area of a regular pyramid, we need to calculate the area of each face and add them together. A regular pyramid has a base that is a regular polygon, and its lateral faces are triangles that meet at a common vertex. We can use the Pythagorean theorem to find the slant height of the pyramid, which is the height of each lateral face.

Let's assume that the base of the regular pyramid is a square with side length 6 cm, and the slant height is 4 cm.

First, we need to find the area of the base of the pyramid:

Area of the base = (side length)^2

                = 6 cm x 6 cm

                = 36 cm^2

Next, we need to find the area of each triangular lateral face. Since the pyramid is a regular pyramid, all the triangular faces are congruent.

We can find the area of each triangular face using the formula:

Area of a triangle = (1/2) x base x height

The base of each triangular face is equal to the side length of the square base, which is 6 cm. The height of each triangular face is equal to the slant height, which is 4 cm.

Area of each triangular face = (1/2) x 6 cm x 4 cm

                            = 12 cm^2

Since the pyramid has 4 triangular faces, we need to multiply the area of one triangular face by 4 to get the total area of all the triangular faces:

Total area of the triangular faces = 4 x 12 cm^2

                                 = 48 cm^2

Finally, we can find the total surface area of the pyramid by adding the area of the base and the area of the triangular faces:

Total surface area = Area of the base + Total area of the triangular faces

                  = 36 cm^2 + 48 cm^2

                  = 84 cm^2

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a) The mean monthly rainfall amount for City A is 334.17 inches and for City B is 5.83 inches.

b) The MAD for City A is 464.28 inches and for City B is 0.46 inches.

c) The median for City A is 5 inches and for City B is 6 inches.

(a) To find the mean monthly rainfall amount for each city, we need to add up all the rainfall amounts and divide by the number of months:

For City A: (5 + 2.5 + 6 + 2008.5 + 5 + 3) / 6 = 334.17 inches

For City B: (7 + 6 + 5.5 + 6.5 + 5 + 6) / 6 = 5.83 inches

(b) To find the mean absolute deviation (MAD) for each city, we need to find the absolute deviations from the mean for each data point, then calculate the average of those absolute deviations:

For City A:

Mean = 334.17 inches

Absolute deviations from the mean: |5 - 334.17| = 329.17, |2.5 - 334.17| = 331.67, |6 - 334.17| = 328.17, |2008.5 - 334.17| = 1674.33, |5 - 334.17| = 328.17, |3 - 334.17| = 331.17

MAD = (329.17 + 331.67 + 328.17 + 1674.33 + 328.17 + 331.17) / 6 = 464.28 inches

For City B:

Mean = 5.83 inches

Absolute deviations from the mean: |7 - 5.83| = 1.17, |6 - 5.83| = 0.17, |5.5 - 5.83| = 0.33, |6.5 - 5.83| = 0.67, |5 - 5.83| = 0.83, |6 - 5.83| = 0.17

MAD = (1.17 + 0.17 + 0.33 + 0.67 + 0.83 + 0.17) / 6 = 0.46 inches

(c) To find the median for each city, we need to arrange the data points in order and find the middle value:

For City A: {2.5, 3, 5, 5, 6, 2008.5}

Median = 5 inches

For City B: {5, 5.5, 6, 6, 6.5, 7}

Median = 6 inches

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The maximum value of f on the ellipse is approximately 1.779 and the minimum value is approximately -1.779.

To find the maximum and minimum values of f(x, y) = 3x + y on the ellipse x^2 + 4y^2 = 1, we can use the method of Lagrange multipliers.

First, we define the Lagrangian function as L(x, y, λ) = 3x + y - λ(x^2 + 4y^2 - 1). We then find the partial derivatives of L with respect to x, y, and λ and set them equal to zero:

∂L/∂x = 3 - 2λx = 0

∂L/∂y = 1 - 8λy = 0

∂L/∂λ = x^2 + 4y^2 - 1 = 0

Solving these equations simultaneously, we obtain the critical points (±1/3√5, ±1/√20). We can then evaluate f at these critical points to find the maximum and minimum values:

f(1/3√5, 1/√20) ≈ 0.593

f(1/3√5, -1/√20) ≈ -0.593

f(-1/3√5, 1/√20) ≈ 1.779

f(-1/3√5, -1/√20) ≈ -1.779

Intuitively, the Lagrange multiplier method allows us to optimize a function subject to a constraint, which in this case is the ellipse x^2 + 4y^2 = 1.

The critical points of the Lagrangian function are the points where the gradient of the function is parallel to the gradient of the constraint, which correspond to the maximum and minimum values of the function on the ellipse.

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The function f(z) = 1 + 6x + 96x^-1 has one local minimum and one local maximum. This function has a local maximum at x. The function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.

To find the local maximum of the function f(x) = 1 + 6x + 96x^(-1), we first need to find the critical points. We do this by finding the first derivative of the function and setting it equal to zero.

Step 1: Find the derivative of the function
f'(x) = d/dx (1 + 6x + 96x^(-1))
f'(x) = 6 - 96x^(-2)

Step 2: Set the derivative equal to zero and solve for x
6 - 96x^(-2) = 0

Step 3: Solve for x
96x^(-2) = 6
x^(-2) = 6/96
x^(-2) = 1/16
x^2 = 16
x = ±4

Step 4: Determine which of the critical points is a local maximum. To do this, we will examine the second derivative of the function.

f''(x) = d^2/dx^2 (1 + 6x + 96x^(-1))
f''(x) = 192x^(-3)

Now we will evaluate the second derivative at each critical point:

f''(4) = 192(4)^(-3) = 3 > 0, so x = 4 is a local minimum.
f''(-4) = 192(-4)^(-3) = -3 < 0, so x = -4 is a local maximum.

Therefore, the function f(x) = 1 + 6x + 96x^(-1) has a local maximum at x = -4.

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The value of the angle of the arc FE⏜ is calculated as: 20°

The angle of an arc is identified by its two endpoints. The measure of an arc angle is found by dividing the arc length by the circle's circumference, then multiplying by 360 degrees. Formulas for calculating arcs and angles vary based on where they are in reference to the circle.

Now, from the given image, we see that:

FGC⏜ = 220°

∠B = 30°

∠OEC = ∠OCE = 30°

CDE⏜ = 220°

Thus:

FE⏜ = 360° - 220° - 120°

FE⏜ = 20°

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