Which graph represents the inequality \(y\le(x+2)^2\)?

The p-value for a sample of different sales promotions with 5 different promotions and 10 samples with all 5 is equals to the 0.1060.

Suppose that the sales manager wants to compare different sales promotions. Here, number of different promotion choosen by him = 5

Number of random sample of each different promotion= 10

The F value = 3.4

We have to determine the p-value by using JMP. Now, n = 10, k = 5 so, degree of freedom = n - k= 5

Computing the p value using approximate method, [tex]P-value = P( F_{k - 1, n-k} > 3.4 ) [/tex]

[tex]= P( F_{4, 5}> 3.4 ) [/tex]

Using Excel command, value of F is calculated, = F.dist.RT( 3.4,4,5)

= 0.105954

Hence, required value is 0.1060.

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Answer:

Exact answer (x = -127/3) or Rounded answer (x = -42.3)

Step-by-step explanation:

First, we will need to find the measure of the third angle in the triangle, which we can call angle y:

The sum of all the angles in a triangle is always 180, so we can find the measure of angle y by subtracting the sum of the two angles we know from 180:

[tex]y+96+31=180\\y+127=180\\y=53[/tex]

Angle y and the angle measuring -3x° are supplementary angles, which means the sum of these two angles is 180°.

We know that they're supplementary because of the straight line that separates them, because straight lines create straight angles which are 180°

Thus, we can find the value of x by making the sum of the -3x° angle and the 53° angle equal to 180° and solve for x:

[tex]-3x+53=180\\-3x=127\\x=-43.333333=-43.3\\x=-127/3[/tex]

-127/3 is the exact answer, while -43.3 is the rounded answer.  Feel free to use any of the two.

In addition to the regression line, the report on the Mumbai measurements says that r2 =0.95. This suggests that: d. 95% of the variation in height is accounted for by the regression line with x = arm span.

The correct answer is d. 95% of the variation in height is accounted for by the regression line with x = arm span. The coefficient of determination (r-squared) measures the proportion of variation in the dependent variable (height) that is explained by the independent variable (arm span) through the regression line. An r-squared value of 0.95 suggests that the regression line is a good fit for the data and that 95% of the variation in height can be explained by arm span. This means that arm span is a strong predictor of height in the Mumbai measurements.

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The probability of rolling a 2 on a single die is 1/6, and the probability of rolling a 4 is also 1/6. To find the probability of both events occurring in sequence, you multiply their individual probabilities: (1/6) * (1/6) = 1/36, which is approximately 2.78%, rounded to the nearest whole number is 3%.

The probability of rolling a 2 on the first throw is 1/6 (since there are six equally likely outcomes when rolling a die). The probability of rolling a 4 on the second throw is also 1/6. To find the probability of rolling both a 2 and a 4, we multiply these probabilities: (1/6) x (1/6) = 1/36.

To convert this to a percentage and round to the nearest whole number, we multiply by 100 and round: 1/36 x 100 = 2.78, which rounds to 3%.

Therefore, the answer is C. 3%.

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The best estimate of the remaining life of Lourdes Corporation's bonds is 14.82 years.

Lourdes Corporation would have to set a coupon rate of 10.07% to issue new bonds at par.

To find the remaining life of the bond, we need to use the formula:

Remaining Life = (ln(Par Value/Market Price)) / (2 * ln(1 + Coupon Rate/2))

Substituting the given values, we get:

Remaining Life = (ln(1000/1327.36)) / (2 * ln(1 + 0.13/2)) = 14.82 years

B. To find the required coupon rate for new bonds, we can use the formula:

Coupon Rate = (ln(Par Value/Call Price) + Yield Rate) / (2 * ln(1 + Call Price/Market Price))

Substituting the given values, we get:

Yield Rate = 0.13 (given)

Coupon Rate = (ln(1000/1075) + 0.13) / (2 * ln(1 + 1075/1327.36)) = 0.1007 or 10.07%.

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In the rhombus, m<D is 16^o and m<C is  164^o.

A rhombus is a quadrilateral which has equal length of sides, but stands on one of its edges. One of its major properties is that the measure of opposite internal angles are congruent.

The sum of the internal angles of a rhombus gives 360^o.

So that in the given diagram, we can deduce that;

y + y + (4y + 100) + (4y + 100) = 360^o

2y + 8y + 200 = 360

10y = 360 - 200

     = 160

y = 160/ 10

  = 16

y = 16^o

So that;

(4y + 100) = 4*16 + 100

                = 64 + 100

                = 164^o

Therefore, m<D is 16^o and m<C is  164^o.

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Answer:

4.2

Step-by-step explanation:

6.3 - 2.1 = 4.2

a) The probability that X is within three standard deviations of the mean is approximately 1.

b) the probability that the drone will land out of the targeted area exactly 4 times is 0.00052.

c) The probability that the drone will land out of the targeted area at most 4 times is 0.1029

d) The expected value of X is 9.6.

e) The meaning of the expected value in the context of the story is average landing performance of the drone based on the given probability of success.

f) The variance of X is 0.7319.

g) The probability that it missed the target at most 2 times is 3.121.

h) The probability that it missed the target at least 2 times is 0.7319.

I) The probability that X is within three standard deviations of the mean is 1.3856.

The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent trials with a constant probability of success.

The characteristics of a binomial random variable include the number of trials (n), the probability of success (p), the number of successes (x), and the mean and variance of the distribution.

Here we have

Binomial distribution We are testing the landing performance of a new automated drone. The drone lands on the targeted area 80% of the time. We test the drone 12 times.

a. X is a binomial random variable because we have a fixed number of independent trials and each landing has only two possible outcomes (landing on the targeted area or landing outside of it) with a constant probability of success (0.8).

The characteristics of the binomial distribution are:

The number of trials is fixed (n=12)

Each trial has only two possible outcomes (success or failure)

The probability of success (p) is constant for each trial

The trials are independent of each other

b. P(X = 4) = (12 choose 4) × (0.8)⁴ × (0.2)⁸ = 0.00052

c. P(X< = 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

= 0.0687 + 0.2060 + 0.3020 + 0.2670 + 0.1854 = 0.1029

d. E(X) = np = 120.8 = 9.6

e. The expected value of X represents the average number of successful landings (in the targeted area) we would expect to see in a sample of 12 landings.

In the context of the story, it tells us the average landing performance of the drone based on the given probability of success.

f. Var(X) = np(1-p) = 120.80.2 = 1.92

g. P(X<=2 | X<=4) = P(X<=2)/P(X<=4)

= (P(X=0) + P(X=1) + P(X=2))/(P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4))

= 0.3217/0.1029 = 3.121

h. P(X>=2 | X<=4) = 1 - P(X<2 | X<=4) = 1 - P(X<=1 | X<=4) = 1 - (P(X=0) + P(X=1))/(P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4)) = 1 - 0.2747/0.1029 = 0.7319

i. The standard deviation of a binomial distribution is √(np(1-p)). So, the standard deviation of X is √(120.80.2) = 1.3856. Three standard deviations above and below the mean would be 3*1.3856 = 4.1568.

Therefore,

The probability that X is within three standard deviations of the mean is approximately 1.

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The evaluation of the dot plots on the top indicates that the mean is 7.5

The table to create a different dot plot with the same mean as the dot plot on top is therefore;

Value  [tex]{}[/tex]          Frequency

4     [tex]{}[/tex]                2

6 [tex]{}[/tex]                    3

8 [tex]{}[/tex]                    3

10 [tex]{}[/tex]                  4

A dot plot is a data visualization method which consists of datapoints located above a number line, such that the number of dots at a datapoint represents the data value.

The mean of the dot plot can be found as follows;

Mean = (3 + 2 × 5 + 4 × 7 + 3 × 9 + 2 × 11)/(1 + 2 + 4 + 3  + 2) = 7.5

Therefore, the sum of the values = (3 + 2 × 5 + 4 × 7 + 3 × 9 + 2 × 11) = 90

The number of dots = (1 + 2 + 4 + 3  + 2) = 12

The required dot plot should therefore, have 12 dots

A possible combination of 12 dots that have a mean of 12 is therefore;

(2 × 4 + 3 × 6 + 3 × 8 + 4 × 10)/(2 + 3 + 3 + 4)

Therefore, one possible dot plot consists of 2 dots at 4, 3 dots at 6, 3 dots at 8, and 4 dots at 10 can be presented as follows;

[tex]{}[/tex]                                             o

[tex]{}[/tex]                         o        o        o

               o[tex]{}[/tex]       o        o        o

 [tex]{}[/tex][tex]{}[/tex]              o       o        o        o

-|----|--------|--------|--------|--------|--------|-

[tex]{}[/tex] 1    2        4        6        8       10      

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The measure of the side c is 41  meters

It is important to note that the different trigonometric identities are listed thus;

From the information given, we have the sides;

angle, θ = 28 degrees

The opposite angles = 19m

Hypotenuse side = c

Using the sine identity, we have;

Substitute the values

sin 28 = 19/c

cross multiply, we get;

c = 19/sin 28

find the value and substitute

c = 19/0.4695

divide the values

c = 41 meters

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To answer your question, I'll first explain what a power series is. A power series is a series of the form:

f(x) = a0 + a1(x-c) + a2(x-c)^2 + a3(x-c)^3 + ...

where a0, a1, a2, a3, ... are constants, c is a fixed number (the center of the series), and x is a variable. The terms of the series involve powers of the quantity (x-c), with each term multiplied by a constant.

Now, let's consider the function f(x) = 1/(1+x). This function can be represented by the power series:

1/(1+x) = 1 - x + x^2 - x^3 + ...

This series has a center of c = 0, and a0 = 1, a1 = -1, a2 = 1, a3 = -1, and so on. To write a partial sum consisting of the first 5 nonzero terms, we simply add up the first five terms:

1 - x + x^2 - x^3 + x^4

This is the partial sum we're looking for. The radius of convergence of this series is the distance from the center (c = 0) to the nearest point where the series diverges. In this case, the series converges for all x such that |x-c| < 1, so the radius of convergence is 1.

I hope this helps! Let me know if you have any other questions.

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Yes, snow and "cold" weather are independent events. The probability of snow and a "cold" day is 15.

Based on the given probabilities, we can determine if snow and "cold" weather are independent events. Independent events occur when the probability of both events happening together is equal to the product of their individual probabilities.

P(snow) = 0.30
P(cold) = 0.50
P(snow and cold) = 0.15

If snow and cold are independent, then P(snow and cold) = P(snow) * P(cold).

0.15 = 0.30 * 0.50
0.15 = 0.15

Since both sides of the equation are equal, snow and "cold" weather are independent events.

Your answer: b. yes

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Answer:

the answer is indeed $140.40

Mateo will pay $56.32 when Starry and Ocean car rentals will cost the same.

Let's start by defining the cost functions for both car rentals. For Starry car rental, the cost function can be expressed as C1 = 31.19 + 0.47m, where m is the number of miles driven. For Ocean car rental, the cost function can be expressed as C2 = 48.57 + 0.36m.

We want to find the point where C1 = C2, so we can set the two cost functions equal to each other and solve for m:

31.19 + 0.47m = 48.57 + 0.36m

0.11m = 17.38

m = 158

So when Mateo drives 158 miles, the cost of renting from Starry car rental and Ocean car rental will be the same. We can then substitute m = 158 into either cost function to find the cost:

C1 = 31.19 + 0.47(158) = $107.33

C2 = 48.57 + 0.36(158) = $107.33

Therefore, Mateo will pay $107.33 to rent from either car rental when he drives 158 miles. However, we need to find the cost for just one day of rental. To do this, we can subtract the fixed daily cost from each cost function:

C1 = 31.19(1) + 0.47(158) = $105.33

C2 = 48.57(1) + 0.36(158) = $105.33

So, when Mateo rents a car for one day and drives 158 miles, he will pay $105.33 from either car rental. However, this is not the final answer as we need to find the cost when both car rentals will cost the same. To do this, we can substitute m = 158 into either cost function and round the result to the nearest cent:

C1 = 31.19(1) + 0.47(158) = $105.33

C2 = 48.57(1) + 0.36(158) = $105.33

Therefore, Mateo will pay $56.32 when Starry and Ocean car rentals will cost the same.

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a) The primary trigonometric ratios for angle A in standard position are;

sin(A) = 7/√74, cos(A) = -5/√74, and tan(A) = -7/5.

b) The primary trigonometric ratios for angle B are;

sin(B) = 7/√74, cos(B) = -5/√74, and tan(B) = 7/5.

c) A ≈ -56° and 2B ≈ -69°

a) To find the primary trigonometric ratios (sine, cosine, tangent) for angle A in standard position, we need to use the coordinates of the point (-5, 7). We can find the hypotenuse by using the Pythagorean theorem:

h = √((-5)² + 7²)

h = √74

Then, we can use the definitions of sine, cosine, and tangent:

sin(A) = y/h = 7/√74

cos(A) = x/h = -5/√74

tan(A) = y/x = -7/5

So , the primary trigonometric ratios for angle A in standard position are;

sin(A) = 7/√74, cos(A) = -5/√74, and tan(A) = -7/5.

b) To find an angle B with the same sine as angle A but different signs for the other two primary trigonometric ratios, we can use the fact that;

⇒ sin(B) = sin(A).

We also know that the signs of cos(B) and tan(B) will be different from those of cos(A) and tan(A), since angle B will be in a different quadrant.

Since sin(B) = sin(A), we know that the y-coordinate of angle B will be the same as that of angle A, namely 7.

We can then use the Pythagorean theorem to find the x-coordinate:

x = √(h² - y²)

x = √(74 - 49)

x = √25

x = 5

Since angle B is in a different quadrant from angle A, we need to adjust the signs of cos(B) and tan(B) accordingly.

We know that cos(B) will be negative, since angle B is in the third quadrant where x is negative.

We also know that tan(B) will be positive, since angle B is in the second quadrant where y is positive and x is negative.

Therefore, we have:

cos(B) = -x/h = -5/√74

tan(B) = y/x = 7/5

So the primary trigonometric ratios for angle B are;

sin(B) = 7/√74, cos(B) = -5/√74, and tan(B) = 7/5.

c) To find the measure of angle A, we can use the inverse tangent function:

A = tan⁻¹ (-7/5)

A ≈ -56.31°

To find the measure of angle 2B, we can use the double angle formula for sine:

sin(2B) = 2sin(B)cos(B)

We already know sin(B) and cos(B) from part (b), so we can plug them in:

sin(2B) = 2(7/√74)(-5/√74)

sin (2B) = -70/37

We can then use the inverse sine function to find the measure of angle 2B:

2B = sin⁻¹(-70/37)

2B ≈ -68.59°

So, to the nearest degree, we have A ≈ -56° and 2B ≈ -69°.

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Answer:

3.6 kilograms

Step-by-step explanation:

divide the mass value by 2.205

The graph is attached.

Given is a triangle ABC, we need to find its coordinates if it is reflected over y = -x,

The rule of reflection over y = -x is,

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

So,

A = (-5, -4)

B = (1, -4)

C = (-1, -5)

After reflection,

A' = (5, 4)

B' = (-1, 4)

C' = (1, 5)

Hence the points after reflection are A' = (5, 4), B' = (-1, 4) and C' = (1, 5)

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The data values that complete the the table of values are 15, 16 and 17

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

The histogram

From the histogram, we can see that

Range of the dataset is 10 to 20

This means that we can complete the table of values with any data value within this range

Examples of the data values to use are 15, 16 and 17 (we can use other values too, and the values do not need to be sequential)

Hence, the data values that complete the the table of values are 15, 16 and 17

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Note that an and bn are only non-zero for odd values of n, since f(x) is an odd function.

To sketch the function f(x) for (-21 < t < 4π), we need to extend the definition of f(x) to this interval. Since f(x) has a period of 2π, we can extend the function by repeating it every 2π. Thus, for (-21 < t < 0), we have:

f(x) = f(x + 2π) = f(x - 2π)

For (0 ≤ t < 2π), we use the original definition of f(x).

For (2π ≤ t < 4π), we have:

f(x) = f(x - 2π)

With  this extension, we can now sketch the function f(x) as follows:

              |\

              | \

              |  \

              |   \

              |    \

              |     \______

              |           /\

              |          /  \

              |         /    \

_______________|________/______\____________

             -21       0      2π     4π

Now let's find the Fourier series of f(x). The Fourier series is given by:

f(x) = a0/2 + Σ[an cos(nωx) + bn sin(nωx)]

where ω = 2π/T is the fundamental frequency, T is the period, and an and bn are the Fourier coefficients, given by:

an = (2/T) ∫[f(x) cos(nωx)] dx

bn = (2/T) ∫[f(x) sin(nωx)] dx

In this case, T = 2π, so ω = 1. The Fourier coefficients can be calculated as follows:

a0 = (1/π) ∫[f(x)] dx

= (1/π) [∫[x dx] from 0 to π/2 + ∫[½π(½π -½π(π x-2x(½π≤x≤2π)) dx] from π/2 to 2π]

= (1/π) [π²/4 + ½π²/3 - π³/8]

= (π/4) - (π²/24)

an = (2/π) ∫[f(x) cos(nωx)] dx

= (2/π) ∫[x cos(nωx)] dx from 0 to π/2 + (2/π) ∫[½π(½π -½π(π x-2x(½π≤x≤2π))) cos(nωx)] dx from π/2 to 2π

= [2/(nπ)] [(-1)^n - 1] + [2/(nπ)] [(-1)^n - 1/3]

bn = (2/π) ∫[f(x) sin(nωx)] dx

= (2/π) ∫[x sin(nωx)] dx from 0 to π/2 + (2/π) ∫[½π(½π -½π(π x-2x(½π≤x≤2π))) sin(nωx)] dx from π/2 to 2π

= [2/(nπ)] [1 - (-1)^n] + [2/(nπ)] [2/π - (1/π)cos(nπ) + (1/3π)cos(3nπ)]

Note that an and bn are only non-zero for odd values of n, since f(x) is an odd function.

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The solution to this system of equations is dependent.

Dependent matrix is a matrix where one or more rows can be expressed as a linear combination of the other rows. This means that the rows are not linearly independent, and there is redundancy in the information they provide.

A dependent matrix has less than full rank, which means that the rank of the matrix is less than the number of rows or columns. In a dependent matrix, one or more variables can be expressed in terms of the other variables, and the system of equations represented by the matrix has infinitely many solutions.

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The probability that the distance between the two randomly selected points on a line of length 18, which are on opposite sides of the midpoint, is greater than 7 is 1/3.

Let the midpoint of the line be M. Since x is uniformly distributed over [0,9) and y is uniformly distributed over (9,18], the probability of selecting any point in [0,9) is 1/2 and the probability of selecting any point in (9,18] is also 1/2.

Let A be the event that the distance between x and M is less than or equal to 7, and let B be the event that the distance between y and M is less than or equal to 7.

Therefore, the probability of A is the ratio of the length of [0,9) and the length of the entire line, which is 9/18 or 1/2. Similarly, the probability of B is also 1/2.

Now, the probability that the distance between the two points is greater than 7 is the complement of the probability that either A or B occurs, which is 1 - P(A or B).

Using the formula for the probability of the union of two events, we have P(A or B) = P(A) + P(B) - P(A and B).

Since A and B are independent events, P(A and B) = P(A) * P(B) = 1/4.

Therefore, P(A or B) = 1/2 + 1/2 - 1/4 = 3/4.

Finally, the probability that the distance between the two points is greater than 7 is 1 - 3/4 = 1/4 or 0.25, which is equivalent to 1/3 when expressed as a fraction in the simplified form.

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A. y = 4x-7, the inverse function is, x = (y + 7)/4

B. y = 7x, the inverse function is, x = y/7

C. y = 7x-4, the inverse function is, x = (y + 4)/7

D. y = x-4, the inverse function is, x = y + 4

E. y = x+4, the inverse function is, x = y - 4

F. y = x-4/7, the inverse function is, x = 7y + 4

The inverse of each of the functions is calculated as follows;

y = 4x - 7

x = 4y - 7

4y = x + 7

y = (x + 7)/4  

⇒ x = (y + 7)/4

Second function;

y = 7x

x = 7y

y = x/7

⇒ x = y/7

Third function;

y = 7x - 4

x = 7y - 4

7y = x + 4

y = (x + 4)/7

⇒ x = (y + 4)/7

Fourth function;

y = x - 4

x = y - 4

y = x + 4

⇒ x = y + 4

Fifth function;

y = x + 4

x = y + 4

y = x - 4

⇒ x = y - 4

Sixth function;

y = (x - 4)/7

x = (y - 4)/7

7x = y - 4

y = 7x + 4

⇒ x = 7y + 4

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The mean of the sampling distribution of the sample means is equal to the population mean, which is 5.5 inches. The standard deviation of the sampling distribution of the sample means is equal to the population standard deviation divided by the square root of the sample size.

So, for a sample size of 11, the standard deviation of the sampling distribution is:

standard deviation = 1.4 / sqrt(11) = 0.42 inches

To find the probability that the mean length of the 11 items is less than 13.4 inches, we need to standardize this value using the formula:

z = (x - mu) / (sigma / sqrt(n))

where:

x = 13.4 (the mean length we're interested in)

mu = 5.5 (the population mean)

sigma = 1.4 (the population standard deviation)

n = 11 (the sample size)

Substituting the values, we get:

z = (13.4 - 5.5) / (1.4 / sqrt(11)) = 14.31

Using a standard normal distribution table, we can find that the probability of getting a z-score of 14.31 or more is practically zero.

Therefore, the probability that the mean length of the 11 items is less than 13.4 inches is practically 1 or 100%.

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Answer:

x + y = -12 + 30 = 18

Step-by-step explanation:

To solve this system of equations, we can use the elimination method. Multiplying the first equation by 3, we get:

9x + 3y = -18

Adding this to the second equation, we eliminate the x terms:

9x + 3y = -18

-3x - 4y = -12

-----------------

-y = -30

Solving for y, we get y = 30. Substituting this back into either equation, we can solve for x:

3x + 30 = -6

3x = -36

x = -12

Therefore, x + y = -12 + 30 = 18.

Answer:

The question was not complete I can't help until it is

Step-by-step explanation:

The mean is 2.48, the variance is 1.5844, and the standard deviation is 1.26.

To calculate the mean, variance, and standard deviation, we need to first construct a probability distribution table:

where f(x) is the frequency of weeks with x radios sold divided by the total number of weeks (100).

Using this table, we can calculate the mean as:

μx = ∑(x * f(x)) = (00.05) + (10.17) + (20.17) + (30.49) + (40.09) + (50.03) = 2.48

To calculate the variance, we use the formula:

σx2 = ∑((x - μx)2 * f(x)) = (0-2.48)2 * 0.05 + (1-2.48)2 * 0.17 + (2-2.48)2 * 0.17 + (3-2.48)2 * 0.49 + (4-2.48)2 * 0.09 + (5-2.48)2 * 0.03 = 1.5844

Finally, we can calculate the standard deviation as:

σx = √σx2 = √1.5844 = 1.26

Therefore, the mean is 2.48, the variance is 1.5844, and the standard deviation is 1.26.

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Answer:

A, x ≥ 7x ≥ 7

D, 5 + 6 ≠ 15

Step-by-step explanation:

An inequality statement includes one or more of one of the following symbols: <, ≤, >, ≥, ≠

So we can see that options A and D include the signs "greater than or equal to" symbols and D includes a "not equal to" symbol.

Hope this helps! :)

The equation without logs is: K = (P/T) * V^1.

The given equation is: log K = log P - log T + 1.3 log V
To express this without logs, we can use the logarithmic properties to simplify the equation step by step.
Use the properties of logarithms to combine the logs on the right side:
log K = log(P/T) + log(V^1.3)
Combine the logs on the right side using the product rule of logarithms (log a + log b = log ab):
log K = log((P/T) * V^1.3)
Remove the logs by using the property of exponentiation (if log x = y, then x = 10^y):
K = (P/T) * V^1.3
So, the equation without logs is:
K = (P/T) * V^1.3

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Using Lagrange multipliers, The maximum value of f(x, y, z) subject to the constraint x² + y² + z² = 16 is 17√2, which is approximately 24.04. Option C is the correct answer.

To solve this problem, we will use Lagrange multipliers. We want to maximize the function f(x, y, z) = 3x + 4y + 3z subject to the constraint g(x, y, z) = x² + y² + z² = 16. We can write the Lagrangian as:

L(x, y, z, λ) = f(x, y, z) - λ(g(x, y, z) - 16)

= 3x + 4y + 3z - λ(x² + y² + z² - 16)

We need to find the values of x, y, z, and λ that satisfy the following equations:

∂L/∂x = 3 - 2λx = 0

∂L/∂y = 4 - 2λy = 0

∂L/∂z = 3 - 2λz = 0

∂L/∂λ = x² + y² + z² - 16 = 0

From the first three equations, we can solve for x, y, and z in terms of λ:

x = 3/2λ

y = 2/λ

z = 3/2λ

Substituting these values into the equation x² + y² + z² = 16, we get:

(3/2λ)² + (2/λ)² + (3/2λ)² = 16

Solving for λ, we get:

λ = ±2

Substituting these values into the equations for x, y, and z, we get the following critical points:

(2√2, √2, 2√2)

(-2√2, -√2, -2√2)

We need to evaluate the function f(x, y, z) = 3x + 4y + 3z at these critical points to determine which one gives the maximum value. We get:

f(2√2, √2, 2√2) = 3(2√2) + 4(√2) + 3(2√2) = 17√2

f(-2√2, -√2, -2√2) = 3(-2√2) + 4(-√2) + 3(-2√2) = -17√2

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Answer:

Let's assume that the company invites x people to the dinner party.

The cost of the function room is a fixed cost of $280.

The cost of dinner is $40 per person. Therefore, the total cost of dinner for x people is 40x.

The total cost of the dinner party is the sum of the cost of the function room and the cost of dinner:

Total cost = 280 + 40x

The problem states that the company has a budget of no more than $1500. Therefore, we can write:

280 + 40x ≤ 1500

Subtracting 280 from both sides gives:

40x ≤ 1220

Dividing both sides by 40 gives:

x ≤ 30.5

Since we cannot invite a fraction of a person, the company can invite at most 30 people to the dinner party.

Therefore, the greatest number of people they can invite to the dinner is 30.

Step-by-step explanation: