You frop a ball off a 50-foot roof to see how long it will bounce. Wach bounce looses 10% of the height of its previous bounce. How high will the 8th bounce be in feet?

The probabilities are given as follows:

a. P(number greater than 10) = 1/6.

b. P(number less than 5) = 1/3.

c. The solid is fair, as each side of the dice has the same probability of coming up.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The total number of outcomes for this problem is given as follows:

12.

2 of the numbers are greater than 10, which are 11 and 12, hence the probability is given as follows:

p = 2/12

p = 1/6.

4 of the numbers, which are 1, 2, 3 and 4, are less than 5, hence the probability is given as follows:

p = 4/12

p = 1/3.

More can be learned about probability at https://brainly.com/question/24756209

#SPJ1

Answer:

Step-by-step explanation:

Key March Highlights: Travel spending totaled $93 billion in February—5% above 2019 levels and 9% above 2022 levels. Leisure travel demand does not appear to be abating with America’s excitement to travel at record highs and more than half (55%  Data Question 2 The following table shows the number of visitors to a park from January to April: Month January

Pierre's question is a common philosophical and scientific question about the nature of prediction and probability.

In the 18th century, there was not a comprehensive understanding of the scientific laws that govern the natural world as we have today. Therefore, the most reasonable response to Pierre's question would be that it depends on the probability model used. The probability of the sun rising tomorrow would be based on various factors such as astronomical observations, scientific knowledge of celestial mechanics, and weather patterns.

While we cannot predict the future with absolute certainty, we can use available data and knowledge to make informed predictions about the likelihood of the sun rising tomorrow.

You can learn more about probability at: brainly.com/question/23704607

#SPJ11

The probability that the randomly selected points is in the bullseye is 1/4

Concentric circles are circles with the same or common center.

To calculate the probability that the randomly selected points is in the bullseye, we use the formula below

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Learn  more about concentric circles here: https://brainly.com/question/23838979

#SPJ1

The sample space refers to option (c) the set of all possible experimental outcomes. In probability theory and statistics, a sample space represents all possible outcomes of an experiment or a random event.

The correct answer is c. The sample space refers to the set of all possible experimental outcomes. This includes every possible outcome that could occur in an experiment, whether or not it actually occurs. For example, if you flip a coin, the sample space would be {heads, tails}. This encompasses every possible outcome of the experiment. It provides a foundation for calculating probabilities and understanding the range of results that may occur in a given situation. Sample spaces can vary in size and complexity, depending on the nature of the experiment or event being studied. Understanding the sample space is crucial for making accurate predictions and informed decisions based on data.

Option a is also correct to some extent, as any particular experimental outcome can be considered a part of the sample space. However, it is not a complete definition of the sample space as it only focuses on one outcome and not all possible outcomes.

Option b is incorrect, as the sample space is not limited to just one particular experimental outcome. It is the set of all possible outcomes.

Option d is also incorrect as the sample space has nothing to do with the sample size or the number of participants in the experiment. It is solely based on the set of all possible outcomes of the experiment.

In conclusion, the sample space is the set of all possible experimental outcomes, including both successful and unsuccessful outcomes. It is an important concept in probability theory and is used to calculate the probability of specific events occurring in an experiment.

To learn more about probability, click here:

brainly.com/question/30034780

#SPJ11

The value of k, for k > 0, such that the slope of the line tangent to the graph off at x = 0 is -2.

We are given a family of functions f(x) = x² - 2x + k, where k is a constant. This family of functions includes all the possible quadratic functions of the form x² - 2x + k. To find the value of k, we need to use the given condition that the slope of the tangent line to the graph of the function at x = 0 equals 6.

To find the slope of the tangent line at x = 0, we need to take the derivative of the function f(x) and evaluate it at x = 0. Taking the derivative of f(x), we get:

f'(x) = 2x - 2

Evaluating f'(x) at x = 0, we get:

f'(0) = 2(0) - 2 = -2

This gives us the slope of the tangent line to the graph of the function at x = 0, which is -2.

Therefore, the answer to the problem is that there is -2 of k, for k > 0, such that the slope of the line tangent to the graph of the function at x = 0 equals 6.

To know more about function here

https://brainly.com/question/28193995

#SPJ1

Complete Question;

Consider the family of functions f(x) = where k is a constant. x^2 - 2x +k

Find the value of k, for k > 0, such that the slope of the line tangent to the graph off at x = 0

The statement "The normal force is equal to the perpendicular component of the object's weight, which decreases as the angle of inclination increases" is true.

As the angle of inclination increases, the object's weight can be divided into two components: one perpendicular to the inclined surface (the normal force) and one parallel to it. As the angle increases, the perpendicular component (normal force) decreases, while the parallel component increases.

So to directly answer your question, the normal force is never equal to the weight of the object on an inclined plane (unless you count the limiting case of level ground). It is equal to the weight of the object times the cosine of the angle the inclined plane makes with the horizontal.

Learn more about the inclination: https://brainly.com/question/4598225

#SPJ11

The vertex of an angle would have the same coordinates after a rotation if it is rotated at angle of 360 degrees.

In Mathematics and Geometry, a rotation refers to a type of transformation which moves every point of the object through a number of degrees around a given point, which can either be clockwise or counterclockwise (anticlockwise) direction.

Generally speaking, when a point (x, y) is rotated about the center or origin (0, 0) in a counterclockwise (anticlockwise) direction by an angle θ, the coordinates of the new point (x′, y′) formed include the following:

x′ = xcos(θ) − ysin(θ)

y’ = xsin(θ) + ycos(θ).

(x′, y′) → (x, y) ⇒ (360 degrees rotation).

Read more on angle of rotation here: brainly.com/question/12428507

#SPJ1

Answer:

15.8

Step-by-step explanation:

mean is the average in math so 9 + 14 + 11 + 31 + 14 =79 then you have to count how many numbers there is and minus it from the total which there is 5 numbers so 79 divided by 5 = 15.8

Answer:

To complete the construction described:

Place the midpoint I of segment [AB]. Draw the circle C of diameter [AB]. Draw the perpendicular bisector of segment [AB]. Label the point where it intersects circle C as E and F. Draw half-lines [AE) and [BE). Draw an arc with center A and radius [AB] that passes through point B. Label the points where the arc intersects half-line [AE) as H and J. Draw an arc with center B and radius [BA] that passes through point A. Label the points where the arc intersects half-line [BE) as G and K. Draw the quarter circle with center E and radius [EG] that is bounded by points G and H. This completes the construction.

The final figure should consist of circle C, perpendicular bisector EF, half-lines [AE) and [BE), arcs passing through points B and A, and the quarter circle with center E, radius [EG], and bounded by points G and H.

Step-by-step explanation:

We have shown that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

To show that cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ, and c>0, we will use the CDF method:

1. Define the transformation: Let Y = cX be a function of the random variable X, where c > 0.

2. Find the CDF of Y: We want to find P(Y ≤ y), which is equal to P(cX ≤ y) or P(X ≤ y/c).

3. Express CDF of Y in terms of X: Since P(X ≤ y/c) is the CDF of X at y/c, we have FY(y) = FX(y/c).

4. Find the PDF of X: The exponential distribution has the PDF fX(x) = λ * exp(-λx) for x ≥ 0.

5. Differentiate the CDF of Y to find its PDF: To find fY(y), we differentiate FY(y) with respect to y. Using the chain rule, we have:

fY(y) = d(FX(y/c))/dy = fX(y/c) * (1/c)

6. Substitute the PDF of X: Now, we replace fX(y/c) with its exponential form λ * exp(-λ(y/c)):

fY(y) = (λ * exp(-λ(y/c))) * (1/c)

7. Simplify the expression: fY(y) = (λ/c) * exp(-λ(y/c))

This is the PDF of an exponential distribution with parameter λ/c. Therefore, cX is exponential with parameter λ/c when X is an exponential random variable with parameter λ and c > 0.

Learn more about "variable": https://brainly.com/question/82796

#SPJ11

Part A: A line graph is the leading graphical representation to show the given information. The line chart is an fitting show since it makes a difference in visualizing the drift of the team's execution over time. It too highlights any outliers and makes a difference in recognizing designs within the information.

Part B: To make a line graph for the given information, take after the steps underneath:

The coming about line chart would appear the team's execution over the course of the primary 13 recreations, highlighting any patterns, crests, or plunges in their execution.

Learn more about graphical representation from

https://brainly.com/question/30154311

#SPJ1

The radius of convergence R is 1/7 and the interval of convergence I is: I = (-1/7, 5/7)

To find the radius of convergence R, we can apply the ratio test:

[tex]lim_n→∞ |(7x-4)(n+1)/7(n+1)| = lim_n→∞ |7x-4|/7 = |7x-4|[/tex]

The series converges when the limit is less than 1, so we have: |7x - 4| < 1

Solving for x,

we get: -1/7 < x < 5/7

This means that the series converges for all values of x within the interval (-1/7, 5/7) and diverges for values of x outside that interval. The interval is open on the left endpoint and closed on the right endpoint because the limit at x=-1/7 and x=5/7 needs to be tested separately.

Learn more about radius here:

https://brainly.com/question/811328

#SPJ4

Finally, we can use the fact that 3 is a zero of multiplicity 1 to determine: f(0) = 0 = -81ac.

A polynomial function of degree 7 with -3 as a zero of multiplicity 3, 0 as a zero of multiplicity 3, and 3 as a zero of multiplicity 1 can be written as:

f(x) = [tex]a(x + 3)^3 * b(x)^3 * c(x - 3)[/tex]

where a, b, and c are constants to be determined.

Since -3 is a zero of multiplicity 3, we know that (x + 3) appears in the function three times as a factor, so we can write:

f(x) =[tex]a(x + 3)^3 * g(x)[/tex]

Here g(x) is some function of degree 4 (since we have accounted for 3 of the 7 total factors). Similarly, since 0 is a zero of multiplicity 3, we know that [tex]x^3[/tex] appears in the function three times as a factor, so we can write:

g(x) = [tex]b(x)^3 * h(x)[/tex]

Here h(x) is some function of degree 1 (since we have accounted for 3 of the remaining 4 factors). Finally, we know that 3 is a zero of multiplicity 1, so we can write:

h(x) = c(x - 3)

Putting it all together, we have:

[tex]f(x) = a(x + 3)^3 * g(x)\\= a(x + 3)^3 * b(x)^3 * h(x)\\= a(x + 3)^3 * b(x)^3 * c(x - 3)[/tex]

Substituting h(x) into g(x), we get:

[tex]g(x) = b(x)^3 * h(x)\\= b(x)^3 * c(x - 3)[/tex]

Substituting g(x) into f(x), we get:

[tex]f(x) = a(x + 3)^3 * g(x)\\= a(x + 3)^3 * b(x)^3 * h(x)\\= a(x + 3)^3 * b(x)^3 * c(x - 3)\\= a(x + 3)^3 * b(x)^3 * c(x - 3)\\[/tex]

Expanding the terms, we get:

[tex]f(x) = a(x^3 + 9x^2 + 27x + 27) * b(x^3)^3 * c(x - 3)\\= a(x^3 + 9x^2 + 27x + 27) * b(x^6) * c(x - 3)\\\\= a(x^3 + 9x^2 + 27x + 27) * b(x^6) * c(x) - 3c(x^5)[/tex]

Now, we can use the fact that -3 is a zero of multiplicity 3 to determine the value of a:

[tex]f(-3) = a(-3 + 3)^3 * b(0)^3 * c(-3) = 0[/tex]

= 0

Since [tex](-3 + 3)^3 = 0,[/tex] we can simplify this equation to:

f(-3) = 0 = [tex]b(0)^3 * c(-3)[/tex]

Since 0 is a zero of multiplicity 3, we can also determine the value of b:

f(0) = [tex]a(0 + 3)^3 * b(0)^3 * c(0 - 3) = 0[/tex]

= 27a * 0 * (-3c)

Simplifying, we get:

f(0) = 0 = -81ac

Learn more about polynomial function visit: brainly.com/question/1600696

#SPJ4

The first positive root of the function f(x) = x² - 3 is located between x = 1 and x = 2.

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on values f(a) and f(b) with opposite signs, then there exists at least one root (zero) of the function between a and b.

In this case, we have f(x) = x² - 3. To find the first positive root of the function, we need to look for a positive value of x where f(x) = 0.

We can start by evaluating f(0) and f(2), which are the values of the function at the endpoints of the interval [0, 2]:

f(0) = 0² - 3 = -3

f(2) = 2² - 3 = 1

Since f(0) is negative and f(2) is positive, by the Intermediate Value Theorem, there must be at least one root of the function between x = 0 and x = 2.

To further narrow down the location of the root, we can evaluate f(1), which is the midpoint of the interval [0, 2]:

f(1) = 1² - 3 = -2

Since f(1) is negative, we know that the root is between x = 1 and x = 2.

To summarize, the first positive root of the function f(x) = x² - 3 is located between x = 1 and x = 2.

Learn more about function here,

https://brainly.com/question/16634496

#SPJ11

The probability that 49% or less of these companies outsourced some part of their manufacturing process in the past two to three years is approximately 0.0094.

a) To solve this problem, we need to use the binomial distribution formula:

P(X ≥ 336) = 1 - P(X < 336)

where X is the number of companies that outsourced some part of their manufacturing process.

We know that n = 555, p = 0.54, and q = 1 - p = 0.46.

Using the binomial distribution formula, we get:

P(X < 336) = Σ (nCx) * p^x * q^(n-x) from x = 0 to 335

However, computing this sum directly can be very time-consuming. Instead, we can use the normal approximation to the binomial distribution since n is large and p is not too close to 0 or 1.

Using the normal approximation, we can calculate the mean and standard deviation of the binomial distribution:

μ = np = 555 * 0.54 = 299.7

σ = sqrt(npq) = sqrt(555 * 0.54 * 0.46) ≈ 11.85

Then, we can transform the binomial distribution to a standard normal distribution:

Z = (X - μ) / σ

P(X < 336) ≈ P(Z < (336 - μ) / σ) = P(Z < (336 - 299.7) / 11.85) ≈ P(Z < 3.05)

Using a standard normal distribution table or a calculator, we find that P(Z < 3.05) ≈ 0.9983.

Therefore, P(X ≥ 336) = 1 - P(X < 336) ≈ 1 - 0.9983 = 0.0017.

b) We can use the same approach as in part (a):

P(X ≥ 286) = 1 - P(X < 286)

μ = np = 555 * 0.54 = 299.7

σ = sqrt(npq) = sqrt(555 * 0.54 * 0.46) ≈ 11.85

Z = (X - μ) / σ

P(X < 286) ≈ P(Z < (286 - μ) / σ) = P(Z < (286 - 299.7) / 11.85) ≈ P(Z < -1.15)

Using a standard normal distribution table or a calculator, we find that P(Z < -1.15) ≈ 0.1251.

Therefore, P(X ≥ 286) = 1 - P(X < 286) ≈ 1 - 0.1251 = 0.8749.

c) We want to find P(X ≤ 0.49n) = P(X ≤ 0.49 * 555) = P(X ≤ 271.95).

We can again use the normal approximation to the binomial distribution:

μ = np = 299.7

σ = sqrt(npq) ≈ 11.85

Z = (X - μ) / σ

P(X ≤ 271.95) ≈ P(Z < (271.95 - 299.7) / 11.85) ≈ P(Z < -2.34)

Using a standard normal distribution table or a calculator, we find that P(Z < -2.34) ≈ 0.0094.

Therefore, the probability that 49% or less of these companies outsourced some part of their manufacturing process in the past two to three years is approximately 0.0094.

To learn more about manufacturing visit:

https://brainly.com/question/21892860

#SPJ11

Answer:

Step-by-step explanation:

a. Correlation: This method measures the strength and direction of the relationship between two continuous variables.
b. Linear Regression: This method predicts the value of one continuous variable based on the value of another continuous variable.
c. Independent t-test: This method compares the means of two independent groups to determine if there is a significant difference between them.
d. Dependent t-test: This method compares the means of two related groups (e.g., pre-test and post-test) to determine if there is a significant difference between them.

a. Correlation - This statistical method is used when the research question involves examining the relationship between two continuous variables. For example, "Is there a correlation between hours spent studying and GPA?"

Research question type: "Is there a relationship between variable A and variable B?"
b. Linear Regression - This statistical method is used when the research question involves predicting a continuous dependent variable based on one or more continuous independent variables. For example, "Can we predict income based on years of education and work experience?"

Research question type: "Can we predict variable A based on variable B?"
c. Independent t-test - This statistical method is used when the research question involves comparing the means of two independent groups on a continuous variable. For example, "Is there a difference in salaries between male and female employees?"

Research question type: "Is there a significant difference in variable A between Group 1 and Group 2?"
d. Dependent t-test - This statistical method is used when the research question involves comparing the means of two related groups on a continuous variable. For example, "Is there a significant difference in test scores before and after a study intervention?"

Research question type: "Is there a significant difference in variable A between the pre-test and post-test results?"

To learn more about Linear Regression, click here:

brainly.com/question/29665935

#SPJ11

The format of £ + E.E___+S___. Therefore, the answer is £36.6 + E. E14.284 + S0.

To lea

To estimate the mean amount a typical parent would spend on their child's birthday gift, we can use a confidence interval with the given information. Since the sample size is relatively small (n=13) and the population standard deviation is unknown, we can use a t-distribution with n-1 degrees of freedom.

The formula for a confidence interval for the population mean is:

x ± t*(s/√n)*

where x is the sample mean, s is the sample standard deviation, n is the sample size, and t* is the critical t-value from the t-distribution for a given level of confidence and degrees of freedom.

For a 98% confidence level and 12 degrees of freedom (n-1), the critical t-value is 2.681.

Plugging in the given values, we get:

36.6 ± 2.681*(14.7/√13) ≈ 36.6 ± 14.284

So the 98% confidence interval for the mean amount a typical parent would spend on their child's birthday gift is £22.316 to £50.884, or in the format of £ + E.E___+S___. Therefore, the answer is £36.6 + E. E14.284 + S0.

To learn more about confidence visit:

https://brainly.com/question/22989641

#SPJ11

The sum of the m integers between ai and aj is si - s(i-1) + s(i-1) - s(j-1) = si - sj, which is divisible by m since si, sj have same remainder. So, there exists a consecutive subsequence of the original sequence with a sum divisible by m, namely the integers between ai and aj.

We can prove this using the Pigeonhole Principle.

Consider the sequence of m integers a1, a2, ..., am. Let's compute the prefix sums of this sequence, which we'll denote by s0, s1, s2, ..., sm. That is, we define si = a1 + a2 + ... + ai-1 for i = 1, 2, ..., m, and s0 = 0.

Note that there are m + 1 prefix sums, but only m possible remainders when we divide a sum by m (namely, 0, 1, 2, ..., m-1).

Therefore, by the Pigeonhole Principle, at least two of the prefix sums must have the same remainder when divided by m. Let's say these are si and sj, where i < j.

Then, the sum of the m integers between ai and aj (inclusive) is si - s(i-1) + s(i-1) - s(j-1) = si - sj, which is divisible by m since si and sj have the same remainder when divided by m.

Therefore, there exists a consecutive subsequence of the original sequence with a sum divisible by m, namely the integers between ai and aj.

Know more about integers here:

https://brainly.com/question/929808

#SPJ11

The number of ways there are for her to select the starting line-up is 68,640 ways.

1. Center: There are 4 players who can play this position, so there are 4 choices.
2. Right Forward: Since one player has been selected as Center, there are now 13 players remaining. So, there are 13 choices for this position.
3. Left Forward: After selecting the Center and Right Forward, 12 players remain, resulting in 12 choices for this position.
4. Right Guard: With three players already chosen, there are 11 players left to choose from, giving us 11 choices.
5. Left Guard: Finally, after selecting players for the other four positions, 10 players remain, providing 10 choices for this position.

Now, we can calculate the total number of ways to select the starting line-up using the counting principle by multiplying the number of choices for each position:

4 (Center) × 13 (Right Forward) × 12 (Left Forward) × 11 (Right Guard) × 10 (Left Guard) = 68,640 ways

So, there are 68,640 ways for the coach to select the starting line-up.

Learn more about counting principle here: https://brainly.com/question/29774420

#SPJ11

Answer:

  see attached

Step-by-step explanation:

You want a graph of the line y = -3x +2.

The infinite number of points that satisfy the equation y = -3x +2 will form a line on the coordinate plane. It will cross the y-axis at y = 2, and will have a slope (rise/run) of -3 units for each unit to the right. The attachment shows the graph.

<95141404393>

The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

We have to given that;

To find distance between (2 + i) and (4 +3i).

Now, We can formulate;

Two points are (2, 1) and (4, 3).

We know that;

The distance between two points (x₁ , y₁) and (x₂, y₂) is,

⇒ d = √ (x₂ - x₁)² + (y₂ - y₁)²

Hence, The distance between (2 + i) and (4 +3i) would be,

⇒ d = √(4 - 2)² + (3 - 1)²

⇒ d = √4 + 4

⇒ d = √8

⇒ d = 2√2

Thus, The distance between (2 + i) and (4 +3i) would be,

⇒ d = 2√2

Learn more about the coordinate visit:

https://brainly.com/question/24394007

#SPJ1

By using the Chinese Remainder Theorem separate the statement into two congruences we have x(p-1)(q−1)+1 (mod pq) for all x € Z.

Under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1), the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can uniquely determine the remainder of the division of n by the product of these integers.

It suffices to show pq divides x(x(p-1)(q-1) - 1) for all x e Z. We consider three cases. Consider gcd (x, pq). It has 3 possibilities.

Case 1: If gcd(x, pq) 1. Then applying using Euler's Theorem we have

= x(pq) = 1 (mod pq)

= x(p-1)(−1) = 1 (mod pq)

= x(p-1)(q-1)+1 (mod pq)

and so the result holds if gcd(x, pq) = 1. EX

Case 2: If gcd(x, pq) p. This means x = 0 (mod p). In this case we have

= 0 = x (mod p).

Since gcd(x, pq) = p therefore qx and = 1 (mod q) by Fermat's Little Theorem. This gives us that x(p-1)(q-1)+1 so we have x9-1 x(p−1)(q−1) = 1 (mod q) = x(p-1)(q-1)+1 = x (mod q).

We have shown that x(p-1)(q-1)+1 = x (mod p) and x(p-1)(q-1)+1 = x (mod q). Using the Chinese Remainder Theorem we get x(p-1)(q−1)+1 = x (mod pq).

Case 3: If gcd(x, pq) = q. This case is same as Case 2, with p being replaced by q.

Thus we have extinguished all cases and we have shown x(p-1)(q−1)+1 (mod pq) for all x € Z.

Learn more about Chinese Remainder Theorem:

https://brainly.com/question/30811240

#SPJ4

Complete question:

Let р and q be distinct primes. Show that for all x € Z, we have the congruence x(p-1)(9–1)+1 x (mod pq). (Hint: Use the Chinese Remainder Theorem/Sun Ze's Theorem to separate the statement into two congruences.)

The surface area of the pyramid is 179 sq. m.

The surface area of a given shape is the summation of all the area of each figure that forms its sides called surfaces.

The given pyramid has triangular shaped surfaces, so that;

area of a triangle = 1/2 *base*height

To determine the area of one of the surfaces, we have;

area of the triangular surface = 1/2x base x height

base = 8 m, and slant height of the surface = 11.2 m

So that;

the area of one triangular surface = 1/2*8*11.2

                                                    = 44.8 sq. m.

Thus since the pyramid has 4 equal triangular surfaces, then;

the surface area of the pyramid = 4 x 44.8

                                            = 179.2

The surface area of the pyramid is 179 sq. m.

Learn more about surface area at https://brainly.com/question/31507420

#SPJ1

The improper fraction can be written as:

44/7

An improper fraction is a fraction where the numerator is larger than the denominator.

Here we want to write.

6 + 2/7 as an improper fraction, to do so, we just need to write 6 as a fraction with a denominator of 7 and then add them.

We know that:

6 = 6*(7/7) = 42/7

Then we can write:

6 + 2/7 = 42/7 + 2/7 = 44/7

That is the improper fraction.

Learn more about improper fractions at:

https://brainly.com/question/387381

#SPJ1

Answer:

Based on the histogram displaying the ages of 50 randomly selected users of an online music service, it is more likely that advertising on the service will reach people who are younger than 30, as the frequency (or height) of the bars appears to be higher in the younger age group compared to the 30 and older age group.

Step-by-step explanation:

The KKT pair is given by:

λ = 0, y = x (when x is inside the ball)

λ = 1/2, y = m + (x – m)/2 (when x is outside the ball)

a) The Lagrangian function for the optimization problem is given by:

L(y, λ) = 1/2 ||y – x||² + λ (r² – ||y – m||²)

where λ is the Lagrange multiplier.

The KKT conditions for the problem are:

Stationarity condition: ∇y L(y, λ) = 0

∇y L(y, λ) = y – x – 2λ (y – m) = 0

Primal feasibility condition: ||y – m||² ≤ r²

Dual feasibility condition: λ ≥ 0

Complementary slackness condition: λ (r² – ||y – m||²) = 0

b) To show that the KKT conditions have a unique solution, we can use the second-order sufficiency conditions. The Hessian matrix of the Lagrangian function is given by:

∇²L(y, λ) = I – 2λ I = (1 – 2λ)I

where I is the identity matrix. Since λ ≥ 0, we have 1 – 2λ ≤ 1, which means that the Hessian matrix is positive definite. Therefore, the KKT conditions have a unique solution.

To calculate the KKT pair, we need to solve the stationarity and primal feasibility conditions. From the stationarity condition, we have:

y – x – 2λ (y – m) = 0

y – 2λy = x – 2λm

y = (I – 2λ)⁻¹(x – 2λm)

Substituting this into the primal feasibility condition, we have:

||(I – 2λ)⁻¹(x – 2λm) – m||² ≤ r²

Expanding this expression, we get:

||x – m||² – 4λ (x – m)ᵀ(I – λ(I – 2λ)⁻¹)(x – m) + 4λ² ||(I – 2λ)⁻¹(m – x)||² ≤ r²

Let A = (I – λ(I – 2λ)⁻¹). Then, the above expression can be written as:

||x – m||² – 4λ (x – m)ᵀA(x – m) + 4λ² ||A(m – x)||² ≤ r²

Since λ ≥ 0, we have A = (I – λ(I – 2λ)⁻¹) ≥ 0, which means that A is positive semidefinite. Therefore, the minimum value of the expression on the left-hand side is achieved when λ = 0 or λ = 1/2.

If λ = 0, then we get:

y = x

If λ = 1/2, then we get:

y = m + (x – m)/2

Therefore, the KKT pair is given by:

λ = 0, y = x (when x is inside the ball)

λ = 1/2, y = m + (x – m)/2 (when x is outside the ball)

To learn more about minimum visit:

https://brainly.com/question/21426575

#SPJ11