Object 2: Pinecone 3D shape: Cone Dimensions: radius = 4 inches height = 6.5 inches



find the formula for area, and base area.

There is not enough evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years.

What is the mean and standard deviation?

The standard deviation is a summary measure of the differences of each observation from the mean. If the differences themselves were added up, the positive would exactly balance the negative and so their sum would be zero. Consequently, the squares of the differences are added.

To find the standardized test statistic (Z-score), we need to first calculate the sample mean and standard error of the mean.

Sample mean:

x = (17.8 + 21.1 + 19.7 + 9.5 + 18.5 + 13.2 + 13.6 + 10.7 + 19.1 + 7.5 + 11.3 + 7.6 + 21.8 + 9.2 + 21.6 + 9.8 + 19.3 + 21.9 + 22.6 + 15.5 + 12.4 + 9.5 + 14.9 + 7.7 + 12.9 + 17.6 + 14.1 + 19.4 + 17.1 + 17.3 + 15.4 + 22.5) / 32

x = 15.91875

Standard error of the mean:

SE = σ /√(n)

SE = 6.1 / √(32)

SE = 1.0823

Now we can calculate the Z-score:

Z = (x - μ) / SE

Z = (15.91875 - 13) / 1.0823

Z = 2.5707

we can find that the p-value associated with a Z-score of 2.5707 is approximately 0.0051.

Since the significance level (α) is 0.06, which is larger than the p-value, we fail to reject the null hypothesis.

Therefore, there is not enough evidence to reject the claim that the mean time it takes smokers to quit smoking permanently is 13 years.

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The expression to define the width of the roof garden is 2W

A rectangle is a four-sided shape with opposite sides that are parallel and equal in length. The perimeter of a rectangle is the sum of the lengths of all its sides. If we let the width of the rectangle be "w" and the length be "l," then the perimeter is given by the formula:

Perimeter = 2w + 2l

In this problem, we are told that Darcie has 50 meters of railing. We can use this information to set up an equation to find the length of the rectangle. Let the length of the rectangle be "L." Then we have:

2w + 2L = 50

Simplifying this equation, we get:

w + L = 25

Since the roof garden is a rectangle, we know that the perimeter is given by:

Perimeter = 2w + 2L

Let the width of the garden be "W" and the length be "L." Then the perimeter of the garden is given by:

Perimeter = 2W + 2L

Since we want to know how much more railing Darcie needs, we can set up an equation:

2W + 2L - 50 = X

where X is the amount of additional railing Darcie needs.

We can simplify this equation by substituting 2w + 2L = 50:

2W + (2w + 2L - 50) = X

Simplifying further, we get:

2W - 50 + 2w + 2L = X

Since w + L = 25, we can substitute 2w + 2L = 50:

2W - 50 + 50 = X

Simplifying, we get:

2W = X

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Complete Question:

Darcie has 50m of railing. How much more railing does Darcie need so that she can put the railing all the way around the roof garden. Find the expression to define the width of the roof garden.

a) The probability distribution are P(S₂ = -2) = 1/9, P(S₂ = -1) = 2/9, P(S₂ = 0) = 1/9, P(S₂ = 1) = 2/9, P(S₂ = 2) = 2/9, P(S₂ = 3) = 2/9, P(S₂ = 4) = 1/9

b) The probability that the particle's position S90,000 is less than or equal to 29,500 is approximately 0.0571.

In this problem, we are given a particle's initial position S₀ = 0, and its position Sₙ after n time intervals, where the position is a sum of n independent and identically distributed random displacements X₁, X₂, ..., Xₙ. Each displacement Xᵢ can take on one of three values: -1, 0, or 2, with equal probability 1/3 for each.

(a) To find the probability distribution of S₂ = X₁ + X₂, we can enumerate all possible values of S₂ and compute their probabilities. The possible values of S₂ are -2, -1, 0, 1, 2, 3, 4, and their respective probabilities are:

P(S₂ = -2) = P(X₁ = -1, X₂ = -1) = (1/3)² = 1/9

P(S₂ = -1) = P(X₁ = -1, X₂ = 0) + P(X₁ = 0, X₂ = -1) = 2(1/3)² = 2/9

P(S₂ = 0) = P(X₁ = 0, X₂ = 0) = (1/3)² = 1/9

P(S₂ = 1) = P(X₁ = 2, X₂ = -1) + P(X₁ = -1, X₂ = 2) = 2(1/3)² = 2/9

P(S₂ = 2) = P(X₁ = 2, X₂ = 0) + P(X₁ = 0, X₂ = 2) = 2(1/3)² = 2/9

P(S₂ = 3) = P(X₁ = 2, X₂ = 1) + P(X₁ = 1, X₂ = 2) = 2(1/3)² = 2/9

P(S₂ = 4) = P(X₁ = 2, X₂ = 2) = (1/3)² = 1/9

(b) To compute P(S90,000 ≤ 29,500), we can use the Central Limit Theorem (CLT) to approximate the distribution of S90,000. The CLT states that the sum of a large number of independent and identically distributed random variables tends to follow a normal distribution, regardless of the underlying distribution of the individual variables. For n large enough, we have:

S90,000 ≈ N(μ, σ²), where μ = nE(X) and σ² = nVar(X)

Here, n = 90,000, E(X) = (-1 + 0 + 2)/3 = 1/3, and Var(X) = [(2-1/3)² + (-1-1/3)² + (0-1/3)²]/3 = 10/9. Therefore, we have:

μ = 90,000(1/3) = 30,000

σ² = 90,000(10/9) ≈ 100,000

Now, we can standardize the variable Z = (S90,000 - μ)/σ and use a standard normal table or calculator to find the probability:

P(S90,000 ≤ 29,500) = P(Z ≤ (29,500 - 30,000)/√100,000) ≈ P(Z ≤ -1.58) ≈ 0.0571 (rounded to four decimal places)

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The statement "There is a 60 percent chance of rain and a 40 percent chance of pure sunshine" illustrates the definition of a probability distribution.

A probability distribution is a function that describes the likelihood of different outcomes in a random event or experiment. It assigns probabilities to each possible outcome, where the probabilities add up to 1 (or 100%).

In the given options, the statement "There is a 60 percent chance of rain and a 40 percent chance of pure sunshine" is a clear example of a probability distribution because it assigns probabilities to two possible outcomes - rain and sunshine - with a total probability of 1. Specifically, the statement is saying that there is a 60% chance of rain and a 40% chance of sunshine. This statement describes the likelihood of different outcomes for the weather, making it an example of a probability distribution.

The other two statements do not illustrate a probability distribution because they only provide information about specific events that have already occurred (i.e., "it rained three-quarters of the day yesterday" and "the sun is shining today and is supposed to shine tomorrow") or possible events that may occur in the future without any mention of their likelihood (i.e., "it may snow either today or tomorrow").

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Answer:

An odd number of data values will always have one middle value

B, D

Step-by-step explanation:

123

2

12345

3

1234567

4

ALWAYS

The answer is option 3: at least 22%, but less than 29%.

Let J be the random variable representing Jordan's arrival time, and let T be the random variable representing Taylor's arrival time.

Then, J is uniformly distributed between 8:00AM and 8:50AM, which means that J ~ U(480, 530) (measured in minutes past 12:00AM).

Similarly, T is uniformly distributed between 8:00AM and 8:30AM, which means that T ~ U(480, 510).

We want to find the probability that they meet, which means that Jordan arrives before Taylor leaves (within 10 minutes of Taylor's arrival time). Let's assume that Taylor arrives at time t (measured in minutes past 12:00AM). Then, Jordan needs to arrive between t-10 and t in order to meet Taylor.

The probability of this happening is:

P(J ∈ [t-10, t]) = (t - (t-10)) / (530 - 480) = 10/50 = 0.2

Since Taylor's arrival time is also uniformly distributed, we need to take the average of this probability over all possible values of t between 480 and 510:

P(meeting) = E[P(J ∈ [t-10, t])] for t ∈ [480, 510]

P(meeting) = (1/31) ∫₄₈₀ ⁴⁹⁰ [0.2] dt + (1/31) ∫₄₉₀ ⁵₀₀ [0.2] dt + (1/31) ∫₅₀₀ ⁵₁₀ [0.2] dt

P(meeting) = (1/31) [((0.2)(10)) + ((0.2)(20)) + ((0.2)(30))]

P(meeting) = 0.1935

Therefore, the probability that Jordan and Taylor meet is approximately 0.1935, which is between 16% and 22%.

Therefore, the answer is option 3: at least 22%, but less than 29%.

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The depth of the water increased between Monday and Tuesday because the value moved to the right on a number line.

To understand on what day there was an increase, let's analyze how the water level changed:

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The graph in the attachment shows that the system of equations, 3y + 3x = 2 and y + x = 8 has: C. No solution

The solution to a system of equations can be determined by finding the coordinate of the point where the lines that represent each of the equations intersect on a coordinate plane when graphed.

Given the system of equations as:

3y + 3x = 2

y + x = 8

The image below shows a red line which represents 3y + 3x = 2 and a blue line that represents y + x = 8. Both lines do not intersect at any point.

Therefore, there is no solution.

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One possible three-dimensional figure that fits this description is a rectangular prism. Another possible option is a cylinder. A third option is a pyramid with a square base.

Based on your question, the three-dimensional figure could be one of the following three options:

1. Cone: If two cross sections are taken parallel to the base, they will both be circles but with different radii, making them similar but not congruent.

2. Pyramid: If two cross sections are taken parallel to the base, they will both be the same shape as the base (e.g., triangles, squares) but with different side lengths, making them similar but not congruent.

3. Frustum: A frustum is a section of a cone or pyramid with the top cut off parallel to the base. The two cross sections taken parallel to the base would be the same shape but with different dimensions, making them similar but not congruent.

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To test whether the type of animal used for animal crackers is not uniformly distributed, we can use a chi-squared goodness-of-fit test.

The null hypothesis is that the type of animal used for animal crackers is uniformly distributed, while the alternative hypothesis is that it is not uniformly distributed.

We first calculate the expected counts under the assumption of a uniform distribution, which would be 686/6 = 114.33 for each animal.

Animal Observed Count Expected Count

Lion 100 114.33

Tiger 110 114.33

Bear 120 114.33

Rhino 116 114.33

Hippo 130 114.33

Zebra 110 114.33

The test statistic is the chi-squared statistic, which can be calculated as:

χ² = Σ(observed count - expected count)² / expected count

Using the table above, we calculate:

χ² = [(100-114.33)²/114.33] + [(110-114.33)²/114.33] + [(120-114.33)²/114.33] + [(116-114.33)²/114.33] + [(130-114.33)²/114.33] + [(110-114.33)²/114.33] = 7.956

The degrees of freedom for this test is (6-1) = 5.

Using a chi-squared distribution table or calculator, the p-value associated with this test statistic and degrees of freedom is approximately 0.1603.

Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. Therefore, we do not have convincing evidence to suggest that the type of animal used for animal crackers is not uniformly distributed.

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The length of the radius of the circle is [tex]4\sqrt{73}[/tex].

Let, AB is the chord with a length of 24 centimeters. Line OD is the distance between the chord and the center of the circle. Line r is the radius.

By the chord bisector theorem, line OD bisects the chord AB and it is perpendicular to AB.

Thus, the length of segment AD would be half of the length of AB. i.e,

AD = 12.

As triangle AOD is a right triangle, use Pythagoras theorem in triangle AOD to find the radius of the circle.

[tex]OA^{2} =AD^{2} +OD^{2} \\r^{2} =12^{2} +32^{2} \\r^{2} =144+1024[/tex]

Further simplify,

[tex]r^{2} =\sqrt{1168}\\ r=4\sqrt{ 73[/tex]

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A Pythagorean Triple is a set of three integers that are positive whole numbers, namely a, b, and c, such that a²+b²=c².

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle

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A. statistically significant difference in the mean blood hemoglobin levels between breastfed infants and formula-fed infants at α = 0.05.

B. the 95% confidence interval for the difference in population means between breast-fed infants and formula-fed infants is (−0.06, 1.18).

C. Both the hypothesis test and the confidence interval lead to the same conclusion that there is a difference in the mean blood hemoglobin levels between the two feeding regimens.

What is null hypothesis?

In statistics, the null hypothesis (H0) is a statement that assumes that there is no significant difference between two or more groups, samples, or populations.

(a) To test the hypothesis that there is a difference in the population means between breast-fed infants and formula-fed infants, we can use a two-sample t-test with equal variances. The null hypothesis is that the population means are equal, and the alternative hypothesis is that they are not equal. Using α = 0.05 as the significance level, the critical value for a two-tailed test with 40 degrees of freedom is ±2.021.

The test statistic can be calculated as:

t = (x1 - x2) / (Sp * √(1/n1 + 1/n2))

where x1 and x2 are the sample means, Sp is the pooled standard deviation, and n1 and n2 are the sample sizes. The pooled standard deviation can be calculated as:

Sp = √(((n1 - 1) * s1² + (n2 - 1) * s2²) / (n1 + n2 - 2))

where s1 and s2 are the sample standard deviations.

Plugging in the values from the table, we get:

t = (13.3 - 12.4) / (1.776 * √(1/23 + 1/19)) = 2.21

Since the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference in the mean blood hemoglobin levels between breastfed infants and formula-fed infants at α = 0.05.

(b) To construct a 95% confidence interval for the difference in population means between breast-fed infants and formula-fed infants, we can use the formula:

(x1 - x2) ± tα/2,Sp * √(1/n1 + 1/n2)

where tα/2,Sp is the critical value of the t-distribution with (n1 + n2 - 2) degrees of freedom and α/2 as the significance level.

Plugging in the values from the table, we get:

(x1 - x2) ± tα/2,Sp * √(1/n1 + 1/n2)

= (13.3 - 12.4) ± 2.021 * 1.776 * √(1/23 + 1/19)

= 0.56 ± 0.62

Therefore, the 95% confidence interval for the difference in population means between breast-fed infants and formula-fed infants is (−0.06, 1.18).

(c) The hypothesis test and the confidence interval both lead to the conclusion that there is a difference in the mean blood hemoglobin levels between breast-fed infants and formula-fed infants. In part (a), we rejected the null hypothesis that the population means are equal, which means we concluded that there is a difference. In part (b), the confidence interval does not contain 0, which means we can reject the null hypothesis that the difference in means is 0 at the 95% confidence level.

Therefore, both the hypothesis test and the confidence interval lead to the same conclusion that there is a difference in the mean blood hemoglobin levels between the two feeding regimens.

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Answer:  H

Step-by-step explanation:

The rule for a graph to be a function is if there are not 2 points that have the same x.

You can do a vertical line test. If you held up a vertical line(line that goes straight up and down) to both of these curves, across the entire curves(everywhere).  The curves do not hit that vertical line 2 or more times.  

Since both curves pass the vertical line test.  They are both functions.

Answer: There is nothing there

Step-by-step explanation:

Answer:

their it is nothing their

Using the amplitude of the motion, After approximately 0.89 seconds, the bobber will be at the midpoint between its high point and low point for the first time.

The bobber will be at the midpoint between its high point and low point for the first time after 1.5/4 seconds.

To find the answer, we need to first find the amplitude of the motion, which is half of the total distance the bobber travels, so amplitude = 1.5/2 = 0.75 inches.

Next, we can use the formula for the period of a simple harmonic motion: T = 2π√(m/k), where T is the period, m is the mass, and k is the spring constant. In this case, we can use the formula T = 3 seconds.

Solving for k, we get k = (4π²)m/T². Since we don't know the mass of the bobber, we can assume it's negligible and use k = 4π²/T². Plugging in T = 3 seconds, we get k = 4π ²/⁹.

Now we can use the formula for the displacement of a simple harmonic motion at time t: x = Acos(ωt), where A is the amplitude and ω is the angular frequency (ω = 2π/T). We want to find when the displacement x = 0.5A (i.e. the midpoint between the high and low points), so we can solve for t:

0.5A = Acos(ωt)

0.5 = cos(2πt/3)

2πt/3 = arccos(0.5)

t = 3arccos(0.5)/2π

t ≈ 0.89 seconds

So after approximately 0.89 seconds, the bobber will be at the midpoint between its high point and low point for the first time.

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Answer:

130 square metres

Step-by-step explanation:

The shaded area is a trapezium.

Therefore, the rule for its area is base1+base2/2xh

base1=6

base2=14

height = 13

Therefore, 6+14/2 x 13

= 130

Hope that helped!!! k

Reject the null hypothesis; there is strong evidence to suggest that the Sharpies last longer than a mean of 14 hours.

What is null hypothesis?

In statistics, a null hypothesis is a statement that suggests there is no significant difference between two sets of data, or that any observed difference is due to chance or sampling variation.

Reject the null hypothesis; there is strong evidence to suggest that the Sharpies last longer than a mean of 14 hours.

The calculated t-value of 2.635 is greater than the critical t-value at the 5% significance level, which indicates that the sample mean is significantly different from the hypothesized population mean of 14 hours.

The p-value of 0.006 is also less than the significance level of 0.05, providing strong evidence against the null hypothesis.

Therefore, we can conclude that the Sharpies last longer than a mean of 14 continuous hours, based on the given data.

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Antonio started watching the movie at 5:17pm

If the movie ended at 7:00pm, and we know that the movie was 1 hour and 43 minutes long, we can then subtract 1 hour and 43 minutes from 7:00pm to find out what time Antonio started the movie.

To subtract 1 hour and 43 minutes from 7:00pm:

= 19:00 - 1:43

= 17:17

As a 24-hour mode, when we convert 17:17 to 12 hours, this gives us 5:17pm.

Note: The movie ended at 7:00pm

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The statement is true. Therefore, we have shown that [tex](rA)^{(-1)} = A^{(-1)}/r,[/tex]which implies that [tex](rA)^{(-1)} = r^{(-1)}\times A^{(-1)[/tex]. Hence, [tex](rA)^{(-1) }= rA^{(-1)[/tex], since r is nonzero.

To prove this, we can start with the definition of the inverse of a matrix:

If A is an invertible matrix, then its inverse, denoted as [tex]A^{(-1),[/tex] is the unique matrix such that [tex]A\times A^{(-1)} = A^{(-1)} \times A = I[/tex], where I is the identity matrix.

Now, let's consider the matrix rA, where r is a nonzero scalar. We want to find its inverse, denoted as [tex](rA)^{(-1)[/tex].

We can start by multiplying both sides of the equation [tex]A\times A^{(-1)} = I[/tex] by r:

[tex]rA\times A^{(-1)} = rI[/tex]

Next, we can multiply both sides of this equation by A from the left:

[tex]rA\times A^{(-1)}A = rIA\\rAI = rA = rA(A\times A^{(-1)})[/tex]

Now, we can use the associative property of matrix multiplication to rearrange the right-hand side of this equation:

[tex]rA\times(AA^(-1)) = (rAA)\times A^{(-1)}\\rA\times I = (rA)\times A^{(-1)}\\rA = (rA)\times A^{(-1)}[/tex]

Finally, we can multiply both sides of this equation by [tex](rA)^{(-1)[/tex] from the left to obtain:

[tex](rA)^{(-1)}rA = (rA)^{(-1)}(rA)\times A^{(-1)}\\I = A^{(-1)}[/tex]

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a. In a random sample of 1,000 births, the expected number of births that take place by Caesarian section is:

E(X) = n*p = 1,000 * 0.32 = 320 births

Therefore, on average, 320 births out of 1,000 will take place by Caesarian section.

b. The variance of the number of Caesarian section births in a sample of 1,000 births is:

Var(X) = np(1-p) = 1,000 * 0.32 * (1-0.32) = 217.60

The standard deviation is the square root of the variance:

SD(X) = sqrt(Var(X)) = sqrt(217.60) = 14.76

Therefore, the standard deviation of the number of Caesarian section births in a sample of 1,000 births is 14.76.

c. To form an interval that is likely to contain the number of Caesarian section births in a sample of 1,000 births, we can use the normal distribution and the central limit theorem. Since n*p = 320 is greater than 10, we can assume that the distribution of the number of Caesarian section births in a sample of 1,000 births is approximately normal.

The 95% confidence interval for the number of Caesarian section births is:

320 ± 1.96*(14.76) = (291.16, 348.84)

Therefore, we can be 95% confident that the number of Caesarian section births in a sample of 1,000 births will be between 291 and 349.

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The amount she will need to pay is $6.95 .

The amount Anye will need is calculated as follows;

The total cost of the two coffees is 2 x $3.55 = $7.10

The total cost of the coffees and muffin is $7.10 + $2.85 = $9.95

Adding the $2 tip, the total cost becomes $9.95 + $2 = $11.95.

If Anye has a $5 gift card, the amount she will need to pay is calculated as;

= $11.95 - $5

= $6.95

Therefore, Anye will need $6.95 in cash to pay for the rest of the order.

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The error in the set of instructions can be resolved by rearranging the steps so they are in sequential order. The Option A is correct.

The most effective solution would be to rearrange the steps so they follow a logical order.

For example, the steps could be rearranged as follows:

Therefore, by doing this would make the instructions clearer and easier to follow. The Option A is correct.

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True. The Monte Carlo simulation is a technique that uses random sampling to simulate real-life scenarios. It can be used to represent a long period of real-time by using a short period of simulated time.

This is because the simulation can generate a large number of random samples, which can provide a comprehensive picture of the possible outcomes in a shorter time frame.
The Monte Carlo simulation is a technique that can be used to represent a long period of real-time by a short period of simulated time. The statement is true. Monte Carlo simulations use random sampling and statistical models to estimate the probabilities of different outcomes, allowing for the analysis of complex systems over a long period of real-time in a shorter simulated time.

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The measures of the angle and side is given by:

Blank 1: (x) = 53

Blank 2: AB =

We know that the measure of semi circular central angle is 90 degrees.

So here angle BDA is 90 degrees. [Since the sum of all interior angles of a triangle is 180 degrees according to the Angle Sum Property]

So the sum of angle DAB and angle ABD is 90 degrees.

So, 37 + x = 90

x = 90 - 37

x = 53 degrees.

Now according to trigonometry, AB is Hypotenuse and DB is Base with respect to angle DBA.

Given that the length of side DB is 15 units.

cos(angle DBA) = DB/AB

cos 37 = 15/AB

AB = 15/cos 37

AB = 18.8 (approximated to one decimal place)

Hence the angle x is 53 degrees and side AB will be 18.8 (rounded off to one decimal place) units.

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A male at the 70th percentile has a body temperature of 98.26°F.

The body temperature of a male at the 70th percentile, we need to use the cumulative distribution function (CDF) of the normal distribution.

The CDF gives the probability that a random variable (in this case, body temperature) is less than or equal to a certain value.

A standard normal distribution table or a calculator to find the corresponding z-score for the 70th percentile, and then use the formula:

z = [tex](x - \mu) / \sigma[/tex]

x is the body temperature we want to find, mu is the mean, sigma is the standard deviation, and z is the z-score corresponding to the 70th percentile.

Using a standard normal distribution table, we find that the z-score for the 70th percentile is approximately 0.52.

Plugging in the values we have:

0.52 = (x - 98.1) / 0.30

Solving for x, we get:

x = 98.1 + 0.30 × 0.52

x = 98.26°F

To draw a well-labeled sketch to support the answer, we can start by drawing a normal distribution curve with the mean of 98.1°F and a standard deviation of 0.30°F.

The point on the x-axis corresponding to the body temperature of a male at the 70th percentile, which is 98.26°F.

The area under the curve to the left of this point, which represents the probability that a male has a body temperature less than or equal to 98.26°F.

The resulting sketch would look like this:

Normal Distribution Curve with 70th Percentile

The shaded area under the curve represents the probability that a male has a body temperature less than or equal to 98.26°F, is approximately 0.70 or 70%.

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Answer:

x = [tex]\frac{4}{9}[/tex]

Step-by-step explanation:

3[tex]\sqrt{x}[/tex] = 2 ( divide both sides by 3 )

[tex]\sqrt{x}[/tex] = [tex]\frac{2}{3}[/tex] ( square both sides )

([tex]\sqrt{x}[/tex] )² = ([tex]\frac{2}{3}[/tex] )²

x = [tex]\frac{2^2}{3^2}[/tex]

x = [tex]\frac{4}{9}[/tex]

(L3) If there is no indication of congruent or equal segments, you are dealing with a(n) orthocenter .

The altitude is a piece of a perpendicular line that connects the triangle's vertex to either its opposite side or an extension of that side.The orthocenter is located inside the triangle if the triangle is acute, on the vertex that is farthest from the base if the triangle is obtuse, and on the base if the triangle is right-angled. The orthocenter is an important point in the study of triangles, as it has a variety of properties that are useful in problem-solving, such as its relationship with the circumcenter and centroid of a triangle.

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Required general solution is [tex]x = c_1 v_1 e^{(λ_1 t)} + c_2 v_2 e^{(λ_2 t)}[/tex] where [tex]c_1 \: and \: c_2 [/tex] are constants determined by the initial conditions.

To find the general solution of the given system X' = Ax.

We first need to find the eigenvalues and eigenvectors of the matrix A.

The characteristic polynomial of A is given by lA - λI =|10 - λ -20||8 -18 - λ|

= (10 - λ)(-18 - λ) - (-20)(8)

= λ^2 - 8λ + 4

The roots of this polynomial are

[tex]λ_1 = 4 +\sqrt{12} \\ λ_2 = 4 - \sqrt{12} [/tex]

Next, we need to find the eigenvectors associated with each eigenvalue. For

[tex]λ_1 = 4 + √12[/tex]

, we have:

[tex](A - λ_1 I) x =[/tex]

|10 - (4 + √12) -20|

|8 -18 - (4 + √12)|

| 2 + √12 20 - (4 + √12)|

Reducing this augmented matrix to row echelon form, we get:

|0 -2/(2+√12)|

|1 (10-4-√12)/(2+√12)|

Thus, the eigenvector associated with [tex]λ_1[/tex] is:[tex]v_1[/tex]=|2/(2+√12)|

|-(10-4-√12)/(2+√12)|

Simplifying, we get:

[tex]v_1[/tex] =|(√3 - 1)/2||1 |

Similarly, for [tex]λ_2 = 4 - √12[/tex]

, we have:

[tex](A - λ_2 I) x[/tex]

=|10 - (4 - √12) -20|

|8 -18 - (4 - √12)|

| 2 - √12 20 - (4 - √12)|

Reducing this augmented matrix to row echelon form, we get:

|0 -2/(2-√12)|

|1 (10-4+√12)/(2-√12)|

Thus, the eigenvector associated with

[tex]λ_2[/tex] is:[tex]v_2[/tex] =|-(√3 + 1)/2||1 |

Now we can write the general solution of the system as:

[tex]x = c_1 v_1 e^{(λ_1 t)} + c_2 v_2 e^{(λ_2 t)}[/tex] where [tex]c_1 \: and \: c_2 [/tex] are constants determined by the initial conditions.

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