A sample in which every possible combination of items in the population has an equal chance of constituting the sample is a:a. random sample.b. statistical sample.c. judgment sample.d. representative sample.

This is the exact answer in symbolic notation and fractions .The integral of (6/((11+t^2)(1+t^2))) dt is a bit complicated and requires partial fraction decomposition.

We can factor the denominator:
11 + t^2 = (11 - i^2)(1 + t^2/(11-i^2)) = 10(1 + t^2/10) + i^2(1 - t^2/11)
1 + t^2 = (1 + i^2)(1 + t^2/(1+i^2))
So we have:
6/((11+t^2)(1+t^2)) = A/(10(1 + t^2/10) + i^2(1 - t^2/11)) + B/(1 + i^2)(1 + t^2/(1+i^2))
6 = A(1 + i^2)(11)(1) + B(10)(11)(1)

The first integral can be evaluated using the substitution u = (10/11)t and then the identity 1 + tan^2(x) = sec^2(x) for the denominator. We get:
-3/(11+10i) * arctan(t/√(110)) + 3/(110 + 11i) * arctan(t/√2)
Evaluating from 0 to 15, we get:
-3/(11+10i) * arctan(15/√(110)) + 3/(110 + 11i) * arctan(15/√2)

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The exact length of the curve represented by x = 6 + 3t^2 and y = 4 + 2t^3, where 0 ≤ t ≤ 3, can be found by evaluating the integral ∫[0,3] 6t√(1 + t^2) dt.

To find the exact length of the curve represented by the parametric equations x = 6 + 3t^2 and y = 4 + 2t^3, where 0 ≤ t ≤ 3, we will use the arc length formula. The length of the curve is the integral of the square root of the sum of the squares of the derivatives of x and y with respect to t.

We start by finding the derivatives of x and y with respect to t:

dx/dt = 6t

dy/dt = 6t^2

Next, we calculate the integrand for the arc length formula:

√[(dx/dt)^2 + (dy/dt)^2] = √[(6t)^2 + (6t^2)^2] = √(36t^2 + 36t^4) = 6t√(1 + t^2)

Now, we can set up the integral to find the length of the curve:

Length = ∫[0,3] 6t√(1 + t^2) dt

To evaluate this integral, we can use a suitable technique such as substitution or integration by parts. After evaluating the integral, we will have the exact length of the curve between t = 0 and t = 3.

In conclusion, the exact length of the curve represented by x = 6 + 3t^2 and y = 4 + 2t^3, where 0 ≤ t ≤ 3, can be found by evaluating the integral ∫[0,3] 6t√(1 + t^2) dt.

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Answer:

Slope = 5/2

Step-by-step explanation:

Slope = (y2-y1)/(x2-x1)

In this case the two points are (6,7) and (4,2)
Using the formula we find that the slope is:
slope = (2-7)/(4-6)
slope = -5/-2
The negatives cancel out and therefore the final answer or slope is 5/2.

we have proved that any set of 11 distinct integers must contain a pair of integers with a difference that is a multiple of 10.

To prove that any set of 11 distinct integers must contain a pair of integers with a difference that is a multiple of 10, we can use the Pigeonhole Principle.

The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least one pigeonhole must contain more than one pigeon.

In this case, we have 11 distinct integers, which can be thought of as the pigeons, and we have 10 possible remainders when dividing an integer by 10, which can be thought of as the pigeonholes.

Consider the possible remainders when dividing an integer by 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. If we place each of the 11 integers into one of these 10 pigeonholes based on their remainder when divided by 10, then by the Pigeonhole Principle, at least one pigeonhole must contain more than one integer.

This means that there must be at least two integers in the same pigeonhole, which implies that their difference is a multiple of 10 (since they have the same remainder when divided by 10).

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Answer: 2520 groups

Step-by-step explanation:

To calculate the number of groups that can be chosen with four SE students and three CPRE students, we need to consider the number of ways to select the students from each group separately.

The number of ways to choose four SE students from the nine available is given by the combination formula, denoted as "9 choose 4" or C(9, 4), and can be calculated as:

C(9, 4) = 9! / (4! * (9 - 4)!) = 9! / (4! * 5!) = (9 * 8 * 7 * 6) / (4 * 3 * 2 * 1) = 126

Similarly, the number of ways to choose three CPRE students from the six available is:

C(6, 3) = 6! / (3! * (6 - 3)!) = 6! / (3! * 3!) = (6 * 5 * 4) / (3 * 2 * 1) = 20

To determine the total number of groups, we multiply the number of choices for SE students and CPRE students:

Total number of groups = C(9, 4) * C(6, 3) = 126 * 20 = 2520

Therefore, there are 2520 different groups of seven students that can be chosen, consisting of four SE students and three CPRE students.

The required shape of the histogram for this distribution is skewed right.

To determine the shape of the histogram based on the given distribution, we can analyze the distribution of probabilities for each number of eggs.

Looking at the probabilities associated with each number of eggs:

0 eggs: 0.02

1 egg: 0.03

2 eggs: 0.07

3 eggs: 0.12

4 eggs: 0.30

5 eggs: 0.27

6 eggs: 0.18

We can observe that the probabilities are not evenly distributed across the different numbers of eggs. The probabilities gradually increase from 0 eggs to 4 eggs, then decrease from 4 eggs to 6 eggs. This indicates that the distribution is skewed right.

Therefore, the shape of the histogram for this distribution is skewed right.

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Answer:

$16

Step-by-step explanation:

We can subtract the amount he spent from the amount he earned to calculate how much Jim had at the end of the day.

The amount earned: $5 for each of four hours of work

So, the amount earned is: 4 hours * $5 per hour = $20

Amount spent: $4

We subtract the amount spent from the amount earned, to find out how much Jim had at the end of the day,

Amount at the end of the day = Amount earned - Amount spent

Amount at the end of the day = $20 - $4

Amount at the end of the day = $16

Therefore, Jim had $16 at the end of the day.

The type of sampling that requires giving every part an equal chance of being selected for the sample is called "simple random sampling."

Simple random sampling is a method of selecting a sample from a population where each member has an equal probability of being chosen. In this type of sampling, every part or element of the population is assigned a number or label, and a random process, such as a lottery or a random number generator, is used to select the sample. The main characteristic of simple random sampling is that each member of the population has an equal and independent chance of being selected, without any bias or preference.

This sampling method ensures that every part of the population has an equal opportunity to be included in the sample, making it a fair representation of the entire population. By giving each part an equal chance of being selected, simple random sampling helps to minimize sampling errors and increase the generalizability of the findings to the larger population. It is often used in research studies and surveys when the goal is to obtain an unbiased and representative sample.

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The given points are (2, 4), (4, 8), (6, 12). To determine the polynomial function of least degree whose graph passes through the given points, we will follow the below-given steps:Let's consider the given points: (2, 4), (4, 8), (6, 12).

The function we want is a polynomial function of least degree. So let's assume that the function we want is a second-degree polynomial function and is of the form:p(x) = ax² + bx + cwhere a, b, and c are constants. Since p(x) passes through the point (2, 4), so by substituting x = 2 and p(x) = 4 in the above equation, we get:4 = a(2)² + b(2) + c     ⇒ 4 = 4a + 2b + c ... (1)Since p(x) passes through the point (4, 8), so by substituting x = 4 and p(x) = 8 in the above equation, we get:8 = a(4)² + b(4) + c     ⇒ 8 = 16a + 4b + c  ... (2)Since p(x) passes through the point (6, 12), so by substituting x = 6 and p(x) = 12 in the above equation, we get:12 = a(6)² + b(6) + c     ⇒ 12 = 36a + 6b + c  ... (3)We have three variables a, b, and c, so we need three equations to solve them. Solving equation (1), (2) and (3) for a, b, and c, we get:a = 1b = 0c = 0Hence, the polynomial function of least degree whose graph passes through the given points is:p(x) = x².

The given points are (2, 4), (4, 8), (6, 12). The function we want is a polynomial function of least degree. So let's assume that the function we want is a second-degree polynomial function and is of the form:p(x) = ax² + bx + c, where a, b, and c are constants. Since p(x) passes through the point (2, 4), so by substituting x = 2 and p(x) = 4 in the above equation, we get 4 = a(2)² + b(2) + c  ⇒ 4 = 4a + 2b + c. Since p(x) passes through the point (4, 8), so by substituting x = 4 and p(x) = 8 in the above equation, we get 8 = a(4)² + b(4) + c  ⇒ 8 = 16a + 4b + c. Since p(x) passes through the point (6, 12), so by substituting x = 6 and p(x) = 12 in the above equation, we get 12 = a(6)² + b(6) + c  ⇒ 12 = 36a + 6b + c. We have three variables a, b, and c, so we need three equations to solve them. Solving equation (1), (2) and (3) for a, b, and c, we get a = 1, b = 0 and c = 0. Hence, the polynomial function of least degree whose graph passes through the given points is p(x) = x².

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A correlation of .60 is twice as strong as a correlation of .30 when strength is measured by r2 or the coefficient of determination.

This is because the coefficient of determination represents the proportion of variance in one variable that is predictable from the other variable, and it is calculated by squaring the correlation coefficient. Therefore, if the correlation coefficient doubles, the coefficient of determination increases by a factor of four (i.e., it becomes four times as large).

To calculate the coefficient of determination, we square the correlation coefficient (r). In this case:

For a correlation of 0.60, the coefficient of determination is 0.60^2 = 0.36.

For a correlation of 0.30, the coefficient of determination is 0.30^2 = 0.09.

Comparing the two coefficients of determination, 0.36 is indeed twice as large as 0.09. Therefore, a correlation of 0.60 is twice as strong as a correlation of 0.30 when measured by the coefficient of determination (r^2).

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We can calculate the value of c in this expression to be: 4513.

In the expression, we are told that the value of n is 250. To resove the question, we will simply substitute the value, 250 in the expression to have:

c=18n+13

c = 18(250) + 13

c = 4500 + 13

= 4513

So, the answer to c is 4513.

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The magnitude of the upward force on the table due to the leg at (lx, 0) is Fx=W.

To find the upward force (Fx) on the table due to the leg at position (Lx, 0), you will need to consider the forces acting on the table. In this scenario, the force Fx is the force exerted by the leg to support the weight of the table and maintain equilibrium.  

To calculate Fx, you'll need to take into account the weight of the table (W) and any additional forces acting on it, such as objects placed on the table. Since the table is in equilibrium, the sum of all forces in the vertical direction should be equal to zero.

Using the principle of equilibrium:

ΣFy = 0

Where ΣFy represents the sum of all vertical forces acting on the table.

In this case, the upward force exerted by the leg (Fx) should be equal to the total downward force acting on the table:

Fx = W

Once you have determined the weight of the table and any additional forces, you can calculate the magnitude of the upward force Fx exerted by the leg at position (Lx, 0).

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the estimated maximum absolute error is approximately 0.1624.

To estimate the maximum absolute error using the Remainder Estimation Theorem, we need to find the value of the remainder term for the polynomial approximation.

The Remainder Estimation Theorem states that if R(n) is the remainder term for the polynomial approximation of degree n, then the maximum absolute error (E) is given by:

E ≤ |R(n)|

In this case, the polynomial approximation is 1 + 8x + 28x^2 and the original function is f(x) = (1 + x)^8. The interval of interest is -0.01 ≤ x ≤ 0.01.

To find the remainder term R(n), we use the Binomial Expansion of (1 + x)^8:

f(x) = (1 + x)^8 = C(8, 0) + C(8, 1)x + C(8, 2)x^2 + ... + C(8, 8)x^8

Where C(n, k) is the binomial coefficient.

The remainder term R(n) can be obtained by subtracting the polynomial approximation from the original function:

R(n) = f(x) - (1 + 8x + 28x^2)

Now we need to estimate the maximum absolute value of R(n) on the interval -0.01 ≤ x ≤ 0.01.

To do this, we can find the maximum value of |R(n)| by evaluating R(n) at the endpoints of the interval and taking the larger value:

|R(n)| = max(|R(-0.01)|, |R(0.01)|)

Substituting the values of x into R(n), we have:

|R(n)| = max(|f(-0.01) - (1 + 8(-0.01) + 28(-0.01)^2)|, |f(0.01) - (1 + 8(0.01) + 28(0.01)^2)|)

Now we evaluate the expressions:

|R(n)| = max(|f(-0.01) - (1 - 0.08 + 0.00028)|, |f(0.01) - (1 + 0.08 + 0.00028)|)

To simplify further, we calculate the values of f(-0.01) and f(0.01):

f(-0.01) = (1 + (-0.01))^8 ≈ 1.0824

f(0.01) = (1 + 0.01)^8 ≈ 1.1859

Substituting these values, we have:

|R(n)| = max(|1.0824 - (1 - 0.08 + 0.00028)|, |1.1859 - (1 + 0.08 + 0.00028)|)

Simplifying the expressions inside the absolute values, we get:

|R(n)| = max(|0.1624|, |0.1059|)

Taking the larger value, we find:

|R(n)| ≈ 0.1624

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The probability of getting a yellow, glass marble is 1/6

Probability is the likelihood of a desired event happening.

There are 4 yellow marbles and half of them are glass.

Thus, the yellow marbles made of glass will be:

4/2 = 2

There are a total of 8 green marbles and 4 yellow marbles in the bag. That is:

total = 8 + 4 = 12

The probability of getting a yellow, glass marble will be:

2/12 = 1/6

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The expression for the area of triangle ABC, given ∠A = 48°, ∠C = 67°, and AC = 15, is 83.57 square units.

The area of a triangle can be calculated using the following formula:

Area = (1/2)×AB ×AC ×sin(C)

We are given the angles ∠A = 48°, ∠C = 67°, and the side AC = 15.

To find the area, we need to find the length of side AB and the sine of angle C.

By law of sines

AB/sin(A) = AC/sin(C)

Rearranging the equation, we get:

AB = (AC × sin(A))/sin(C)

Substituting the given values, we have:

AB = (15 × sin(48°))/sin(67°)

AB = (15 × 0.7431) / 0.9217

AB = 12.09

Now that we have the length of side AB, we can substitute it into the formula for the area:

Area = (1/2)× AB × AC × sin(C)

Substituting the values AB = 12.09, AC = 15, and sin(C) = 0.9217:

Area = (1/2) × 12.09 × 15 × 0.9217

Area = 83.57

Therefore, the expression for the area of triangle ABC, given ∠A = 48°, ∠C = 67°, and AC = 15, is 83.57 square units.

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Answer and Explanation:

1. Describe the relationship shown in the graph.

The graph shows that as the temperature outside gets hotter, the heating cost gets lower.

This makes sense because when it is 90°F outside, there's no need to heat inside; at that point most people would want air conditioning to cool down the room.

2. Draw the best-fit line.

See the attached image.

The best-fit line goes through the average of the data points.

3. Does the scatter plot show a positive, negative, or no relationship?

The scatter plot shows a negative relationship because as the graph advances horizontally, it decreases vertically.

The area of the figure is 198 mm².

We have,

We can consider from the figure that there are two types of shapes.

- 2 Rectangle

- 1 Triangle

Now,

Rectangle:

Area = Length x Width

= (10 + 5) x 9

= 15 x 9

= 135 mm²

Rectangle:

Area = Length x Width

= 7 x 5

= 53 mm²

Triangle:

Area = 1/2 x base x height

= 1/2 x (15 - 11) x 5

= 1/2 x 4 x 5

= 2 x 5

= 10 mm²

Now,

The area of the figure.

= 135 + 53 + 10

= 198 mm²

Thus,

The area of the figure is 198 mm².

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Here are the steps to find the values:

(i) To find f(-2), substitute -2 into the function f(x):

[tex]\sf\: f(-2) = 3(-2)^2 - 2 \\ \sf\: f(-2) = 3(4) - 2 \\ \sf\: f(-2) = 12 - 2 \\ \sf\: f(-2) = 10[/tex]

Therefore, f(-2) = 10.

(ii) To find g⁻¹(-2), we need to find the value of x that satisfies g(x) = -2.

[tex]\sf\: g(x) = x + 3 \\ \sf\: -2 = x + 3 \\ \sf\: x = -2 - 3 \\ \sf\: x = -5[/tex]

Therefore, g⁻¹(-2) = -5.

(iii) To find the value of x for which f[g(x)] = g[f(x)], we need to set the two functions equal to each other and solve for x:

[tex]\sf\: f[g(x)] = g[f(x)] \\ \sf\: 3[g(x)]^2 - 2 = g(x) + 3 \\ \sf\: 3(x + 3)^2 - 2 = x + 3 \\ \sf\: 3(x^2 + 6x + 9) - 2 = x + 3 \\ \sf\: 3x^2 + 18x + 27 - 2 = x + 3 \\ \sf\: 3x^2 + 17x + 25 = x + 3 \\ \sf\: 3x^2 + 16x + 22 = 0[/tex]

You can then solve this quadratic equation to find the value(s) of x.

(iv) To find f⁻¹[g(4)], we need to find the value of x that satisfies f(x) = 4.

[tex]\sf\: f(x) = 3x^2 - 2 \\ \sf\: 4 = 3x^2 - 2 \\ \sf\: 3x^2 = 6 \\ \sf\: x^2 = 2 \\ \sf\: x = \pm \sqrt{2}[/tex]

Therefore, f⁻¹[g(4)] = ±√2.

the volume of the tetrahedron bounded by the coordinate planes and the plane x/8 + y/10 + z/5 = 1 is 200 cubic units.

What is Volume?

Volume is the amount of three-dimensional space enclosed by a closed surface, such as the space occupied or contained by a substance or shape.

To find the volume of the tetrahedron bounded by the coordinate planes and the plane x/8 + y/10 + z/5 = 1, we can use the concept of a triangular pyramid.

First, let's visualize the tetrahedron and the given plane. The equation x/8 + y/10 + z/5 = 1 represents a plane in three-dimensional space. By examining the equation, we can determine that the plane intersects the coordinate axes at the points (8, 0, 0), (0, 10, 0), and (0, 0, 5).

Now, we can consider the tetrahedron formed by the coordinate planes and the points of intersection. This tetrahedron has three triangular faces: a base formed by the x-axis and y-axis, a front face formed by the x-axis and z-axis, and a side face formed by the y-axis and z-axis.

To find the volume of the tetrahedron, we can calculate the volume of each triangular pyramid and sum them up. The formula for the volume of a pyramid is (1/3) * base area * height.

Let's calculate the volume of each pyramid:

Base pyramid: The base is a triangle with base length 8 and height 10 (formed by the x-axis and y-axis).

The base area is (1/2) * 8 * 10 = 40 square units. The height is 5 units (the z-coordinate of the plane).

Therefore, the volume of the base pyramid is (1/3) * 40 * 5 = 200/3 cubic units.

Front pyramid: The base is a triangle with base length 8 and height 5 (formed by the x-axis and z-axis).

The base area is (1/2) * 8 * 5 = 20 square units. The height is 10 units (the y-coordinate of the plane). Therefore, the volume of the front pyramid is (1/3) * 20 * 10 = 200/3 cubic units.

Side pyramid: The base is a triangle with base length 10 and height 5 (formed by the y-axis and z-axis). The base area is (1/2) * 10 * 5 = 25 square units. The height is 8 units (the x-coordinate of the plane). Therefore, the volume of the side pyramid is (1/3) * 25 * 8 = 200/3 cubic units.

Finally, we add up the volumes of all three pyramids to find the total volume of the tetrahedron:

Total volume = (200/3) + (200/3) + (200/3) = 600/3 = 200 cubic units.

Therefore, the volume of the tetrahedron bounded by the coordinate planes and the plane x/8 + y/10 + z/5 = 1 is 200 cubic units.

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(a) For t ≤ 0, F(t) = 0, and for t > a, F(t) = 1.

(b) [tex]F (or) t < = 0, R(t) = 1, and f(or) t > a, R(t) = 0.[/tex]

(c)The hazard rate h(t) for a uniform distribution is a decreasing function.

(d) [tex]t_median = (0 + a) / 2 = a / 2[/tex]

(e)[tex]p(T > 1/2a + 1/4a | T > 1/2a) = length of interval (1/2a + 1/4a, a] / length of interval (1/2a, a] = (a - (1/2a + 1/4a)) / (a - 1/2a) = (3/4).[/tex]

(f)The uniform distribution does not exhibit the memoryless property because the remaining time to failure is dependent on the time already passed.

What is Probability?

Probability is a branch of mathematics that deals with the study of uncertainty and the likelihood of events occurring. It provides a quantitative measure of the likelihood or chance of an event happening. Probability is used to analyze and predict outcomes in various fields, including statistics, physics, economics, and engineering.

In probability theory, an event is defined as a specific outcome or set of outcomes of an experiment or random process. The probability of an event is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain or guaranteed event. Intermediate probabilities indicate varying degrees of likelihood.

(a) To find the cumulative distribution function (CDF) F(t), we need to determine the probability that the time to failure T is less than or equal to a given value t.

Since T follows a uniform distribution over (0, a], the probability density function (PDF) f(t) is constant within this interval:

f(t) = 1/a, for 0 < t ≤ a

To find the CDF, we integrate the PDF from 0 to t:

[tex]F(t) = \int[0 to t] f(x) dx[/tex]

For 0 < t ≤ a, the integral evaluates to:

[tex]F(t) = \int[0.. t] (1/a) dx = (t/a), for 0 < t < = a[/tex]

For t ≤ 0, F(t) = 0, and for t > a, F(t) = 1.

(b) The reliability function R(t) represents the probability that the time to failure T is greater than a given value t. It can be calculated as the complement of the CDF:

R(t) = 1 - F(t)

For 0 < t ≤ a, we have:

[tex]R(t) = 1 - (t/a)[/tex]

[tex]F (or) t < = 0, R(t) = 1, and f(or) t > a, R(t) = 0.[/tex]

(c) The hazard rate h(t) represents the instantaneous rate of failure at time t. It can be calculated as the ratio of the PDF to the reliability function:

[tex]h(t) = f(t) / R(t)[/tex]

For a uniform distribution, the PDF is constant, so:

[tex]h(t) = (1/a) / (1 - (t/a)) = 1 / (a - t)[/tex]

The hazard rate h(t) for a uniform distribution is a decreasing function.

(d) The mean time to failure (MTTF) represents the average time until failure occurs. For a uniform distribution, the MTTF can be calculated as the average of the lower and upper bounds:

[tex]MTTF = (0 + a) / 2 = a / 2[/tex]

The median time to failure (t_median) is the value of t such that the probability of failure is 0.5. For a uniform distribution, the median time is simply the midpoint between the lower and upper bounds:

[tex]t_median = (0 + a) / 2 = a / 2[/tex]

(e) To find the conditional probability[tex]p(T > 1/2a + 1/4a | T > 1/2a)[/tex], we need to calculate the probability that T is greater than 1/2a + 1/4a, given that it is already known to be greater than 1/2a.

Since T follows a uniform distribution, the conditional probability is determined by the relative lengths of the intervals. The interval (1/2a + 1/4a, a] represents the remaining possible values of T when T > 1/2a. The interval (1/2a, a] represents the possible values of T when T > 1/2a + 1/4a.

Therefore, [tex]p(T > 1/2a + 1/4a | T > 1/2a) = length of interval (1/2a + 1/4a, a] / length of interval (1/2a, a] = (a - (1/2a + 1/4a)) / (a - 1/2a) = (3/4).[/tex]

(f) The uniform distribution does not have the memoryless property. The memoryless property states that the conditional probability of an event occurring in the future does not depend on how much time has already passed.

In the case of a uniform distribution, knowing that T > t does affect the probability of T > t + Δ for any positive value Δ. The uniform distribution does not exhibit the memoryless property because the remaining time to failure is dependent on the time already passed.

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If every individual in a population has an equal chance of being selected for a sample, the sample is said to be a "random sample" or a "simple random sample."

A random sample is a sampling technique used in statistics to gather data from a population. It is considered one of the most unbiased and reliable methods for selecting a sample. In a random sample, each individual in the population has an equal probability of being selected, regardless of their characteristics or attributes.

This equal chance of selection eliminates any systematic bias and ensures that the sample is representative of the population.

The process of obtaining a random sample involves assigning a unique identifier to each individual in the population and using a randomization method to select the desired number of individuals.

Random sampling can be conducted through various techniques, such as simple random sampling, where individuals are selected directly from the population using a random number generator or a randomization table.

The goal of a random sample is to reduce the potential for selection bias and increase the generalizability of the findings to the larger population. By giving every individual an equal opportunity to be included in the sample, researchers can make more accurate inferences and draw conclusions about the population as a whole.

This type of sampling method ensures that each member of the population has an equal opportunity to be included in the sample, making it a representative subset of the population.

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(X+3)(5-x)≤0

Therefore, collecting like terms,

x+3 ≤ 0

x ≤ -3

OR 5 - × ≤ 0

-x ≤ -5

Dividing both sides by -1,

x ≥ 5

The area of the trapezoid is 5610 square meters and the radius of the circle is approximately 24 feet.

To find the area of a trapezoid, we can use the formula:

Area = (base1 + base2) × height / 2

Given: Height = 51 meters, Base1 = 43 meters, Base2 = 67 meters

Plugging in the values into the formula:

Area = (43 + 67) × 51 / 2

Calculating this expression:

Area = 110 × 51 / 2

Area = 5610 square meters

Therefore, the area of the trapezoid is 5610 square meters.

To find the radius of a circle when the circumference is known, we can use the formula:

Circumference = 2πr

Given: Circumference = 150.72 feet

Plugging in the value into the formula:

150.72 = 2πr

Dividing both sides of the equation by 2π:

r = 150.72 / (2π)

Calculating this expression:

r ≈ 150.72 / (2 × 3.14159)

r ≈ 24 feet

Therefore, the radius of the circle is approximately 24 feet.

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Answer:

20

Step-by-step explanation:

let the missing number = x,

6:15 = 8:x

6/15 = 8/x

6x = 8*15

x = 120/6

x = 20

The minute hand of the analog clock turns 5400 degrees from 17:50 on Monday to 08:50 on Tuesday.

To calculate the number of degrees the minute hand of an analog clock turns between two given times, we need to determine the elapsed time between those two times and then convert it into degrees.

From 17:50 on Monday to 08:50 on Tuesday, we can break it down into the following steps:

Calculate the number of minutes from 17:50 to 00:00 (midnight).

Monday has 24 hours, and from 17:50 to 00:00, it's 6 hours and 10 minutes (since 60 minutes minus 50 minutes).

So, the total minutes from 17:50 to 00:00 are

6 * 60 + 10 = 370 minutes.

Calculate the number of minutes from 00:00 (midnight) to 08:50 on Tuesday.

Tuesday has 24 hours, and from 00:00 to 08:50, it's 8 hours and 50 minutes.

So, the total minutes from 00:00 to 08:50 are

8 * 60 + 50 = 530 minutes.

Add up the total minutes from step 1 and step 2.

370 minutes (from 17:50 to midnight) + 530 minutes (from midnight to 08:50) = 900 minutes.

Now, let's calculate the number of degrees the minute hand turns in 900 minutes:

An analog clock has 60 minutes on the dial, which corresponds to 360 degrees (a complete circle).

Therefore, in one minute, the minute hand turns 360/60 = 6 degrees.

The number of degrees the minute hand turns in 900 minutes is:

900 minutes * 6 degrees/minute = 5400 degrees.

So, the minute hand of the analog clock turns 5400 degrees from 17:50 on Monday to 08:50 on Tuesday.

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Answer:

The constant of proportionality is the value by which one variable is multiplied to obtain the other variable in a proportional relationship. For example, if we have two variables, x and y, that are proportional, we can write the relationship as y=kx, where k is the constant of proportionality.

Step-by-step explanation:

Answer:

[tex]y=\frac{17}{13} e^{-x}+\frac{48}{13} e^{\frac{1}{2}x }\cos(x)+\frac{6}{13} e^{\frac{1}{2}x }\sin(x)[/tex]

Step-by-step explanation:

Given the third-order differential equation with initial conditions.

[tex]4y'''+y'+5y=0; \ y(0)=5, \ y'(0)=1, \ y''(0)=-1[/tex]

(1) - Find the characteristic equation

[tex]4y'''+y'+5y=0\\\\\Longrightarrow 4m^3+0m^2+m+5=0\\\\\Longrightarrow \boxed{4m^3+m+5=0}[/tex]

(2) - Solve the characteristic equation for "m." First using the rational root theorem

[tex]4m^3+m+5=0\\\\\rightarrow \frac{p}{q} = \pm \frac{1,5}{1,2,4} \\\\\text{\underline{Trying m=-1 first: }}\\\\ \begin{array}{c|rrrr}\vphantom{\dfrac12}-1 & 4 & 0 & 1 & 5\\\cline{1-1} & \downarrow & -4 & 4 & -5\\\cline{2-5} & 4 & -4 & 5 & 0\end{array}\\\\\text{The remainder is 0} \ \therefore \boxed{m=-1} \ \text{is a root.}\\\\\text{Now we are left with}\rightarrow \boxed{4m^2-4m+5=0} \ \text{Use the quadratic equation}[/tex]

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{The Quadratic Equation:}}\\ax^2+bx+c=0\\\\x=\frac{-b \pm\sqrt{b^2-4ac} }{2a} \end{array}\right}[/tex]

[tex]4m^2-4m+5=0\\\\\Longrightarrow m=\frac{4 \pm\sqrt{(-4)^2-4(4)(5)} }{2(4)} \\\\\Longrightarrow m=\frac{4 \pm\sqrt{-64} }{8}\\\\\Longrightarrow m=\frac{4}{8} \pm \frac{8i}{8} \\\\\Longrightarrow \boxed{m=\frac{1}{2} \pm i}[/tex]

Thus, we have found three roots.

[tex]m=-1, \frac{1}{2} \pm i[/tex]

(3) - Form the solution.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Solutions to Higher-order DE's:}}\\\\\text{Real,distinct roots} \rightarrow y=c_1e^{m_1t}+c_2e^{m_2t}+...+c_ne^{m_nt}\\\\ \text{Duplicate roots} \rightarrow y=c_1e^{mt}+c_2te^{mt}+...+c_nt^ne^{mt}\\\\ \text{Complex roots} \rightarrow y=c_1e^{\alpha t}\cos(\beta t)+c_2e^{\alpha t}\sin(\beta t)+... \ ;m=\alpha \pm \beta i\end{array}\right}[/tex]

Notice that we have one real, distinct root and complex roots. Thus, we can form the solution in the following manner.

[tex]\therefore \text{the general solution}\rightarrow\boxed{\boxed{y=c_1e^{-x}+c_2e^{\frac{1}{2}x }\cos(x)+c_3e^{\frac{1}{2}x }\sin(x)}}[/tex]

(4) - Use the given initial conditions to find the arbitrary constants "c_1," "c_2," and "c_3"

[tex]\text{Recall...} \ y(0)=5, \ y'(0)=1, \ y''(0)=-1[/tex]

Take two derivatives of the general solution.

[tex]y=c_1e^{-x}+c_2e^{\frac{1}{2}x }\cos(x)+c_3e^{\frac{1}{2}x }\sin(x)\\\\\\ \Rightarrow y'=-c_1e^{-x}+c_2\frac{1}{2}e^{\frac{1}{2}x}\cos(x)-c_2e^{\frac{1}{2}x}\sin(x)+c_3e^{\frac{1}{2}x }\cos(x)+c_3\frac{1}{2}e^{\frac{1}{2}x }\sin(x) \\\\\Longrightarrow \boxed{y'=-c_1e^{-x}+(c_3+\frac{1}{2} c_2)e^{\frac{1}{2}x}\cos(x)+(-c_2+\frac{1}{2} c_3)\frac{3}{2}e^{\frac{1}{2}x }\sin(x)}\\\\\\[/tex]

[tex]\Rightarrow y''=c_1e^{-x}+\frac{1}{2} (c_3+\frac{1}{2} c_2)e^{\frac{1}{2}x }\cos(x)+(c_3+\frac{1}{2} c_2) e^{\frac{1}{2}x}\sin{x}+(-c_2+\frac{1}{2} c_3)e^{\frac{1}{2}x } \cos(x)+\frac{1}{2} (-c_2+\frac{1}{2} c_3)e^{\frac{1}{2}x }\sin(x)\\\\\Longrightarrow \boxed{y''=c_1e^{-x}+(-\frac{3}{4} c_2+c_3)e^{\frac{1}{2}x }\cos(x)+(- c_2-\frac{3}{4}c_3) c_3e^{\frac{1}{2}x }\sin(x)}[/tex]

Plug in the initial conditions and form a system of equations.

[tex]\left\{\begin{array}{ccc}5=c_1+c_2\\1=-c_1+\frac{1}{2}c_2+c_3 \\-1=c_1-\frac{3}{4}c_2+c_3 \end{array}\right[/tex]

Creating a matrix and using a calculator to row-reduce,

[tex]\Longrightarrow\left[\begin{array}{ccc}1&1&0\\-1&\frac{1}{2}&1\\1&-\frac{3}{4}&1\end{array}\right] =\left[\begin{array}{ccc}5\\1\\-1\end{array}\right] \\\\\Longrightarrow \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] =\left[\begin{array}{ccc}\frac{17}{13} \\\frac{48}{13} \\\frac{6}{13} \end{array}\right] \\\\\therefore \boxed{c_1=\frac{17}{13} , c_2=\frac{48}{13} , \ and \ c_3=\frac{6}{13} }[/tex]

(5) - Thus, the given differential equation is solved with the given initial conditions

[tex]\boxed{\boxed{y=\frac{17}{13} e^{-x}+\frac{48}{13} e^{\frac{1}{2}x }\cos(x)+\frac{6}{13} e^{\frac{1}{2}x }\sin(x)}}[/tex]

The speed of the bug clinging to the rim of the disk is 1 m/s, and the magnitude of its acceleration is 10 m/[tex]s^2[/tex].

The speed of an object moving in a circular path can be calculated using the formula v = ω * r, where v is the speed, ω (omega) is the angular speed, and r is the radius of the circular path. In this case, the radius of the disk is 0.1 m, and the angular speed is 10 rad/sec. Plugging these values into the formula, we get v = 10 rad/sec * 0.1 m = 1 m/s.

The acceleration of an object moving in a circular path is given by the formula a = [tex]ω^2[/tex] * r, where a is the acceleration, ω (omega) is the angular speed, and r is the radius of the circular path. Substituting the given values, we have a = [tex](10 rad/sec)^2[/tex]* 0.1 m = 100 m/[tex]s^2[/tex].

Therefore, the bug's speed is 1 m/s and the magnitude of its acceleration is 10 m/[tex]s^2[/tex].

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Answer:

2x(3+7)8 divided by 5-9

2x( 10)8 divided by 5-9

   (20)1.6-9

       32-9

        23  

Step-by-step explanation:

PEMDAS

P- Parentheses

E- Exponents

M- Multiplication

D- Division

A- Addition

S- Subtraction

Hope this helped