a sphere is inscribed in a unit cube. a smaller cube is then inscribed within the sphere. what is the side length of the smaller cube?

a) The logistic equation for this problem is L dP/dt = P(1 - P/L), where L = 2000 and Po = 100.

b)  The maximum reproduction rate is 0.03465 times the current population.

a. The logistic equation for this problem is:

L dP/dt = P(1 - P/L)

where L is the carrying capacity of the lake, P is the current population, and dP/dt is the rate of change of the population over time.

We know that at t = 0, P = 100, and one year later at t = 1, P = 200. So we can use this information to find k, which is the growth rate coefficient:

P(t) = L / (1 + (L / Po - 1) * exp(-kt))

200 = L / (1 + (L / 100 - 1) * exp(-k))

200 = L / (1 + (L - 100) * exp(-k))

200 + 200L - 20000 = L * (1 + (L - 100) * exp(-k))

200L -[tex]L^2[/tex] * exp(-k) + 200L * exp(-k) - 10000 * exp(-k) = 0

[tex]L^2[/tex] - 400L + 5000 = (L - 200)^2 - 30000

[tex](L - 200)^2[/tex] = 35000

L = 200 + sqrt(35000) ≈ 223.6

So L ≈ 223.6, and we can use this to find k:

2000 = 223.6 / (1 + (223.6 / 100 - 1) * exp(-k))

20000 + 2000L - 2236 = L * (1 + (L - 100) * exp(-k))

2236 - [tex]L^2[/tex]  * exp(-k) + 2236 * exp(-k) - 100 * exp(-k) = 0

[tex]L^2[/tex] - 4472L + 220000 = 0

(L - 2000)(L - 100) = 0

So either L = 2000 or L = 100. We know that L ≠ 100, since we know that the population stabilizes at 2000 after a long time. Therefore, L = 2000, and we can solve for k:

k = -ln((L / Po - 1) / (1 + (L / Po - 1))) / t

k = -ln((2000 / 100 - 1) / (1 + (2000 / 100 - 1))) / 1

k ≈ 0.0693

Therefore, the logistic equation for this problem is:

L dP/dt = P(1 - P/L)

dP/dt = 0.0693P(1 - P/2000)

b. The maximum reproduction rate occurs when the population is halfway to the carrying capacity, or P = L/2. At this point, the equation becomes:

dP/dt = 0.0693P(1 - 0.5)

dP/dt = 0.03465P

Therefore, the maximum reproduction rate is 0.03465 times the current population. For example, if the current population is 1000, the maximum reproduction rate is 34.65 fish per year.

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Full Question : Logistic Equation: L dP dt P(1-2). P() = L - Po Po 1+ 4e ki Where A 1. Fish population in a lake grows according to the logistic law. The initial population of 100 fish and year later it was 200. After a long time fish population stabilized at 2000.

a. Write down the logistic equation for this problem.

b. What is the maximum reproduction rate (fish/year)?

a) from 2 to infinity (∞). (b) from 2 to 4. (c) from 0 to infinity (∞). (d) from 0 to infinity (∞), and subtract the square of the mean from the result. (e) from 0 to the variable. (f) CDF of 0.5 (g) CDF of 0.6

To find the probability that a pump lasts more than two years, you need to integrate the probability density function (PDF) from 2 to infinity (∞). To find the probability that a pump lasts between two and four years, integrate the PDF from 2 to 4. The mean lifetime can be found by taking the expected value of the random variable, which is the integral of the product of the variable and the PDF, from 0 to infinity (∞).

The variance of the lifetimes can be calculated by taking the expected value of the square of the variable, minus the square of the mean. Integrate the product of the square of the variable and the PDF, from 0 to infinity (∞), and subtract the square of the mean from the result. The cumulative distribution function (CDF) of the lifetime can be found by integrating the PDF from 0 to the variable.

This represents the probability that the lifetime is less than or equal to a specific value. The median lifetime can be found by solving for the value of the variable that corresponds to a CDF of 0.5. To find the 60th percentile of lifetimes, solve for the value of the variable that corresponds to a CDF of 0.6.

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(a) cos t = sin t, Step 1: Convert the equation into a single trigonometric function.
sin t = cos (90° - t)

Step 2: Substitute the given equation.
cos t = cos (90° - t)

Step 3: Find the general solution.
t = 90° - t + 360°k, where k is an integer.

Step 4: Solve for t.
2t = 90° + 360°k

Step 5: Find the solutions between 360° and 720°, inclusive.
For k=0: t = 45° (not in the given interval)
For k=1: t = 405° (within the given interval)
For k=2: t = 765° (not in the given interval)

Answer: t = 405°

(b) ta t = -4.3315

(c) sin t = -0.9397

Step 1: Find the reference angle.
t_ref = arcsin(0.9397) ≈ 70°

Step 2: Find the general solution.
t = 180°k - 70°, where k is an integer.

Step 3: Find the solutions between 360° and 720°, inclusive.
For k=2: t = 290° (not in the given interval)
For k=3: t = 470° (within the given interval)
For k=4: t = 650° (within the given interval)

Answer: t = 470°, 650°

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To make the function continuous on the interval (-infinity,infinity), we need to make sure that the two expressions for f(x) match at x=7. In other words, we need to have:

lim as x approaches 7 from the left of f(x) = lim as x approaches 7 from the right of f(x)

Using the given expressions for f(x), we can calculate these limits as:

lim as x approaches 7^- of f(x) = lim as x approaches 7^- of (cx+8) = 7c + 8
lim as x approaches 7^+ of f(x) = lim as x approaches 7^+ of (cx^2-8) = c(7^2)-8 = 49c - 8

Setting these equal to each other and solving for c, we get:

7c + 8 = 49c - 8
56c = 16
c = 2/7

Therefore, the value of the constant c that makes the function continuous on (-infinity,infinity) is c = 2/7.

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The proof is completed, and we have shown that for any odd integer N, ⌈N/24⌉ = N/2 + 3

Here's a step-by-step proof that for any odd integer N:

⌈N/24⌉ = N/2 + 3/4

Proof:

Step 1: Assume N is an odd integer.

Let N be an arbitrary odd integer. This means N can be expressed as N = 2k + 1, where k is an integer.

Step 2: Substitute N with 2k + 1 in the left-hand side (LHS) of the equation.

⌈(2k + 1)/24⌉

Step 3: Simplify the expression inside the ceiling function.

Since 2k + 1 is odd, we can rewrite it as 2k + 1 = 2k + 3/4 - 1/4

Step 4: Apply the properties of the ceiling function.

The ceiling of a sum is equal to the sum of the ceilings of the individual terms.

⌈(2k + 3/4 - 1/4)/24⌉ = ⌈(2k + 3/4)/24⌉ + ⌈(-1/4)/24⌉

Step 5: Simplify the expression inside the first ceiling function.

Since 2k + 3/4 is a positive number, we can write it as ⌈(2k + 3/4)/24⌉ = (2k + 3)/96 + 1 if (2k + 3)/4 is not an integer, or ⌈(2k + 3/4)/24⌉ = (2k + 3)/96 if (2k + 3)/4 is an integer.

Step 6: Simplify the expression inside the second ceiling function.

⌈(-1/4)/24⌉ = ⌈-1/96⌉

Step 7: Apply the ceiling function to the negative fraction.

Since -1/96 is a negative fraction but greater than -1, the ceiling function of -1/96 is -1.

⌈-1/96⌉ = -1

Step 8: Substitute back the simplified expressions into the original equation.

⌈(2k + 3/4)/24⌉ + ⌈(-1/4)/24⌉ = (2k + 3)/96 + 1 + (-1) = (2k + 3)/96

Step 9: Substitute back N with 2k + 1.

(2k + 3)/96 = (2(2k + 1) + 3)/96 = (4k + 2 + 3)/96 = (4k + 5)/96 = k + 1/24

Step 10: Apply the ceiling function to the simplified expression.

The expression k + 1/24 is a positive number, so the ceiling function of k + 1/24 is equal to k + 1.

⌈k + 1/24⌉ = k + 1

Step 11: Substitute back k with (N - 1)/2.

k + 1 = ((N - 1)/2) + 1 = (N - 1)/2 + 2/2 = (N + 1)/2

Step 12: Conclusion.

⌈N/24⌉ = (N + 1)/2

So, the proof is completed, and we have shown that for any odd integer N, ⌈N/24⌉ = N/2 + 3

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When the radius is 1 in, the diameter of the basketball is increasing at a rate of 3/π in/sec.

To find the rate of change of the diameter, we need to use the chain rule of differentiation.

Let's start by finding the formula for the diameter of a sphere in terms of its radius. The diameter (d) is twice the radius (r), so we have:

d = 2r

Now, we can take the derivative of both sides with respect to time (t), using the chain rule:

d/dt (d) = d/dt (2r)

The derivative of the diameter with respect to time (d/dt (d)) is the rate of change we're looking for. The derivative of 2r with respect to time is:

d/dt (2r) = 2 (d/dt (r))

So, we have:

d/dt (d) = 2 (d/dt (r))

We know that the rate of change of the radius (d/dt (r)) is given as 6 inº/sec, but we need to find it when the radius is 1 in. We can substitute these values into the equation to get:

d/dt (d) = 2 (d/dt (r)) = 2 (6 inº/sec) = 12 inº/sec

Therefore, when the radius is 1 in, the diameter of the basketball is increasing at a rate of 12 inº/sec.

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The distance between the centers of circles M and N is 8.

To find the distance between the centers of circles M and N, we can use the properties of intersecting circles and the given information. Since the circles intersect at points A and B with AB = 6, they form an isosceles triangle with AM = BM = 5 (radius of circle M) and AN = BN = 5 (radius of circle N).

Let's call the intersection of the line segment AB with the line segment MN as point P. Triangles AMP and BNP are congruent right triangles (by SAS congruence) with a right angle at P. Using the Pythagorean theorem on one of these triangles, let's say triangle AMP, we can find the length of AP:

AP^2 + MP^2 = AM^2
AP^2 + 3^2 = 5^2 (since AB = 6, and by symmetry, MP = 3)
AP^2 = 25 - 9
AP = √16 = 4

Since AP is half of the length of AB, the distance from the center of M to the center of N is twice the length of AP:

MN = 2 * AP = 2 * 4 = 8

The distance between the centers of circles M and N is 8.

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The probability that all of the first 4 speakers will be history majors is 1/7 or approximately 0.143.

Explain probability

Probability is a mathematical concept used to determine the likelihood of an event occurring. It is represented by a number between 0 and 1, where 0 means the event is impossible and 1 means it is certain. Probability theory is widely applied in fields like science, engineering, and finance to analyze and predict the outcomes of random events.

According to the given information

The total number of ways to choose 4 students from 7 is given by the combination formula:

C(7, 4) = 7! / (4! * 3!) = 35

Out of the 7 candidates, 5 are history majors. So, the number of ways to choose all 4 candidates to be history majors is given by the combination formula:

C(5, 4) = 5! / (4! * 1!) = 5

Therefore, the probability of selecting 4 history majors out of 4 speakers is:

P = 5/35 = 1/7

So the probability that all of the first 4 speakers will be history majors is 1/7 or approximately 0.143.

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The question that is a statistical question is option B: How many students in a middle school class like each type of food? A statistical question is a question that can be answered by collecting and analyzing data. Option B is asking for data on the number of students who like different types of food, which can be collected and analyzed to provide an answer.

Options A, C, and D are not statistical questions. Option A asks for a list of students who can speak another language, which is not a question that requires data analysis. Option C asks for a specific piece of information (which classes the principal is visiting), but it does not involve collecting and analyzing data. Option D asks for a single number (the number of students in a middle school), which does not involve data analysis either.

*IG:whis.sama_ent

In this case the base is going to be the trapezoid (Do you see why? It is a trapezoidal prism.)

The area of the trapezoid is:

[tex]\frac{b_{1}+b_{2}}{2}*h=\frac{4+10}{2}*8=56[/tex]

At 9:00, the distance between the tips of the hands is changing at a rate of 0.00 m/min (rounded to two decimal places).

To solve this problem, we can use the law of cosines to relate the angle between the two clock hands, e, and the distance between the tips of the two hands, c. The law of cosines states that:

[tex]c^2 = a^2 + b^2 - 2abcos(e)[/tex]

Where a and b are the lengths of the clock hands and c is the distance between the tips of the hands.

At 9:00, the hour hand is pointing directly at the 9 and the minute hand is pointing directly at the 12. This means that the angle between the hands is:

e = 90 degrees

Substituting this into the law of cosines, we get:

[tex]c^2 = 2.5^2 + 2^2 - 2(2.5)(2)cos(90)\\c^2 = 6.25 + 4 - 0\\c^2 = 10.25[/tex]

c = sqrt(10.25)
c = 3.2 meters (approximately)

Now we can differentiate both sides of the equation with respect to time, t:

2c(dc/dt) = 2a(da/dt) + 2b(db/dt) - 2ab(sin(e))(de/dt)

At 9:00, we know that da/dt = 0 and db/dt = 0, since the lengths of the clock hands are not changing. We also know that sin(90) = 1. Substituting these values and solving for dc/dt, we get:

2(3.2)(dc/dt) = 2(2.5)(0) + 2(2)(0) - 2(2.5)(2)(1)(de/dt)
6.4(dc/dt) = -10(de/dt)
dc/dt = -10/6.4
dc/dt = -1.56 meters per hour (approximately)

Therefore, the distance between the tips of the clock hands is changing at a rate of approximately -1.56 meters per hour (or about 1.56 meters per hour in the opposite direction).
At 9:00, the hour hand is at the 9 on the clock face, and the minute hand is at the 12. Let the length of the hour hand be 2.5m (A) and the length of the minute hand be 2m (B). We want to find the rate at which the distance between the tips of the hands (C) is changing with respect to time (t).

Using the Law of Cosines, we can write an equation relating the angle between the clock hands (θ) and the distance between the tips of the hands (C):

C² = A² + B² - 2AB * cos(θ)

At 9:00, the angle θ is 90 degrees, so cos(θ) = 0.

C² = (2.5)² + (2)² - 2(2.5)(2)(0) = 6.25 + 4

Now we differentiate both sides of the equation with respect to t:

2C * dC/dt = 0

Since we want to find dC/dt at 9:00, we need to calculate C first:

C = √(6.25 + 4) = √10.25 ≈ 3.20m

Now, using the derived equation:

2(3.20) * dC/dt = 0

dC/dt = 0 m/min

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a) The probability of all 3 people being born on Tuesday is 1/343, calculated by multiplying the probability of each person being born on Tuesday (1/7) together.

b) The probability of all 3 people being born on different days of the week is 30/343, calculated by multiplying the probability of each person being born on a different day of the week (7/7, 6/7, 5/7) together.

a) To find the probability that all 3 people are born on Tuesday, we need to first determine the probability that one person is born on Tuesday, which is 1/7 (assuming all days of the week are equally likely). Since each person's birthday is independent of the others, the probability that a second person is also born on Tuesday is also 1/7, and the probability that a third person is born on Tuesday is also 1/7. To find the probability that all three people are born on Tuesday, we multiply the probabilities together: (1/7) x (1/7) x (1/7) = 1/343. So the probability that all 3 people are born on Tuesday is 1/343.

b) To find the probability that all 3 people are born on a different day of the week, we need to first determine the probability that one person is born on any given day of the week, which is 1/7. Once one person's birthday has been determined, the probability that the second person is born on a different day of the week is 6/7 (since there are only 6 other days of the week to choose from). Similarly, the probability that the third person is born on a different day of the week from the first two is 5/7. To find the probability that all three people are born on different days of the week, we multiply the probabilities together: (1/7) x (6/7) x (5/7) = 30/343. So the probability that all 3 people are born on different days of the week is 30/343.

a) To find the probability that all 3 people are born on a Tuesday, you simply calculate the probability of each person being born on a Tuesday and then multiply these probabilities together. Since there are 7 days in a week, the probability of being born on any specific day (including Tuesday) is 1/7. So, the probability of all 3 being born on Tuesday is:

(1/7) x (1/7) x (1/7) = 1/343

b) To find the probability that all 3 people are born on different days of the week, first consider the probability for each person. The first person can be born on any day (7/7 chance). For the second person, they have a 6/7 chance of being born on a different day than the first person. Finally, the third person has a 5/7 chance of being born on a different day than the first two. Multiply these probabilities together:

(7/7) x (6/7) x (5/7) = 210/343

So, the probability of all 3 people being born on different days of the week is 210/343.

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The answer is 12×(114)=3960

The number (125)

is the number of ways of choosing groups of 5

people from a pool of 12

You can choose the leader first (12

possibilities) and the remaining team after ((114)

possibilities).

Thereby, the answer is 12×(114)=3960

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To find the indefinite integral of the given function, we need to use the method of partial fractions. First, we factor the denominator:

x-5 = 0
x = 5

Therefore, x-5 is a linear factor, and we can write:

(x^3 - 5x^2 + 2) / (x-5) = Ax^2 + Bx + C + D/(x-5)

where A, B, C, and D are constants that we need to find. To do this, we can multiply both sides by (x-5) and then substitute x=5 to get:

A(5)^2 + B(5) + C + D/(5-5) = 5^3 - 5(5)^2 + 2

This simplifies to:

25A + 5B + C = 108

Next, we can differentiate both sides of the partial fraction equation to get:

x^3 - 5x^2 + 2 = (Ax^2 + Bx + C)(x-5) + D

Expanding and equating coefficients, we get:

A = 1, B = -10, C = 23, D = -113

Therefore, we can write:

∫x^3−5x^2+2/(x−5) dx = ∫(x^2 - 10x + 23) dx - ∫(113/(x-5)) dx

The first integral can be evaluated using the power rule:

∫(x^2 - 10x + 23) dx = (1/3)x^3 - 5x^2 + 23x + C1

where C1 is the constant of integration.

For the second integral, we need to use the logarithmic rule:

∫(113/(x-5)) dx = 113 ln|x-5| + C2

where C2 is the constant of integration. Note that we need to include the absolute value of x-5 to account for the fact that the denominator can be negative for some values of x.

Therefore, the final answer is:

∫x^3−5x^2+2/(x−5) dx = (1/3)x^3 - 5x^2 + 23x + 113 ln|x-5| + C

where C is the constant of integration.

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Answer:

B

Step-by-step explanation:

So in this question, we have two forces and two velocities. We're asked to find the second velocity. From what we know

F₁ = 16N

V₁ = 2m/s

F₂ = 10N

V₂ = ?

  Solution

when F₁ was 16N, V₁ was 2m/s. All that's left is to find the second velocity:

   16N = 2m/s

   10N = V₂

⇒ V₂ = [tex]\frac{2 m/s * 10N}{16N}[/tex]

         = [tex]\frac{20m/s}{16}[/tex]

         = 1.25 m/s

So now when we compare V₁ and V₂, we should get the answer      

         [tex]\frac{V1}{V2} = \frac{2m/s}{1.25m/s}[/tex]  = 1.6

V₁ is 1.6 bigger than V₂. This means the speed will decrease

margin of error = z* * sqrt(pq/n) = 1.55 * sqrt(0.150.85/280) ≈ 0.056

Rounded to 3 decimals, the margin of error is 0.056.

The 88% confidence interval for the percent of men in the general population who are color blind is:

p ± margin of error = 0.15 ± 0.056

Lower Bound: 0.15 - 0.056 = 0.094

Upper Bound: 0.15 + 0.056 = 0.206

Rounded to 3 decimals, the 88% confidence interval is (0.094, 0.206).

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Using this expression Rn = (4/n) Σi=0n-1 [(1 + Δx)/Δx]^(-2i-2), we can now compute the Riemann sum for different values of n. As n gets larger, the Riemann sum gets closer to the exact value of the integral.

To use the definition of the definite integral to evaluate the given integral ∫31(2x^- 2)dx using a right-endpoint approximation to generate the Riemann sum, we first need to split the interval [3,1] into smaller subintervals. Let's choose n subintervals of equal width, where n is a positive integer. Then, the width of each subinterval will be Δx = (3-1)/n = 2/n.
Next, we need to choose the right endpoint of each subinterval as the sample point to evaluate the function. Therefore, the i-th sample point will be xi = 1 + iΔx, where i = 0, 1, 2, ..., n-1.Using these sample points, the Riemann sum for the given integral is given by:
Rn = Σi=0n-1 f(xi)Δx
where f(x) = 2x^-2 is the integrand.
Substituting the expressions for xi and Δx, we get:
Rn = Σi=0n-1 f(1 + iΔx)Δx
Rn = Σi=0n-1 (2/(1 + iΔx)^2)(2/n)
Now, we can simplify this expression using the formula for the sum of a geometric series:
Σi=0n-1 r^i = (1 - r^n)/(1 - r)
where r is the common ratio.
In our case, the common ratio is (1 + Δx)/Δx, so we have:
Rn = (4/n) Σi=0n-1 [(1 + Δx)/Δx]^(-2i-2)

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the smallest one of the smallest one of the reasons you have any ideas for dinner tonight

The value of the limit [tex]\lim _{h \rightarrow 0} \frac{(8+h)^3-512}{h}[/tex] is 192.

Start with the given expression:

[tex] \frac{(8+h)^3-512}{h}[/tex]

To estimate the limit as h approaches 0, substitute smaller and smaller values of h into the expression and observe the results.
For example, let's try h = 0.1, 0.01, and 0.001:

- When h = 0.1:

[tex]\frac{(8+0.1)^3-512}{0.1}=194.41[/tex]

- When h = 0.01:

[tex]\frac{(8+0.01)^3-512}{0.01}=192.2401[/tex]

- When h = 0.001:

[tex]\frac{(8+0.001)^3-512}{0.001}=192.02400[/tex]

Observe that as h gets smaller and smaller, the result of the expression approaches a value around 192.
So, based on our estimations, the limit as h approaches 0 is approximately 192.

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Relational databases are heavily based on the mathematical concept of: Set Theory. The correct option is (A).

Relational databases are based on the principles of set theory, which deals with sets of elements and their relationships with each other. In a relational database, data is organized into tables, with each table representing a set of related data.

The tables are then related to each other through the use of keys, which allow for the establishment of relationships between different sets of data. The principles of set theory also govern the use of operations such as union, intersection, and difference, which can be used to manipulate the data in the tables.

Therefore, the mathematical concept of set theory is a fundamental part of the design and use of relational databases.

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For the 0.5 mm thickness, the two principal stresses are 66.5 MPa (tension) and 0 MPa (compression), and the maximum shear stress is 33.25 MPa. For the 5 mm thickness, the two principal stresses are 3.5 MPa (tension) and 0 MPa (compression), and the maximum shear stress is 1.75 MPa.

To solve this problem, we can use the formula for the hoop stress in a thin-walled cylinder:
Hoop stress = (pressure x radius) / thickness

(i) For the 0.5 mm thickness:
Radius = (10 mm - 0.5 mm) / 2

= 4.75 mm

Hoop stress = (7 MPa x 4.75 mm) / 0.5 mm

= 66.5 MPa

To find the principal stresses, we can use the following formula:
Principal stress = [tex]\frac{(\text{hoop stress} + \text{axial stress}) }{2} \pm \sqrt{((\text{hoop stress} - \text{axial stress}) / 2)^2 + \text{shear stress}^2}[/tex]

Since the tube is thin-walled, we can assume that the axial stress is negligible.

Therefore.
Principal stress = [tex]\frac{(66.5 \ MPa + 0 \ MPa) }{2} \pm \sqrt{((66.5 \ MPa - 0 \ MPa) / 2)^2 + 0^2 \ MPa}[/tex]
Principal stress = 33.25 MPa ± 33.25 MPa
The two principal stresses are 66.5 MPa (tension) and 0 MPa (compression).

The maximum shear stress is equal to half the difference between the two principal stresses, i.e. 33.25 MPa.

(ii) For the 5 mm thickness:

Radius = (10 mm - 5 mm) / 2

= 2.5 mm

Hoop stress = (7 MPa x 2.5 mm) / 5 mm

= 3.5 MPa

Principal stress = [tex]\frac{(3.5 \ MPa + 0 \ MPa) }{2} \pm \sqrt{((3.5 \ MPa - 0 \ MPa) / 2)^2 + 0^2 \ MPa}[/tex]
Principal stress = 1.75 MPa ± 1.75 MPa
The two principal stresses are 3.5 MPa (tension) and 0 MPa (compression).
The maximum shear stress is equal to half the difference between the two principal stresses, i.e. 1.75 MPa.

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The online poll asking the preferred mobile phone type used by school children is an observational study, not an experimental study, and there is no controlled factor involved.

An online poll asking the preferred mobile phone type used by school children would be considered an observational study. This is because the researcher is not actively manipulating any variables or treatments. Instead, they are simply observing and collecting data on the preferences of school children.
If the study were to be designed as an experiment, the controlled factor would be the type of mobile phone being offered as an option in the poll. For example, the researcher could randomly assign some participants to see only options for Apple iPhones, while others would only see options for Samsung Galaxy phones. By controlling the options presented to participants, the researcher could test whether there is a difference in preference for different types of mobile phones among school children.
However, it is important to note that an experiment of this nature may not be feasible or ethical. It may be difficult to limit the options presented to participants in an online poll, and doing so could potentially bias the results. Additionally, it may not be ethical to limit the options presented to participants, as it could be seen as withholding information or forcing a particular preference on them. Therefore, an observational study would likely be a more appropriate and ethical approach to studying the preferences of school children for mobile phones.

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The extreme values of f subject to both constraints is 1.

To find the extreme values of f subject to both constraints, we can use the method of Lagrange multipliers.

Let L(x,y,z,λ,μ) = f(x,y,z) - λ(g(x,y,z)) - μ(h(x,y,z)), where g(x,y,z) = x^2 + y^2 - 2 and h(x,y,z) = x + y - 1.

Taking partial derivatives and setting them equal to zero, we get the following system of equations:

∂L/∂x = 1 - 2λx - μ = 0

∂L/∂y = 1 - 2λy - μ = 0

∂L/∂z = 1 - μ = 0

g(x,y,z) = x^2 + y^2 - 2 = 0

h(x,y,z) = x + y - 1 = 0

From the third equation, μ = 1. Substituting this into the first two equations, we get:

1 - 2λx - 1 = 0 => λx = 0

1 - 2λy - 1 = 0 => λy = 0

Since λ cannot be zero, we must have x = y = 0.5. Substituting this into h(x,y,z) = 0, we get z = 0.

Therefore, the only critical point of f subject to the constraints is (0.5, 0.5, 0), with a function value of f(0.5, 0.5, 0) = 1.

To determine whether this is a maximum or minimum, we need to check the second partial derivatives.

∂^2L/∂x^2 = -2λ, ∂^2L/∂y^2 = -2λ, ∂^2L/∂z^2 = 0,

∂^2L/∂x∂y = ∂^2L/∂y∂x = 0, ∂^2L/∂x∂z = ∂^2L/∂z∂x = 0, ∂^2L/∂y∂z = ∂^2L/∂z∂y = 0.

At the critical point, λ = -1/2, so ∂^2L/∂x^2 = ∂^2L/∂y^2 = 1, which is positive definite, indicating that this critical point is a minimum of f subject to the constraints.

Therefore, the minimum value of f subject to both constraints is 1, which occurs at (0.5, 0.5, 0).

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Step-by-step explanation:

The ratio of the central angle of an arc to the total angle of the circle, 360°, is proportional to the ratio of the arc length to the circumference of the circle. So, given the central angle, find the circumference using the radius of the circle and make the proportion:

central angle/360 = arc length/circumference

Solve for arc length, and you will get either a major or a minor arc.

If the central angle is greater than or equal to 180°, then it is a major arc.

If the central angle is less than 180°, it is a minor arc, and to find the major arc you will subtract the minor arc from 360°.

By function notation Therefore, f(x)=-1/3x+4/3 is the **correct answer*

A function can be written using symbols using function notation. The value of the function at x is denoted by the symbol f(x). In other words, when x is the input, the function's output is f(x).

As an illustration, the function y = 2x + 1 can be written as f(x) = 2x + 1 in function notation. This indicates that the function returns 2x + 1 as its output when we pass an input of x into it.

Thus, a set of symbols or signs that designate items like phrases, integers, sentences, etc. is known as **function notation. Without a lengthy written description, function notation is a quicker way to describe a function.

A function can be written using symbols using function notation.

The value of the function at x is denoted by the symbol f(x). In other words, when x is the input, the function's output is f(x).

The function denoted by f(y) = -1/3y + 4/3 can be written as f(x) = -1/3x + 4/3 in function notation.

This is so because the independent variable x and the dependent variable y are one and the same.

By tradition, we refer to the function's value when the input is x as f(x).

Therefore, f(x)=-1/3x+4/3 is the **correct answer**.

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The constant c that makes f(x, y) a valid pdf is c = 1/6.

In this problem, we are given two continuous random variables X and Y, with a joint pdf given by f(x, y) = cxy for 0 ≤ x ≤ 2, 0 ≤ y ≤ 2 - x and f(x, y) = 0 otherwise. Our first task is to find the value of the constant c that makes f(x, y) a valid pdf.

To do this, we need to use the fact that the integral of the joint pdf over the entire range of the variables must be equal to 1. That is,

∫∫ f(x,y) dxdy = 1

Integrating f(x, y) over the given range, we get:

[tex]\int_0^2 \int_0^{(2-x)}[/tex] cxy dy dx = 1

Using the limits of integration, we can integrate the inner integral first:

[tex]\int_0^2[/tex]cx/2 (2-x)² dx = 1

Expanding and simplifying, we get:

3 [2x³ - 3x² + 2] from 0 to 2 = 1

Substituting the limits of integration and simplifying, we get:

c/3 [2(2)³ - 3(2)² + 2] - c/3 [2(0)³ - 3(0)² + 2] = 1

Simplifying further, we get:

8c/3 - 2c/3 = 1

6c/3 = 1

c = 1/6

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the answer is x = 11.2.Use a proportion to find the missing side

length in the following similar figures

what is proportion  ?

In mathematics, proportion refers to the relationship between two or more quantities that have the same ratio. In other words, when two ratios are equal, they are said to be in proportion.

In the given question,

In a proportion, the product of the means is equal to the product of the extremes. Therefore, we have:

5 x = 4 x 14

Simplifying the right side, we get:

5x = 56

Dividing both sides by 5, we obtain:

x = 11.2

Therefore, the answer is x = 11.2.

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Ron's average speed in bike race was 12miles/hour.

What is speed?

Speed is distance traversed per unit of time. It determines that how fast an object is moving. It is the scalar quantity so it has the magnitude of the velocity vector. It has no  direction. An object moving with higher speed means an object is moving faster. An object with lower speed means it is moving slower. If the object isn't moving at all, it has zero speed that is in rest.

The average speed of an object that travels a distance d in time t is d/t.

Ron finished a 15-mile bike race in 1.25 hours.

Here d= 15 miles and t= 1.25 hours.                                       15/1.25

so, average speed = 15/1.25                                                   = (15×100)/125

Dividing 15 by 1.25 we get,                                                     =12

                               = 12 miles/ hour.

Hence Ron's average speed in bike race was 12miles/hour.

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Answer: 35 square units.

Step-by-step explanation:

- In this question you need to substitute in the given values to the triangle to fulfill the equation for the area of triangle.

Area of a triangle = [tex]\frac{1}{2}[/tex] x base x height.

The base was 'x', which was 14 units.

The height was 'h', which was given as 5 units.

Substitute these into the equation.

The central bright spot will be surrounded by rings of decreasing intensity as you move away from the center. The pattern will be symmetrical around the central axis.

To answer this question, we first need to calculate the number of Fresnel zones "seen" from point P, which is 6.00m away from the opaque screen. The Fresnel zone formula is as follows:

Number of zones = (π * d^2) / (4 * λ * L)

where:
d = diameter of the circular hole (6.00mm)
λ = wavelength of the light (500nm)
L = distance between the screen and point P (6.00m)

First, let's convert the units of d and λ to meters for consistency:

d = 6.00mm * (1m / 1000mm) = 0.006m
λ = 500nm * (1m / 1,000,000,000nm) = 5 * 10^-7m

Now, we can plug the values into the formula:

Number of zones = (π * (0.006)^2) / (4 * 5 * 10^-7 * 6)
Number of zones ≈ 37.68

Since the number of zones must be an integer, there will be 37 Fresnel zones seen from point P.

To determine if point P will be bright or dark, we need to check if the number of zones is odd or even. In this case, 37 is odd, so point P will be bright.

Roughly, the diffraction pattern on a vertical plane containing P will consist of alternating bright and dark concentric rings. The central bright spot will be surrounded by rings of decreasing intensity as you move away from the center. The pattern will be symmetrical around the central axis.

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