if a sample is very large, it need not be randomly selected. true or false

The rectangle can have P = 20 and L = 6 because P = 2(6) + 2(4) would equal 20.

Here, we have,

given that,

L=6 and A=24

so, we get,

W = 24/6 = 4

The formula for the perimeter of a rectangle is P=2L + 2W.

If the width is W = 4 and the length is L=6, then the perimeter becomes:

P = 2(6) + 2(4)

so, we get,

P = 20

Therefore the answer is 20

The rectangle can have P = 20 and L = 6 because P = 2(6) + 2(4) would equal 20,

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Answer:

The value of the integral is $\boxed{\frac{625}{2}\pi-\frac{15625}{3}}$.

Step-by-step explanation:

To change to cylindrical coordinates, we replace $x$ and $y$ by $r\cos\theta$ and $r\sin\theta$, respectively, and $z$ remains the same. We also need to convert the limits of integration.

The region of integration is the upper half of a sphere of radius 5 centered at the origin, and we can express it as $0\leq \theta\leq 2\pi$, $0\leq r\leq 5$, and $0\leq z\leq \sqrt{25-r^2}$. Thus, we have:

∫

−

5

5

∫

0

25

−

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2

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−

25

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2

−

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2

25

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2

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2

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2

+

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2

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=

∫

0

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∫

0

5

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0

25

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2

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2

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∫

−5

5

​

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0

25−x

2

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25−x

2

−y

2

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25−x

2

−y

2

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x

2

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2

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dzdydx=∫

0

2π

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∫

0

5

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∫

0

25−r

2

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​

r

r

2

​

dzdrdθ

Simplifying the integral and evaluating, we get:

\begin{align*}

\int_0^{2\pi}\int_0^5\int_0^{\sqrt{25-r^2}}r\sqrt{r^2},dz,dr,d\theta &= \int_0^{2\pi}\int_0^5r^3\left[\frac{1}{2}z^2\right]_0^{\sqrt{25-r^2}},dr,d\theta \

&= \int_0^{2\pi}\int_0^5r^3\left(\frac{1}{2}(25-r^2)\right),dr,d\theta \

&= \int_0^{2\pi}\left[\frac{1}{4}r^4-\frac{1}{6}r^6\right]_0^5,d\theta \

&= \int_0^{2\pi}\frac{625}{4}-\frac{3125}{6},d\theta \

&= \frac{625}{2}\pi-\frac{15625}{3}

\end{align*}

Therefore, the value of the integral is $\boxed{\frac{625}{2}\pi-\frac{15625}{3}}$.

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It is not possible to accurately estimate the probability that a person will call when you're thinking of them as it is a subjective experience that cannot be quantified. However, we can consider some general factors that may affect the probability:

Likelihood of thinking of the person: This is highly dependent on individual circumstances and varies greatly between people. Some factors that may increase the likelihood include how close you are to the person, how often you interact with them, and recent events or memories involving them.

Likelihood of the person calling: This also depends on individual circumstances and varies based on factors such as the person's availability, their likelihood of initiating communication, and external factors that may prompt them to call.

Assuming both events are independent, we can estimate the combined probability as the product of the individual probabilities:

P(thinking of a person) * P(person calls)

However, since we cannot accurately estimate these probabilities, any calculated value would be purely speculative.

If the combined events were to occur once, it would not necessarily provide compelling evidence that the event was not merely a chance occurrence. However, if it happened multiple times in a day, the probability of it being a chance occurrence would decrease significantly, and it may be reasonable to suspect that there is some underlying factor influencing the events. However, it is still important to consider that coincidences do happen, and it is possible for unrelated events to occur together multiple times.

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Answer:6 inches

Step-by-step explanation:

Set a consists of e2 and e4, set b consists of e1 and e3, and set c consists of e1, e4, and e5.

In the given sample space with five equally likely experimental outcomes: e1, e2, e3, e4, and e5, we have three sets defined as follows:

a = {e2, e4}

b = {e1, e3}

c = {e1, e4, e5}

Set a consists of outcomes e2 and e4, set b consists of outcomes e1 and e3, and set c consists of outcomes e1, e4, and e5.

These sets represent subsets of the sample space, where each element of the sample space belongs to one or more sets. Set a represents the outcomes where e2 or e4 occur, set b represents the outcomes where e1 or e3 occur, and set c represents the outcomes where e1, e4, or e5 occur.

It's important to note that sets a, b, and c are not mutually exclusive. For example, outcome e1 belongs to both sets b and c.

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To find the horizontal asymptote(s) for the given function, we need to examine the behavior of the function as x approaches positive or negative infinity.

Let's denote the given function as f(x). We are given f(x) = 5x^2 / (6x - 8).

To find the horizontal asymptote(s), we can take the limit of the function as x approaches positive or negative infinity.

As x approaches positive infinity (x → +∞):

Taking the limit of f(x) as x approaches positive infinity:

lim(x → +∞) (5x^2) / (6x - 8)

To determine the horizontal asymptote, we can divide the leading terms of the numerator and denominator by the highest power of x, which in this case is x^2:

lim(x → +∞) (5x^2/x^2) / (6x/x^2 - 8/x^2)

lim(x → +∞) 5 / (6 - 8/x^2)

As x approaches infinity, 1/x^2 approaches 0, so we have:

lim(x → +∞) 5 / (6 - 0)

lim(x → +∞) 5 / 6

Therefore, as x approaches positive infinity, the function f(x) approaches the horizontal asymptote y = 5/6.

As x approaches negative infinity (x → -∞):

Taking the limit of f(x) as x approaches negative infinity:

lim(x → -∞) (5x^2) / (6x - 8)

Again, let's divide the leading terms of the numerator and denominator by x^2:

lim(x → -∞) (5x^2/x^2) / (6x/x^2 - 8/x^2)

lim(x → -∞) 5 / (6 - 8/x^2)

As x approaches negative infinity, 1/x^2 also approaches 0:

lim(x → -∞) 5 / (6 - 0)

lim(x → -∞) 5 / 6

Therefore, as x approaches negative infinity, the function f(x) also approaches the horizontal asymptote y = 5/6.

In conclusion, the given function has a horizontal asymptote at y = 5/6 as x approaches positive or negative infinity

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The values of b for which the given inequality will be satisfied are: b = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, 12}

The given inequality which shows the status of the purchase by Amy and her friends is,

0.89 b + 1.82 ≤ 12.50

where b is the number of burgers they can purchase.

Solving the given inequality we get,

0.89 b + 1.82 - 1.82 ≤ 12.50 - 1.82 [Subtracting 1.82 from both sides]

0.89 b ≤ 10.68

(0.89 b)/0.89 ≤ 10.68/0.89 [Dividing 0.89 with both sides]

b ≤ 12

since b represents the number of burgers so it cannot be negative or fraction.

So the values for which the inequality will be satisfied are: b = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9 , 10, 11, 12}.

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The question is incomplete. Complete question will be -

TRUE, a nonlinear function may contain a product of two variables.

A nonlinear function may contain a product of two variables. In fact, nonlinear functions can have a wide variety of terms, including products, powers, and combinations of variables.

A function is considered nonlinear if it does not satisfy the properties of linearity, which include the property of superposition, homogeneity, and additivity.

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Answer:

$4.65

Step-by-step explanation:

2.34=4.6566 USD

x=1.99 ⋅ 2.34

It would take 4 weeks for the cost of the food delivery service to equal the regular weekly grocery bill. The total cost would amount to $320.

- x represents the number of weeks.

- C represents the cost of the food delivery service.

- G represents the regular weekly grocery bill.

Based on the given information, we can establish the following equations:

- For the food delivery service: C = 80 + 70(x - 1)

- For the regular grocery bill: G = 60

We need to find the number of weeks (x) when the cost of the food delivery service (C) is equal to the regular grocery bill (G).

Setting the equations equal to each other, we have:

80 + 70(x - 1) = 60

Now, let's solve for x:

80 + 70(x - 1) = 60

70(x - 1) = 60 - 80

70(x - 1) = -20

x - 1 = -20/70

x - 1 = -2/7

x = 1 - 2/7

x = 5/7

Since x represents the number of weeks, we round up to the nearest whole number, resulting in x = 1 week.

To find the total cost, we substitute x = 1 into the equation for C:

C = 80 + 70(1 - 1)

C = 80

Therefore, it would take 4 weeks for the cost of the food delivery service to equal the regular weekly grocery bill. The total cost over those 4 weeks would amount to $320.

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The chi-square test-statistic for this data is 5.600.

The chi-square test-statistic measures the discrepancy between the observed frequencies and the expected frequencies.

It is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies.

The formula for each category is (observed - expected)[tex]^2[/tex] / expected. By summing up these values for all categories, we obtain the chi-square test-statistic.

This test-statistic helps determine if there is a significant difference between the observed and expected frequencies, indicating whether the data supports the claim of equal likelihood for all categories.

A larger chi-square value indicates a greater deviation from the expected frequencies.

The chi-square test is used to assess the goodness of fit between observed and expected data, with higher values suggesting a poorer fit. The significance of the test-statistic is evaluated using a chi-square distribution and degrees of freedom, typically determined by the number of categories minus one.

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The population after 11 years will be  56,181.

The rate of increase of the population would be represented with an exponential equation.

An exponential equation can be described as an equation with exponents. The exponent is usually a variable.

The general form of exponential equation is f(x) = [tex]e^{x}[/tex]

Where:

Population after t years = [tex]p(1 + r)^{t}[/tex]

Where:

= [tex]32,000(1 + 0.0525)^{11}[/tex]

= [tex]32,000(1.0525)^{11}[/tex]

= 56,181

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Given that dimensions of the rectangular prism are as follows:

length = 36 cmwidth = 18 cmheight = 6 cm

And the interior of the cereal bowl is a half sphere with a radius of 6 cm.

Let us find the volume of the cereal bowl: Volume of hemisphere =

[tex]2/3 πr³= 2/3 × π × 6³= 2/3 × π × 216= 452.389[/tex]

Volume of hemisphere = 1/2 × 452.389= 226.194 cubic cm

Now, find the volume of 9 bowls as follows:

Volume of 1 bowl = 226.194 cubic cm

Volume of 9 bowls = 9 × 226.194= 2035.746 cubic cm

Now, find the volume of the rectangular prism as follows:

Volume of rectangular prism =

[tex]l × b × h= 36 × 18 × 6= 3888 cubic cm[/tex]

Therefore, comparing the volume of the 9 bowls and the rectangular prism, we haveVolume of 9 bowls =

2035.746 cubic cmVolume of rectangular prism =

3888 cubic cm

Since, 3888 > 2035.746

Therefore, Pam has enough cereal to completely fill 9 bowls.

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We can use the binomial distribution to solve this problem.

Let X be the number of people out of 15 who used an online travel website to book their hotel. Then, X follows a binomial distribution with n = 15 and p = 0.7.

The expected value of X is given by:

E(X) = n × p

Substituting the values given in the problem, we get:

E(X) = 15 × 0.7 = 10.5

Therefore, we would expect 10 people (rounding down 10.5 to the nearest whole person) out of 15 to use an online travel website to book their hotel.

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There would be a total of 28 handshakes between the 3 latecomers and the initial group of 5 members.

To calculate the number of combinations, we use the formula:

C(n, r) = n! / (r!(n-r)!)

where "n" represents the total number of items (in this case, people), and "r" represents the number of items to be chosen (in this case, 2 for a handshake).

Let's apply this formula to our scenario. We have 3 latecomers and 5 initial members. We want to select 2 people to form a handshake. Plugging these values into the combination formula, we get:

C(8, 2) = 8! / (2!(8-2)!)

= 8! / (2!6!)

To simplify the calculation, let's break down the factorial terms:

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

2! = 2 * 1

6! = 6 * 5 * 4 * 3 * 2 * 1

Now we can substitute these factorial terms back into the combination formula:

C(8, 2) = (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) / [(2 * 1) * (6 * 5 * 4 * 3 * 2 * 1)]

Simplifying further:

C(8, 2) = (8 * 7) / (2 * 1)

= 56 / 2

= 28

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From the grid, it appears that the length of XY is approximately 10 units.

To find the length of XY, we need to calculate the distance between the points X and Y on the coordinate grid.

From the grid, we can see that the X-coordinate of point X is 1 and the X-coordinate of point Y is 7.

To calculate the horizontal distance between these two points, we subtract the smaller X-coordinate from the larger one: 7 - 1 = 6 units.

Similarly, the Y-coordinate of point X is 2 and the Y-coordinate of point Y is -6. To calculate the vertical distance between these two points, we subtract the smaller Y-coordinate from the larger one: 2 - (-6) = 8 units.

Using the horizontal and vertical distances, we can apply the Pythagorean theorem to find the length of the line segment XY.

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

In this case, the horizontal distance is 6 units and the vertical distance is 8 units. So, applying the Pythagorean theorem:

Length of XY = √(6^2 + 8^2)

Length of XY = √(36 + 64)

Length of XY = √100

Length of XY = 10 units

Therefore, the length of XY is closest to 10 units.

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The drive-thru is able to estimate their wait time more consistently will be Fast Chicken, because it has a smaller IQR.

In descriptive statistics, the interquartile range tells you the spread of the middle half of the distribution. Quartiles segment any distribution that’s ordered from low to high into four equal parts. The interquartile range (IQR) contains the second and third quartiles, or the middle half of the data set.

The correct option here is Fast Chicken, because it has a smaller IQR (Interquartile Range). IQR is the difference between the third quartile and the first quartile, which is represented by the box in the box plot. In this case, the IQR for Fast Chicken is 14.5 - 10 = 4.5, while the IQR for Super Fast Food is 15.5 - 8.5 = 7. A smaller IQR indicates that the data is more consistent and less spread out.

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The first statement is false, The second statement is true , The third statement is false , The fourth statement is false , The fifth statement is true.

The first statement is false. If overhead is underapplied, it means that the actual overhead incurred exceeds the overhead applied to jobs, resulting in a higher cost of goods sold reported.

The second statement is true. Overapplied overhead refers to the situation where the overhead applied to jobs using the predetermined overhead rate is greater than the actual overhead incurred.

The third statement is false. Under- or overapplied factory overhead is not always closed entirely to Cost of Goods Sold. It can be allocated or adjusted based on the accounting policies of the company.

The fourth statement is false. Direct materials and direct labor costs are typically debited to the respective accounts and not to the Factory Overhead account in a job costing system.

The fifth statement is true. A time ticket is a source document used by employees to report the amount of direct and indirect labor performed for a job. It helps determine the allocation of direct labor to the job and indirect labor to the factory overhead.

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If m is a nonzero integer, then m^(1/m) is not always greater than 1.

The statement is false.

To determine if m^(1/m) is greater than 1, we can consider different values of m. For positive values of m, such as m = 2, m^(1/m) = 2^(1/2) = √2, which is approximately 1.414 and greater than 1.

However, if we consider negative values of m, such as m = -2, m^(1/m) = (-2)^(1/(-2)) = (-2)^(-1/2), which is equal to 1/√(-2). Since the square root of a negative number is not defined in the real number system, the value of m^(1/m) is not defined for negative values of m.

Therefore, the statement that m^(1/m) is always greater than 1 for nonzero integers m is false.

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The equation of the graph after translation is y = |2x + 9|

To translate the graph of y = |2x + 7| one unit to the left, we need to replace x with (x + 1) in the equation. This will shift the entire graph one unit to the left. The equation for the translated graph is:

y = |2(x + 1) + 7|

Simplifying this equation, we have:

y = |2x + 2 + 7|

y = |2x + 9|

Therefore, the equation for the translation of the graph of y = |2x + 7| one unit to the left is y = |2x + 9|.

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The goal of this simulation is to generate and transmit a random binary sequence through an AWGN (Additive White Gaussian Noise) channel and evaluate the Bit Error Rate (BER) as a function of Signal-to-Noise Ratio (SNR) for QPSK modulation. The following steps can be taken to achieve this:

Step 1: Generate a random binary sequence {bn} of length N using the MATLAB function rand(1,N) and rounding it to the nearest integer. The length N can be chosen as 210, 215, or 220.

Step 2: Map the binary sequence {bn} to a QPSK symbol sequence {ak} using the MATLAB function cammod(sk,4). Each pair of binary digits is mapped to a QPSK symbol.

Step 3: Add Gaussian noise to the QPSK symbols {ak} using the MATLAB function awgn(Qx,snr) to generate the received QPSK symbols {rx}. The noise level is determined by the SNR value, which is varied from 0 to 16 dB in steps of 2 dB.

Step 4: Demap the received QPSK symbols {rx} to obtain an estimated binary sequence {n} using the MATLAB function qamdemod(T2,4).

Step 5: Calculate the BER for each SNR value and plot it versus SNR. The BER is the ratio of the number of bits in error to the total number of transmitted bits.Finally, the plot of the BER versus SNR can be labeled and titled appropriately to produce a clear and informative figure.

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Answer:

x = 3

Step-by-step explanation:

2(x+4) + 2 = 5x + 1

2x + 8 + 2 = 5x + 1

2x + 10 = 5x + 1

-3x + 10 = 1

-3x = -9

x = 3

To solve for x, we need to simplify the equation and isolate the variable. Let's proceed with the given equation:

2(x + 4) + 2 = 5x + 1

First, distribute the 2 to the terms inside the parentheses:

2x + 8 + 2 = 5x + 1

Combine like terms on the left side:

2x + 10 = 5x + 1

Next, let's move all terms containing x to one side of the equation and the constant terms to the other side. We can do this by subtracting 2x from both sides:

2x - 2x + 10 = 5x - 2x + 1

Simplifying further:

10 = 3x + 1

To isolate the x term, subtract 1 from both sides:

10 - 1 = 3x + 1 - 1

9 = 3x

Finally, divide both sides of the equation by 3 to solve for x:

9/3 = 3x/3

3 = ×

Therefore, the solution to the equation is x = 3.

The expression α − β represents the difference between the two zeroes of the quadratic polynomial f(x).

To evaluate α − β, we need to find the values of α and β. In a quadratic polynomial of form ax^2 + bx + c, the zeroes (or roots) α and β can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Given that the quadratic polynomial is f(x) = ax^2 + bx + c, the zeroes α and β satisfy the equation f(α) = 0 and f(β) = 0.

Substituting α and β into the polynomial, we get:

f(α) = aα^2 + bα + c = 0,

f(β) = aβ^2 + bβ + c = 0.

We can rearrange these equations to isolate the term involving the difference α − β:

f(α) - f(β) = a(α^2 - β^2) + b(α - β) = 0.

Factoring out (α - β) from the equation, we have:

(α - β)(a(α + β) + b) = 0.

Since we know that f(x) = ax^2 + bx + c, the sum of the zeroes α + β is given by:

α + β = -b/a.

Substituting this value into the previous equation, we have:

(α - β)(-b + b) = 0,

(α - β)(0) = 0.

Therefore, α - β = 0.

The final answer is α - β = 0, indicating that the difference between the zeroes of the quadratic polynomial is zero, implying that the zeroes are equal.

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The integral in spherical coordinates is:

∫π 0 ∫π/4 0 ∫√(2-r^2)cos(φ) −√(2-r^2)cos(φ) ρ^2 sin(φ) dρ dφ dθ.

To rewrite the given integral in spherical coordinates, we first need to express the integrand in terms of spherical coordinates. We have:

z = ρ cos(φ)

r = ρ sin(φ) cos(θ)

x^2 + y^2 = ρ^2 sin^2(φ) = ρ^2 - z^2

Solving for ρ, we get:

ρ^2 = x^2 + y^2 + z^2 = r^2 + z^2

ρ = √(r^2 + z^2)

Substituting these expressions, we get:

∫2π 0 ∫√2 1 ∫√2−r^2 −√2−r^2 r dz dr dθ

= ∫π 0 ∫π/4 0 ∫√(2-r^2)cos(φ) −√(2-r^2)cos(φ) ρ^2 sin(φ) dρ dφ dθ

So the integral in spherical coordinates is:

∫π 0 ∫π/4 0 ∫√(2-r^2)cos(φ) −√(2-r^2)cos(φ) ρ^2 sin(φ) dρ dφ dθ.

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Answer:

16

Step-by-step explanation:

First, find the midpoint of the lengths you have. Then multiply by the frequency.

so:

5x5= 25

15x8= 120

25x7= 175

Then add all the numbers you got.

So:

25+120+175= 320

Add all the frequencies: 5+8+7= 20

Answer: 320/20= 16

The emission of an alpha particle, positron, or electron through the radioactive decay of a nucleus always leads to a change in the atomic number of the atom. Therefore, the correct answer is: All of the above.

Alpha particles are composed of two protons and two neutrons, so when an alpha particle is emitted from a nucleus, the atomic number decreases by two.

Positrons are antiparticles of electrons, and their emission from a nucleus leads to a decrease in the number of protons and an increase in the number of neutrons, resulting in a change in the atomic number.

Electron emission, also known as beta-minus decay, occurs when a neutron in the nucleus of an atom decays into a proton, an electron, and an antineutrino. The electron is then emitted from the nucleus, and the atomic number of the atom increases by one, while the mass number remains the same.

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Rewriting the integral f (x,y,z) dz dy dx as an iterated integral, the integral is - ∫(from -1 to 1) ∫(from 0 to 1-y) ∫(from x^2 to 1-z) f(x, y, z) dy dz dx.

To rewrite the integral f(x, y, z) dz dy dx as an iterated integral in the order dx dy dz and dy dz dx, with given limits, follow these steps:

For dx dy dz:

1. Identify the limits for x: -1 to 1

2. Determine the limits for y: x^2 to 1 (from the given limits)

3. Determine the limits for z: 0 to 1-y (from the given limits)

Therefore, ∫(from -1 to 1) ∫(from x^2 to 1) ∫(from 0 to 1-y) f(x, y, z) dx dy dz

For dy dz dx:

1. Identify the limits for x: -1 to 1

2. Determine the limits for z: 0 to 1-y (from the given limits)

3. Determine the limits for y, keeping in mind that y goes from x^2 to 1:
  - For z, solve 1-y = z, which gives y = 1-z
  - So, y goes from x^2 to 1-z

Therefore, ∫(from -1 to 1) ∫(from 0 to 1-y) ∫(from x^2 to 1-z) f(x, y, z) dy dz dx

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The solution is given by the point of intersection of the two lines which is (-1/4, 3/4).

To find the point of intersection of two lines, we need to determine the equations of the lines and then solve them simultaneously.

Finding the equation of the first line passing through the points (-1, 3) and (0, 0).

The slope of the line (m1) can be calculated using the formula:

m1 = (y2 - y1) / (x2 - x1)

Substituting the values (-1, 3) and (0, 0):

m1 = (0 - 3) / (0 - (-1))

= -3 / 1

= -3

Using the point-slope form of the line equation:

y - y1 = m1(x - x1)

Substituting the values (-1, 3):

y - 3 = -3(x - (-1))

y - 3 = -3(x + 1)

y - 3 = -3x - 3

y = -3x

So, the equation of the first line is y = -3x.

Similarly, second line,

The slope of the line (m2) is:

m2 = (2 - 0) / (1 - (-1))

= 2 / 2

= 1

Using the point-slope form with the values (-1, 0):

y - 0 = 1(x - (-1))

y = x + 1

So, the equation of the second line is y = x + 1.

Equating the equations of the lines to find the point of intersection and hence the solution,

-3x = x + 1

0 = 4x + 1

-1 = 4x

x = -1/4

Put x = -1/4 in 2nd equation,

y = x + 1

y = (-1/4) + 1

y = 3/4

Therefore, the point of intersection of the two lines is (-1/4, 3/4).

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The probability that a patient chosen at random from this study took the medication, given that they reported clearer skin, is approximately 0.43.

To find the probability that a patient chosen at random from the study took the medication, given that they reported clearer skin, we can use conditional probability.

Let's denote the events:

A: Patient took the medication.

B: Patient reported clearer skin.

We want to find P(A|B), which is the probability that a patient took the medication given that they reported clearer skin.

From the information given:

Number of patients who received the medication and reported clearer skin = 15

Number of patients who did not receive the medication and reported clearer skin = 20

Total number of patients who reported clearer skin = 15 + 20 = 35

Number of patients who received the medication = 40

Total number of patients in the study = 100

Using these values, we can calculate P(A|B) using the formula for conditional probability:

P(A|B) = P(A ∩ B) / P(B)

P(A ∩ B) is the probability that a patient both took the medication and reported clearer skin, which is given as 15.

P(B) is the probability that a patient reported clearer skin, which is calculated as the number of patients who reported clearer skin divided by the total number of patients in the study:

P(B) = 35 / 100 = 0.35

Therefore, we can now calculate P(A|B):

P(A|B) = P(A ∩ B) / P(B) = 15 / 0.35 ≈ 0.43

Hence, the probability that a patient chosen at random from this study took the medication, given that they reported clearer skin, is approximately 0.43.

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To find the number of children, students, and adults attending the movie theater, we can solve the system of equations based on the given information.

Let's assume the number of children attending the movie theater is C. Since there are half as many adults as children, the number of adults attending is A = C/2. Let's denote the number of students attending as S.

From the seating capacity of the theater, we have the equation C + S + A = 379. Since there are half as many adults as children, we can substitute A with C/2 in the equation, which becomes C + S + C/2 = 379.

To solve for C, S, and A, we need another equation. We know the ticket prices for each category, so the total ticket sales can be calculated as 5C + 7S + 12A. Given that the total ticket sales amount to $2746, we can substitute the variables and obtain the equation 5C + 7S + 12(C/2) = 2746.

Now we have a system of two equations with two variables. By solving this system, we can find the values of C, S, and A, which represent the number of children, students, and adults attending the movie theater, respectively.

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