Exercice 3-(4 pts): Enchaînements d'opérations, simplification de fractions et nombres relatifs. Recopier chaque expression, calculer en rédigeant en colonne, étape par étape. Pour les expressions A, B et C, donnez le résultat sous la forme d'une fraction irréductible. A=4+3x6 48
D = (7 + 3) - 9×2
B = 4x3+6 48 (4 + 3) x 6 ÷ 48
E = (-5+3-4)-(-4+6) × 2
C= (4+3)×6 ÷ 48

The integral of e from -5 to 5 is equal to zero.

The integral from -5 to 5 of e dx can be written as:

∫₋₅⁵ e dx

To evaluate this integral, we can use the fundamental theorem of calculus, which states that the definite integral of a function can be evaluated by finding its antiderivative and evaluating it at the limits of integration. In other words, we need to find the antiderivative of the function e and evaluate it at 5 and -5.

The antiderivative of e is itself, so we have:

∫₋₅⁵ e dx = e|(-5 to 5)

Now, we can evaluate e at the limits of integration:

e|(-5 to 5) = e(5) - e(-5)

Since e is a constant, we have:

e|(-5 to 5) = e - e

Therefore, the integral from -5 to 5 of e dx is equal to zero.

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Complete Question:

Evaluate the integral: integral from -5 to 5 edx

Each polygon has 3 sides, and they are equilateral triangles, since their interior angles of 72 degrees satisfy the equation (n-2) x 180 / n = 72.

A polygon is a two-dimensional closed shape with straight sides, made up of line segments connected end to end, and usually named by the number of its sides.

An equilateral triangle is a polygon with three sides of equal length and three equal angles of 60 degrees, making it a regular polygon.

According to the given information:

Since the two polygons are congruent and joined together, we can imagine them forming a larger regular polygon.

Let's call the number of sides of each polygon "n".

The interior angle of a regular n-gon can be calculated using the formula:

interior angle = (n-2) x 180 / n

For each of the congruent polygons, the interior angle is 72 degrees. Therefore:

72 = (n-2) x 180 / n

Multiplying both sides by n:

72n = (n-2) x 180

Expanding the brackets:

72n = 180n - 360

Simplifying:

108n = 360

n = 360 / 108

n = 10/3

Since n must be a whole number for a regular polygon, we round 10/3 to the nearest whole number, which is 3.

Therefore, each polygon has 3 sides, and they are equilateral triangles.

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Therefore, the probability of selecting a red marble first, followed by another red marble, and then a green marble is 7/55.

The probability of selecting a red marble on the first draw is 7/11 since there are 7 red marbles out of 11 total marbles in the box.

After the first red marble is drawn and not replaced, there are 10 marbles left in the box, including 6 red marbles, 3 green marbles, and 1 grey marble. Therefore, the probability of selecting a second red marble on the next draw is 6/10 or 3/5.

Finally, after the second red marble is drawn and not replaced, there are 9 marbles left in the box, including 5 red marbles, 3 green marbles, and 1 grey marble. Therefore, the probability of selecting a green marble on the third draw is 3/9 or 1/3.

To calculate the probability of these three events occurring in succession, we multiply the individual probabilities together:

[tex](7/11) * (3/5) * (1/3) = 7/55[/tex]

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The road crew will take approximately 0.304 hours, or 18.26 minutes, to repave the 35-mile road. This can be answered by the concept of Time and Distance.

To find the time it takes for the road crew to repave the road, we divide the length of the road (35 miles) by the rate at which they can repave (115 miles per hour). This gives us the following calculation:

Time = Distance / Rate

Time = 35 miles / 115 miles per hour

Simplifying, we get:

Time = 0.304 hours

Converting hours to minutes, we get:

Time = 0.304 hours × 60 minutes per hour

Time = 18.26 minutes

Therefore, the road crew will take approximately 0.304 hours, or 18.26 minutes, to repave the 35-mile road.

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The approximate volume of the given cylindrical container which has 3 balls is 63.62 in³, under the condition that tennis ball has a diameter of about 3 inches. Then, the required answer is 64 in³ which is Option A.

Now

The volume of a tennis ball is approximately

[tex]4/3 * \pi * (diameter/2)^{3}[/tex]

=[tex]4/3 * \pi * (1.5)^{3}[/tex]

= 14.137 in³.

Therefore, 3 balls are present in the container.

The diameter of a tennis ball = 3 inches,

Radius = 1.5 inches.

The height of the cylindrical container can be evaluated by multiplying the diameter of a tennis ball by three

Now,  three tennis balls are kept on top of each other.

Then, the height of the cylindrical container

3 × 3 = 9 inches.

The radius = 1.5 inches.

The volume of a cylinder = [tex]V = \pi * r^2 * h[/tex]

Here,

 V = volume,

r = radius

h = height.

Staging the values

[tex]V = \pi * (1.5)^{2} * 9[/tex]

= 63.62 in³.

The approximate volume of the given cylindrical container which has 3 balls is 63.62 in³, under the condition that tennis ball has a diameter of about 3 inches. Then, the required answer is 64 in³ which is Option A.

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The length of the curve [tex]y = (x^2 + 1)^2[/tex] from x = 0 to x = 2 is approximately 8.019 units.

To discover the length of the curve [tex]y = (x^2 + 1)^2[/tex] from x = to x = 2, able to utilize the equation for bend length of a bend:

[tex]L = ∫[a,b] sqrt[1 + (dy/dx)^2] dx[/tex]

where a and b are the limits of integration.

To begin with, we got to discover the derivative of y with regard to x:

[tex]dy/dx = 2(x^2 + 1)(2x)[/tex]

Following, ready to plug in this derivative and the limits of integration into the circular segment length equation:

[tex]L = ∫[0,2] sqrt[1 + (2(x^2 + 1)(2x))^2] dx[/tex]

We are able to streamline the expression interior of the square root:

[tex]1 + (2(x^2 + 1)(2x))^2[/tex]

= [tex]1 + 16x^2(x^2 + 1)^2[/tex]

Presently able to substitute this back into the circular segment length equation:

[tex]L = ∫[0,2] sqrt[1 + 16x^2(x^2 + 1)^2] dx[/tex]

Tragically, this fundamentally does not have a closed-form arrangement, so we must surmise it numerically.

One way to do this is usually to utilize numerical integration strategies, such as Simpson's Run the Show or the trapezoidal Run the Show.

Utilizing Simpson's run the show with a step measure of 0.1, we get:

L ≈ 8.019

Therefore, the length of the curve [tex]y = (x^2 + 1)^2[/tex] from x = 0 to x = 2 is approximately 8.019 units.

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The equation of the circle centered at the origin with a radius of 12 is x² + y² = 144.

In order to find the equation of a circle centered at the origin with a radius of 12, we need to use the standard form equation of a circle, which is:

(x - h)² + (y - k)² = r²

Where (h,k) represents the center of the circle, and r represents the radius.

In this case, since the circle is centered at the origin, h = 0 and k = 0. Also, since the radius is 12, we can substitute r = 12 in the above equation to get:

x² + y² = 12²

Simplifying further, we get:

x² + y² = 144

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Answer:

a. 6 × (7 - 5) + 4 =

   6 × (2) + 4 =

   12 + 4 = 16

b. (-2 + 24) ÷ (12 - 4) =

   22 ÷ 8 = 2.75

The statement "The sample statistic usually differs from the population parameter because of bias" is false because the differences is due to random sampling variability.

The sample statistic usually differs from the population parameter due to random sampling variability, and not necessarily because of bias. However, bias can also contribute to differences between the sample statistic and population parameter.

Bias refers to a systematic deviation of the sample statistic from the population parameter in one direction. Bias occurs when the sample selection process favors some characteristics of the population and excludes others.

On the other hand, sampling variability is a natural variation that occurs when taking different samples from the same population.

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Answer:

Step-by-step explanation:

The answer is 25% of the circle. If you simplify 25% into a fraction it would be 1/4 or one-fourth. Hope it helps <3

A function from the given data is f(x) is -2(2)^x.

There are many possible functions that can pass through the given data points, but a simple one that fits the pattern is a geometric sequence

f(x) = -2(2)^x

Using this recursive formula, we can find the values in the sequence

f(0) = -2(2)^0 = -2

f(1) = -2(2)^1 = -4

f(2) = -2(2)^2 = -2 * 4 = -8

f(3) = -2(2)^3 = -2 * 8 = -16

Therefore, the function that fits the given data is

f(x) = -2(2)^x

This function generates the sequence -2, -4, -8, -16 when x = 0, 1, 2, 3, respectively. Each term is multiplied by -2, which means the sequence is decreasing and each term is twice the previous term.

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The partial derivative of f(x, y) with respect to x, evaluated at a = (x=a, y=a), is fa = 0.

In this case, since a is not a variable in f, we cannot differentiate with respect to a.

The function f(x, y) is defined as f(x, y) = 5.23 + 8x y2 + sin(y).

The partial derivative of f with respect to x is fx = 15x2 + 40x4, which is not relevant to finding fa.

The partial derivative of f with respect to y is fy = 16xy + cos(y).

However, we are asked to find fa, which is the partial derivative of f with respect to a.

Since a is not one of the variables in f, we cannot take the partial derivative of f with respect to a, and therefore fa is equal to 0.

So, the answer is:

fa = 0.

It is important to note that when finding partial derivatives, we need to differentiate with respect to one variable at a time, holding all other variables constant.

In this case, since a is not a variable in f, we cannot differentiate with respect to a.

A partial derivative is a mathematical concept in multivariable calculus that represents the rate of change of a function with respect to one of its variables, while holding all other variables constant.

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1. The first partial derivatives of the

(a) (a) f(x,y) = [tex]x \sqrt{x^2-y^2}[/tex] is [tex]df/dy = -2y[/tex].

(b) f(x,y) = [tex]e^{-\frac{x}{y} }[/tex] is [tex]df/dy = xe^{(-x/y)}/y^2[/tex]

2. The second partial derivatives of f(x,y) = In [tex](1+x^2y^3)[/tex] is [tex]\frac{d^2f}{dx} dy = x^2/(1+x^2y^3)^2[/tex]

(a) To find the first partial derivatives of [tex]f(x, y) = xVx^2 - y^2[/tex], we differentiate with respect to each variable separately while treating the other variable as a constant:

[tex]df/dx = Vx^2 + 2\times(1/2)x = 3/2\timesVx[/tex]

[tex]df/dy = -2y[/tex]

(b) To find the first partial derivatives of [tex]f(x, y) = e^{(-x/y)[/tex], we differentiate with respect to each variable separately while treating the other variable as a constant:

[tex]df/dx = -e^{(-x/y)} \times (-1/y) = e^{(-x/y)}/y[/tex]

[tex]df/dy = e^{(-x/y)} \times x/y^2 = xe^{(-x/y)}/y^2[/tex]

(11) To find the second partial derivatives of f(x, y) = ln[tex](1+x^2y^3)[/tex], we first find the first partial derivatives:

[tex]\frac{df}{dx}[/tex] = 0

[tex]\frac{df}{dy} =\frac{x^2} {(1+x^2y^3)}[/tex]

Now we differentiate again with respect to each variable separately:

[tex]\frac{d^2f}{dx^2} =0[/tex]

[tex]\frac{d^2f}{dy^2} = -x^2/(1+x^2y^3)^2[/tex]

To find the mixed partial derivatives, we differentiate ∂f/∂x with respect to y and df/dy with respect to x:

[tex]\frac{d^2f}{dy} dx=0[/tex]

[tex]\frac{d^2f}{dx} dy = x^2/(1+x^2y^3)^2[/tex]

Since [tex]\frac{d^2f}{dy}dx = \frac{d^2f}{dx} dy[/tex], we have shown that the mixed partial derivatives fxy and fyx are equal.

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Question :-

1. Find the first partial derivatives of the following functions (it is not necessary to simplify).

(a) f(x,y) = [tex]x \sqrt{x^2-y^2}[/tex]

(b) f(x,y) = [tex]e^{-\frac{x}{y} }[/tex]

2. Find the second partial derivatives of the following function and show that the mixed derivatives [tex]f_{xy}[/tex] and [tex]f_{yw[/tex] are equal.

f(x,y) = In [tex](1+x^2y^3)[/tex]

The greatest possible value of $xy$ is $17\times5=\boxed{85}$. To get the greatest possible value of $xy$, we need to maximize the values of $x$ and $y$. We can start by rearranging the equation $5x+3y=100$ to solve for one of the variables in terms of the other:

$5x+3y=100 \implies 5x=100-3y \implies x=\frac{100-3y}{5}$
Since $x$ must be a positive integer, $100-3y$ must be divisible by 5. The largest multiple of 3 less than 100 is 99, so we can try values of $y$ starting from 1 and working up to 33 (because if $y\geq34$, then $5x\leq0$, which is not positive).
When $y=1$, we get $x=\frac{100-3}{5} = 19.4$, which is not an integer.
When $y=2$, we get $x=\frac{100-6}{5} = 18.8$, which is also not an integer.
When $y=3$, we get $x=\frac{100-9}{5} = 18.2$, still not an integer.
When $y=4$, we get $x=\frac{100-12}{5} = 17.6$, still not an integer.
When $y=5$, we get $x=\frac{100-15}{5} = 17$, which is an integer.
From here, we can continue to increase $y$ and see that the values of $x$ will only decrease. Thus, the greatest possible value of $x$ is 17 and the corresponding value of $y$ is 5.

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The surface area and volume to the nearest hundredth are 276.32 square units and 351.68 cubic units respectively.

In Mathematics and Geometry, the surface area of a cylinder can be determined by using the following mathematical equation (formula):

SA = 2πrh + 2πr²

Where:

By substituting the given parameters, we have the following;

Surface area = 2πrh + 2πr²

Surface area = 2(3.14)(4)(7) + 2(3.14)(4²)

Surface area = 175.84 + 100. 48

Surface area = 276.32 square units.

Next, we would determine the volume of this cylinder by using this formula:

Volume of cylinder, V = πr²h

Volume of cylinder, V = (3.14)(4²) × 7

Volume of cylinder, V = 351.68 cubic units.

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The fourth vertex must have a y-coordinate of 3. Therefore, the coordinate for the fourth vertex is (4, 6).

Coordinates are a set of numerical values that represent the position of a point on a map, graph, or other two-dimensional surface. Coordinates are most commonly expressed using two numbers representing the horizontal and vertical positions of a point. They can also be expressed using three numbers representing the x, y, and z positions of a point in space. Coordinates are used to define the exact location of a point on a map, graph, or other two-dimensional surface.

The correct answer is D. (4, 6). In order to draw a parallelogram on the coordinate plane, the given vertices must form a quadrilateral. The fourth vertex needed to complete the parallelogram must be located so that the opposite sides are congruent and parallel. The three given vertices form two pairs of opposite sides that are congruent and parallel. The coordinate of the fourth vertex must have the same x-coordinate as the given vertex (4, 0). The y-coordinate of the fourth vertex must have the same absolute value as the y-coordinate of the given vertex (-3). This means that the fourth vertex must have a y-coordinate of 3. Therefore, the coordinate for the fourth vertex is (4, 6).

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For a population with a normal distribution, the mean (μ) is 27.5 and the standard deviation (σ) is 71.5. When drawing a random sample of size n=180, the mean of the distribution of sample means (μ¯x) is equal to the population mean (μ). Therefore, μ¯x = 27.5.
The standard deviation of the distribution of sample means (σ¯x) is calculated by dividing the population standard deviation (σ) by the square root of the sample size (n).
σ¯x = σ / √n = 71.5 / √180 ≈ 5.33 (rounded to 2 decimal places)
So, the mean of the distribution of sample means is 27.5, and the standard deviation of the distribution of sample means is 5.33.

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a. The cost at the production level of 1900 is $8,374,600.

b. The average cost at the production level of 1900 is $4,408.95.

c. The marginal cost at the production level of 1900 is $12,800.

d. The production level that will minimize the average cost is 150.

e. The minimal average cost is $3,800.

a) To find the cost at the production level of 1900, we simply substitute x = 1900 into the cost function:

[tex]C(1900) = 19600 + 600(1900) + 2(1900)^2[/tex]

C(1900) = 19600 + 1140000 + 7220000

C(1900) = 8374600.

Therefore, the cost at the production level of 1900 is $8,374,600.

b) The average cost is given by the total cost divided by the production level:

[tex]Average cost = (19600 + 600x + 2x^2) / x[/tex]

Substituting x = 1900, we get:

[tex]Average cost = (19600 + 600(1900) + 2(1900)^2) / 1900[/tex]

Average cost = 8374600 / 1900

Average cost = 4408.95

Therefore, the average cost at the production level of 1900 is $4,408.95.

c) The marginal cost is the derivative of the cost function with respect to x:

Marginal cost = dC/dx = 600 + 4x

Substituting x = 1900, we get:

Marginal cost = 600 + 4(1900)

Marginal cost = 12800

Therefore, the marginal cost at the production level of 1900 is $12,800.

d) To find the production level that will minimize the average cost, we need to take the derivative of the average cost function and set it equal to zero:

[tex]d/dx (19600 + 600x + 2x^2) / x = 0[/tex]

Simplifying this equation, we get:

[tex](600 + 4x) / x^2 = 0[/tex]

Solving for x, we get:

x = 150

Therefore, the production level that will minimize the average cost is 150.

e) To find the minimal average cost, we simply substitute x = 150 into the average cost function:

[tex]Average cost = (19600 + 600(150) + 2(150)^2) / 150[/tex]

Average cost = 3800.

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For a normal distribution, the probability of a value being between a positive z-value and its population mean is indeed the same as that of a value being between a negative z-value and its population mean.

This is due to the symmetric nature of the normal distribution curve, where probabilities are mirrored around the mean.

The normal distribution is characterized by its bell-shaped curve, which is symmetric around the mean. The mean is also the midpoint of the curve, and the curve approaches but never touches the horizontal axis. The standard deviation of the distribution controls the spread of the curve.

In a normal distribution, the probability of a value being between a positive z-value and its population mean is indeed the same as that of a value being between a negative z-value and its population mean.

This is due to the symmetric nature of the normal distribution curve, where probabilities are mirrored around the mean.

This means that if we have a normal distribution with a mean of μ and a standard deviation of σ, the probability of a value falling between μ+zσ and μ is the same as the probability of a value falling between μ-zσ and μ.

This property of the normal distribution makes it easy to compute probabilities for any range of values, by transforming them into standard units using the z-score formula.

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for a series connection, S(t) = 2 and h(t) = 0 for t ≥ 0. For a parallel connection, S(t) = 8 and h(t) = 0 for t ≥ 0.

For a series connection, the system fails if either element fails. Thus, the system lifetime distribution S(t) is the minimum of the individual lifetimes:

S(t) = min(h1(t), h2(t)) = min(2, 4) = 2 for t ≥ 0.

The system hazard rate h(t) is the derivative of the system lifetime distribution:

h(t) = d/dt S(t) = 0 for t > 0.

For a parallel connection, the system fails if both elements fail. Thus, the system lifetime distribution S(t) is the product of the individual lifetimes:

S(t) = h1(t) * h2(t) = 8 for t ≥ 0.

The system hazard rate h(t) is the derivative of the system lifetime distribution:

h(t) = d/dt S(t) = 0 for t > 0.

Therefore, for a series connection, S(t) = 2 and h(t) = 0 for t ≥ 0. For a parallel connection, S(t) = 8 and h(t) = 0 for t ≥ 0.

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The answer is: O 30.813. This can be answered by the concept of critical value.

The critical value(s) of x2 based on the given information can be found using a chi-square distribution table with degrees of freedom (df) = n-1 = 23-1 = 22 and a significance level (α) = 0.10. The critical value(s) of x2 that correspond to the rejection region(s) are those that have a cumulative probability (p-value) of less than or equal to 0.10 in the right-tail of the chi-square distribution.

Using a chi-square distribution table or calculator, we can find that the critical value of x2 for α = 0.10 and df = 22 is 30.813.

Therefore, the answer is: O 30.813.

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The differential volume of the coating is 230.4 cm³

We know that,

Differentials are infinitesimal changes in a function.

we have to find the volume of the coating using differentials:

Since steel cube with sides 24 cm is to be coated with 0.2 cm of copper, its volume is given by V = L³ where L = length of cube.

So, the differential change in volume, V is

dV = (dV/dL)dL where

dV = differential volume of coating

dV/dL = derivative of V with respect to L and and

dL = thickness of coating

So,  dV/dL = dL³/dL

= 2L²

So, dV = (dV/dL)dL

= 2L² dL

Given that

L = 24 cm and

dL = 0.2 cm

Substituting the values of the variables into the differential equation, we have

dV = 2L² dL

= 2(24 cm)² × 0.2 cm

= 2 × 576 cm² × 0.2 cm

= 1152 cm² × 0.2 cm

= 230.4 cm³

So, the volume of the coating is 230.4 cm³

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complete question:

A steel cube with sides 24 cm is to be coated with 0.2 cm of copper. Use differentials to estimate the volume (in cm3) of copper in the coating. Express your answer to one decimal place

The final integral is:

∫1/sin²(2x) dx = -1/2 × cot(2x) + C.

To evaluate the integral, we can use the substitution u = sin(2x), which

implies du/dx = 2cos(2x). Then, we have:

[tex]\int 1/sin^{2} (2x) dx = \int 1/(u^{2} \times (1 - u^{2} )^{(1/2)}) \times (du/2cos(2x)) dx[/tex]

Now, we can simplify the integral using the trigonometric identity 1 -

sin²(2x) = cos²(2x),

which gives us:

∫1/sin²(2x) dx = ∫1/(u² × cos(2x)) du

Using the power rule of integration, we can integrate this expression as:

∫1/sin²(2x) dx = -1/2 × cot(2x) + C

where C is the constant of integration.

Therefore, the answer is:

∫1/sin²(2x) dx = -1/2 × cot(2x) + C.

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The type I error for the hypothesis test of the given claim in the entomologist's article would be rejecting the null hypothesis when it is actually true, i.e., concluding that fewer than 16 in ten thousand male fireflies are unable to produce light due to a genetic mutation, when in fact this claim is not supported by the data.

In hypothesis testing, the null hypothesis (H0) is the assumption that there is no significant difference or effect, while the alternative hypothesis (Ha) is the claim that the researcher is trying to support. In this case, the null hypothesis would be that the proportion of male fireflies unable to produce light due to a genetic mutation is equal to or greater than 16 in ten thousand (p ≥ 0.0016), while the alternative hypothesis would be that the proportion is less than 16 in ten thousand (p < 0.0016).

The type I error, also known as alpha error or false positive, occurs when the null hypothesis is actually true, but the test erroneously leads to its rejection. In other words, the researchers conclude that the proportion of male fireflies unable to produce light is less than 16 in ten thousand, when in reality it could be equal to or greater than 16 in ten thousand.

Therefore, the type I error in this hypothesis test would be rejecting the null hypothesis and concluding that fewer than 16 in ten thousand male fireflies are unable to produce light due to a genetic mutation, when in fact this claim is not supported by the data.

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1. The mean of a binomial distribution is given by 2.4

Therefore, we expect the person to get about 2 or 3 correct answers by guessing.

2. We can expect the person to get about 1 to 2 correct answers, plus or minus 1 standard deviation, if they are guessing on 12 questions.

We can model the situation as a binomial distribution with parameters

n = 12 (number of trials) and p = 1/5 (probability of guessing the correct answer).

The mean of a binomial distribution is given by μ = np, so in this case, the mean is:

μ = 12 x 1/5 = 2.4

Therefore, we expect the person to get about 2 or 3 correct answers by guessing.

The standard deviation of a binomial distribution is given by [tex]\sigma = \sqrt{(np(1-p)), }[/tex]

so in this case, the standard deviation is:

[tex]\sigma = \sqrt{( 12 * 1/5 * 4/5) } = 1.3856.[/tex].

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The area of the shaded region for the circle is derived to be equal to 104.86 square feet

The shaded region is the triangle area in the circle, so it is derived by subtracting the area of the triangle from the area of the circle as follows:

area of the circle = 3.14 × 7 ft × 7 ft

area of the circle = 153.86 ft²

area of triangle = 1/2 × 14 ft × 7 ft

area of triangle = 49 ft²

area of the shaded region = 153.86 ft² - 49 ft²

area of the shaded region = 104.86 ft²

Therefore, the area of the shaded region for the circle is derived to be equal to 104.86 square feet

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To find the mean service time per job, E[Ts], in a system with non-identical service rates (μ and γ) and a limit of N jobs, you can follow these steps:

Step 1: Calculate the mean throughput rate λe
The mean throughput rate λe can be computed as the sum of the product of the steady-state probabilities (pn) and their corresponding service rates (μ or γ).

λe = p1*μ1 + p2*μ2 + ... + pn*μn

Step 2: Determine the mean service time per job E[Ts]
Now that you have the mean throughput rate λe, you can find the mean service time per job E[Ts] using the formula:

E[Ts] = 1 / λe

In summary, to obtain an expression for the mean service time per job E[Ts] in a system with non-identical service rates and a limit of N jobs, you first calculate the mean throughput rate λe as the sum of the product of the steady-state probabilities pn and the corresponding service rates μ and γ. Then, you find the mean service time per job E[Ts] by taking the reciprocal of the mean throughput rate λe.

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Xe2 means "X is an element of the set {2}". So, Xe2 means "the number of utilized stations is 2".
P(X=4) means "the probability that the number of utilized stations is exactly 4".

So, P(X+4)=0.15 means "the probability that the number of utilized stations plus 4 is equal to 4, which is equal to 0.15". This is not a meaningful statement.
The probability that the number of utilized stations is exactly 2 is given by P(X=2), which is equal to 0.2.
An event in which the number of utilized stations is exactly 2 is the event {X=2}.

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On solving the question, we can say that Therefore, the probability that a region was hit at most twice is 0.349.

The likelihood that an event will occur or that a proposition is true is determined by a field of mathematics known as probability theory. An event's probability is expressed as a number between 0 and 1, where 1 indicates certainty and roughly 0 indicates how likely it is that the event will occur. A probability is a numerical expression of the likelihood or potentiality of a given event. Probabilities can alternatively be stated as integers between 0 and 1, percentages between 0% and 100%, or as percentages between 0% and 100%. the fraction of times that all equally probable alternatives occur compared to all potential outcomes.

Given information:

Number of regions = 687

Area of each region = 0.25 km²

Total number of bombs hit = 535

Poisson distribution applies

To find the mean number of hits per region:

λ = mean number of hits per region

λ = total number of hits / total number of regions

λ = 535 / 687

λ ≈ 0.78 (rounded to 2 decimal places)

Therefore, the mean number of hits per region is 0.78.

To find the standard deviation of hits per region:

λ = 0.78 (mean number of hits per region)

σ =standard deviation

σ = sqrt(λ)

σ ≈ sqrt(0.78)

σ ≈ 0.88 (rounded to 2 decimal places)

Therefore, the standard deviation of hits per region is 0.88.

To find the probability that a region was hit exactly twice:

P(X = 2) = (e^-λ * λ^2) / 2!

P(X = 2) = (e^-0.78 * 0.78^2) / 2!

P(X = 2) ≈ 0.146 (rounded to 3 decimal places)

To find the number of regions expected to be hit exactly twice:

Expected number of regions = total number of regions * P(X = 2)

Expected number of regions = 687 * 0.146

Expected number of regions ≈ 100 (rounded to a whole number)

To find the probability that a region was hit at most twice:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X ≤ 2) = (e^-0.78 * 0.78^0) / 0! + (e^-0.78 * 0.78^1) / 1! + (e^-0.78 * 0.78^2) / 2!

P(X ≤ 2) ≈ 0.349 (rounded to 3 decimal places)

Therefore, the probability that a region was hit at most twice is 0.349.

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The answers are:

mean = 0.78 (rounded to 2 decimal places)

standard deviation = 0.88 (rounded to 2 decimal places)

P(X = 2) = 0.140 (rounded to 3 decimal places)

ans = 0 (rounded to a whole number)

P(X ≤ 2) = 0.503 (rounded to 3 decimal places)

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

Given:

Number of regions = 687

Area of each region = 0.25 km²

Total number of bombs hit = 535

Poisson distribution applies

To solve this problem, we can use the Poisson distribution formula:

P(X = x) = ([tex]e^{-λ}[/tex] * [tex]λ^{x}[/tex]) / x!

where:

P(X = x) is the probability of x number of bomb hits in a region

e is the mathematical constant approximately equal to 2.71828

λ is the mean number of hits per region

Mean number of hits per region:

λ = total number of bomb hits / total number of regions

λ = 535 / 687

λ = 0.778

Standard deviation of hits per region:

σ = sqrt(λ)

σ = sqrt(0.778)

σ = 0.881

Probability of a region being hit exactly twice:

P(X = 2) = (e^-0.778 * 0.778^2) / 2!

P(X = 2) = 0.140

Expected number of regions hit exactly twice:

Expected value = λ * P(X = 2)

Expected value = 0.778 * 0.140

Expected value = 0.109

Rounding to a whole number, we get: 0

Probability of a region being hit at most twice:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

P(X ≤ 2) = ([tex]e^{-0.778}[/tex] * 0.778^0) / 0! + ([tex]e^{-0.778}[/tex] * [tex]0.778^{1}[/tex]) / 1! +

([tex]e^{-0.778}[/tex]  * 0.778²) / 2!

P(X ≤ 2) = 0.063 + 0.196 + 0.244

P(X ≤ 2) = 0.503

Therefore, the answers are:

mean = 0.78 (rounded to 2 decimal places)

standard deviation = 0.88 (rounded to 2 decimal places)

P(X = 2) = 0.140 (rounded to 3 decimal places)

ans = 0 (rounded to a whole number)

P(X ≤ 2) = 0.503 (rounded to 3 decimal places)

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