Faiza is studying the population of rabbits in a park.
She wants to estimate the number of rabbits in the park.
On Monday she catches a random sample of 20 rabbits in the park, marks each rabbit
with a tag and releases them back into the park.
On Tuesday she catches a random sample of 42 rabbits in the park.
12 of the rabbits are marked with a tag.
(a) Find an estimate for the number of rabbits in the park.

The area of the larger pin is 62.1 square centimeters.

Let's denote the side length of the smaller pin as "s".

Then, the side length of the larger pin would be 3s.

The formula for the area of an equilateral triangle is:

Area = (√(3) / 4) x side length²

Given that the area of the smaller pin is 6.9 square centimeters, we can set up the following equation:

6.9 = (√(3) / 4) s²  ....(1)

To find the approximate area of the larger pin, we need to calculate the area using the side length 3s:

Area = (√(3) / 4) (3s)²

= (√(3) / 4)  9s²

= 9 [(√(3) / 4)s²]

Substitute the value of s² from equation (1)

= 9 x 6.9

= 62.1

Therefore, the area of the larger pin is 62.1 square centimeters.

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Someone in 1955 paid only 0.2% of today's price for a loaf of bread

To find what percentage of today's price someone in 1955 paid for a loaf of bread, we need to use the concept of inflation. Inflation is the increase in the general price level of goods and services in an economy over a period of time. In other words, the cost of goods and services increases over time due to inflation.

To calculate the inflation rate, we can use the following formula:

Inflation rate = (Current price - Base price) / Base price x 100%

Here, the base price is the price of bread in 1955, and the current price is the price of bread today.

Base price = 20 cents

Current price = $2.50

Using the formula, we get:

Inflation rate = ($2.50 - $0.20) / $0.20 x 100%

Inflation rate = $2.30 / $0.20 x 100%

Inflation rate = 1150%

This means that the price of bread has increased by 1150% since 1955 due to inflation. To find out what percentage of today's price someone in 1955 paid, we can divide the 1955 price by the inflation rate and multiply by 100%.

Percentage of today's price = (Base price / Inflation rate) x 100%

Percentage of today's price = (20 cents / 1150%) x 100%

Percentage of today's price = 0.002 x 100%

Percentage of today's price = 0.2%

Therefore, someone in 1955 paid only 0.2% of today's price for a loaf of bread.

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The answer is option C, 2.88. This means that the data points in the distribution are spread out around the mean of 4.8 with a variance of 2.88.

To calculate the variance for a binomial distribution with 12 trials and a probability of success of 0.4, we can use the formula Var(X) = np(1-p), where n is the number of trials and p is the probability of success.

In this case, n = 12 and p = 0.4, so Var(X) = 12(0.4)(1-0.4) = 2.88.

Therefore, the answer is option C, 2.88. This means that the data points in the distribution are spread out around the mean of 4.8 with a variance of 2.88. A higher variance indicates that the data points are more spread out, while a lower variance indicates that the data points are closer together.

A binomial distribution with 12 trials (n) and a probability of success (p) of 0.4 has a variance (σ²) calculated by the formula σ² = n * p * (1 - p). In this case, σ² = 12 * 0.4 * (1 - 0.4) = 12 * 0.4 * 0.6 = 2.88. Therefore, the variance for this distribution is 2.88, which corresponds to the third option in your multiple-choice question.

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The standard form of the quadratic equation is:

y = 3x² + 24x + 45

To do it, we just need to expand the product in the right side of the given quadratic, here we start with:

y = 3(x + 3)(x + 5)

Expanding the product we will get:

y = 3*(x + 3)*(x + 5)

y = (3x + 9)*(x + 5)

y = 3x*x + 3x*5 + 9x + 9*5

y = 3x² + 15x + 9x + 45

y = 3x² + 24x + 45

That is the standard form.

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The function is :[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

To find y as a function of x, we need to solve the differential equation:

[tex]y′′′ − 17y′′ + 72y′ = 168e^x[/tex]

Step 1: Find the characteristic equation

[tex]r^3 - 17r^2 + 72r = 0[/tex]

Factor out r:

[tex]r(r^2 - 17r + 72) = 0[/tex]

Factor the quadratic:

r(r - 8)(r - 9) = 0

So the roots are:

r₁ = 0, r₂ = 8, r₃ = 9

Step 2: Find the general solution

The general solution will be of the form:

[tex]y(x) = C1 + C2e^8x + C3e^9x + y_p(x)[/tex]

where y_p(x) is a particular solution to the non-homogeneous equation.

Step 3: Find the particular solution

We can use the method of undetermined coefficients to find a particular solution. Since the right-hand side is an exponential function, we can guess that the particular solution is also an exponential function:

[tex]y_p(x) = A e^x[/tex]

[tex]y_p′(x) = A e^x[/tex]

[tex]y_p′′(x) = A e^x[/tex]

[tex]y_p′′′(x) = A e^x[/tex]

Substituting into the differential equation:

[tex]A e^x - 17A e^x + 72A e^x = 168 e^x[/tex]

Simplifying:

[tex]56A e^x = 168 e^x[/tex]

A = 3

So the particular solution is:

[tex]y_p(x) = 3 e^x[/tex]

Step 4: Find the constants using initial conditions

y(0) = C₁ + C₂ + C₃ + 3 = 16

y′(0) = 8C₂ + 9C₃ + 3 = 23

[tex]y′′(0) = 8^2 C2 + 9^2 C3 = 24[/tex]

Solving for the constants, we get:

C₁ = 10, C₂ = 7/8, C₃ = 97/72

Step 5: Write the final solution

Substituting the constants and the particular solution into the general solution, we get:

[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

So the function y(x) is:

[tex]y(x) = 10 + (7/8) e^8x + (97/72) e^9x + 3 e^x[/tex]

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The directional derivative of f(x, y, z) at the point P in the direction pointing to the origin is 3√(14)/14.

To find the directional derivative of f(x, y, z) = xy + z^3 at P = (3, -2, -1) in the direction pointing to the origin, we need to first find the gradient of f at P.

Gradient of f(x, y, z) = ∇f(x, y, z) = (fx, fy, fz) = (y, x, 3z^2)

At P = (3, -2, -1), the gradient of f is:

∇f(3, -2, -1) = (-2, 3, 3)

The direction vector pointing from P to the origin is:

d = <-3, 2, 1>

To find the directional derivative of f at P in the direction of d, we need to take the dot product of the gradient of f at P and the unit vector in the direction of d:

|d| = √((-3)^2 + 2^2 + 1^2) = √(14)

u = d/|d| = <-3/√(14), 2/√(14), 1/√(14)>

Directional derivative of f at P in the direction of d is:

Duf(P) = ∇f(3, -2, -1) · u

Duf(P) = (-2, 3, 3) · <-3/√(14), 2/√(14), 1/√(14)>

Duf(P) = (-6/√(14)) + (6/√(14)) + (3/√(14))

Duf(P) = 3√(14)/14

Therefore, the directional derivative of f(x, y, z) = xy + z^3 at the point P = (3, -2, -1) in the direction pointing to the origin is 3√(14)/14.

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The Richter ranking of this event is 4.10.

I₀ = Intensity of Baval, Pakistan earthquake

I₁= Intensity of 0 level earthquake

I₀/I₁ = 12, 589.

let M is the magnitude and S is the standard intensity of Earthquake.

So, M = log ₁₀ I/S

M = log₁₀ ((I₀ / S) / (I₁/S))

M = Log₁₀ (I₀/ I₁)

M=  Log₁₀(12, 589)

M = 4.099

M= 4.1

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The equations have been solved below

The solutions to the equations is as follows;

1) w + 7/8 = 5

w = 5 - 7/8

w = 40 - 7/8

w = 33/8 or 4 1/8

2) 5h - 2/3 = 6

5h = 6 + 2/3

5h = 18 + 2/3

5h = 20/3

h = 20/3 * 1/5

h = 4/3 = 1 1/3

3) v/9 + 2 = 7

v/9 = 7 - 2

v/9 = 5

v = 9 * 5

v = 45

4) 4r/3 - 1 = 8/3

4r = 8/3 + 1

4r = 8 + 3/3

4r = 11/3

r = 11/3 * 1/4

r = 11/12

5) 5y = 13/4

y = 13/4 * 1/5

y = 13/20

6) 3f/2 + 1/2 = 7/2

3f/2 = 7/2 - 1/2

3f/2 = 6/2

3f/2 = 3

f = 3 * 2/3

f = 2

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The value of angle FCE is 100⁰.

The value of arc DE is 125⁰.

The value of angle DCA is 125⁰.

The value of arc FAE is 180⁰.

The measure of angle FCE is calculated by applying the following formula.

Based on the angle of intersecting chord theorem, we will have the following equation.

m∠ECB = 25⁰ (intersecting chord theorem)

m∠FCE = 180 - (55 + 25) (sum of angles on a straight line)

m∠FCE = 100⁰

Angle DCE = FCE + FCD

FCD = 55 (vertical opposite angles)

Angle DCE = 100 + 55 = 155⁰

Arc DE = 155⁰ (intersecting chord theorem)

Angle DCA = 360 - (155 + 25 + 55) = 125⁰

Arc FAE = 180 (semi circle)

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The maximum volume of the cylinder is 27/40 times the volume of the cone, which simplifies to ⅘.

Let's denote the height of the cylinder as h and the angle of the cone as θ. We can then express the radius of the cone as:

r = R/H * h

Using similar triangles, we can relate the radius of the cylinder to the height of the cone and the angle θ as:

y/h = R/H

Solving for h, we get:

h = H*y/R

Substituting this expression for h into the formula for the radius of the cone, we get:

r = R/H * H*y/R = y

Therefore, the radius of the inscribed cylinder is simply y.

The volume of the cylinder is then given by:

V_cylinder = πy[tex]^2h[/tex]

Substituting the expression for h, we get:

V_cylinder = π[tex]y^2[/tex](H*y/R)

Simplifying, we get:

V_cylinder = π[tex]y^3[/tex]H/R

The volume of the cone is given by:

V_cone = (1/3)π[tex]R^2[/tex]H

We want to find the maximum volume of the cylinder in terms of the volume of the cone. To do this, we can take the ratio of the volume of the cylinder to the volume of the cone:

V_cylinder/V_cone = (π[tex]y^3[/tex]H/R) / (1/3)π[tex]R^2[/tex]H

Simplifying, we get:

V_cylinder/V_cone = 3[tex]y^3[/tex]/[tex]R^2[/tex]

To find the maximum value of this ratio, we can take the derivative with respect to y and set it equal to zero:

d/dy (V_cylinder/V_cone) = 9y^2/R^2 - 6y^3/R^3[tex]9y^2/R^2 - 6y^3/R^3[/tex] = 0

Solving for y, we get:

y = (3/2)R

Substituting this value of y back into the ratio, we get:

V_cylinder/V_cone = [tex]3((3/2)R)^3/R^2[/tex] = (27/8)

Therefore, the maximum volume of the cylinder is 27/40 times the volume of the cone, which simplifies to ⅘.

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To find the derivative of f(x), we can use the power rule for derivatives, which states that if f(x) = x^n, then f'(x) = n*x^(n-1). Using this rule, we get:

f(x) = 3x + 4
f'(x) = 3*(x^(1-1)) = 3

So, the derivative of f(x) is simply 3.

As for the inverse of f, denoted as f^-1(x), we can find it by solving for x in terms of y in the equation y = 3x + 4.

y = 3x + 4
y - 4 = 3x
x = (y - 4)/3

Therefore, f^-1(x) = (x - 4)/3.
To answer your question, we first need to find the derivative of the given function f(x) = 3x + 4. We will use the power rule for differentiation:

f'(x) = d(3x + 4)/dx

Now, let's differentiate each term with respect to x:

d(3x)/dx = 3 (since the derivative of x with respect to x is 1)

d(4)/dx = 0 (since the derivative of a constant is 0)

So, f'(x) = 3 + 0 = 3

Therefore, the derivative of the function f(x) = 3x + 4 is f'(x) = 3.

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The trinomial form of the expression (2x + 8) (х - 3) is 2x² + 2x - 24.

Given the expression in the question:

(2x + 8) (х - 3)

First, we expand using the distributive property.

(2x + 8) (х - 3)

2x(х - 3) + 8(х - 3)

2x×х + 2x×-3 + 8×х +8×-3

Multiplying, we get:

2x² -6x + 8х - 24

Collect and add like terms

Add -6x and 8x

2x² -6x + 8х - 24

2x² + 2x - 24

Therefore, the expanded form is 2x² + 2x - 24.

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Meg can get 1 and 3/4 servings of 1/2 jug from her 7/8 jug of orange juice.

Meg has a 7/8 jug of orange juice, and she wants to know how many 1/2 jug servings she can get from it. To solve this problem, we need to divide the total amount of orange juice by the amount of orange juice in each serving.

First, we need to convert the 7/8 jug to an equivalent fraction with a denominator of 2. To do this, we can multiply both the numerator and denominator of 7/8 by 2, which gives us 14/16.

Next, we can divide 14/16 by 1/2 to find out how many 1/2 jug servings Meg can get from the jug. To divide fractions, we invert the second fraction and multiply. So we have:

14/16 ÷ 1/2 = 14/16 x 2/1 = 28/16

Now, we need to simplify this fraction by dividing the numerator and denominator by their greatest common factor, which is 4. So we have:

28/16 = (28 ÷ 4) / (16 ÷ 4) = 7/4

Therefore, Meg can get 7/4 or 1 and 3/4 servings of 1/2 jug from her 7/8 jug of orange juice.

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The correct option will be option C: Multiply/divide both sides by the same non-zero constant.

To solve the linear equation of one variable;

Step-1: we have to balance each side by simplifying the equation

Step-2: add/subtract constant term on both sides of the equation to separate variable and constant term on both sides

Step-3: divide the coefficient of the variables on both sides.

So according to the question,

the given equations are:

AAA: 3(x+2)=18

BBB: x+2=6

We have to find a way from equation AAA to Equation BBB

from the above equation, it is clear BBB is factor AAA.

to get BBB from equation AAA, we have to just divide 3 on both sides of the equation AAA.

Therefore option C, will be correct as 3 is a non-zero constant. we have to divide both sides by this same non-zero constant.

Therefore The correct option will be option C:

Multiply/divide both sides by the same non-zero constant.

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The area of the region that lies inside the circle r=3sinΘ and outside the cardioid r=1+sinΘ is 9π/4 - √3/2.

To find the area, we first need to determine the values of θ at which the two curves intersect. Setting r=3sinΘ equal to r=1+sinΘ, we get sinΘ = 1/2, which gives Θ = π/6 and Θ = 5π/6.

Next, we can use the area formula for polar coordinates: A=1/2∫βα(f(θ))2dθ. Since the cardioid is inside the circle for Θ between π/6 and 5π/6, we need to find the area of the circle minus the area of the cardioid. Thus, we have:

A = 1/2 [(∫0^(π/6) (3sinΘ)^2 dΘ) + (∫5π/6^π (3sinΘ)^2 dΘ) - (∫π/6^(5π/6) (1+sinΘ)^2 dΘ)]

Simplifying and evaluating the integrals, we get: A = 9π/4 - √3/2

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a. The tension in rope AB is T_AB = (FR/2) + (sqrt((FR/2)^2 + T_AC^2)).

b. The tension in rope AC is T_AC = (FR/2) + (sqrt((FR/2)^2 + T_AB^2)).

To answer this question, we can use the fact that the net force acting on the boater must be zero (assuming they are not accelerating).

Let T_AB be the tension in rope AB and T_AC be the tension in rope AC.

a. In the x-direction: T_AB - T_AC = 0 (since the boater is not moving horizontally).
In the y-direction: T_AB + T_AC - FR = 0 (since the net force acting on the boater must be zero).

Using these two equations, we can solve for T_AB:

T_AB = (FR/2) + (sqrt((FR/2)^2 + T_AC^2))

b. Similarly, we can solve for T_AC:

T_AC = (FR/2) + (sqrt((FR/2)^2 + T_AB^2))

Note that we are given FR = 70 lb, the force exerted by the flowing water on the boater. We can substitute this value into the above equations to find the tensions in the ropes.

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The appropriate confidence interval for determining the percent of American adults who believe in the existence of angels would be a confidence interval for a population proportion. This is because we are interested in the proportion or percentage of American adults who hold a particular belief.

A confidence interval is a range of values that we can be reasonably sure contains the true population parameter. In this case, we want to estimate the proportion of American adults who believe in angels and we can use statistical methods to estimate this parameter.

A confidence interval for a population proportion is typically calculated using the sample proportion and the sample size. The margin of error is also taken into consideration when calculating the interval. This type of interval would allow us to estimate the proportion of American adults.

It is important to note that the confidence interval only gives us an estimate of the population parameter and not an exact value. The confidence level indicates how confident we can be that the true population parameter falls within the interval.

In conclusion, to determine the percent of American adults who believe in the existence of angels, an appropriate confidence interval would be a confidence interval for a population proportion. This would provide us with an estimate of the proportion with a certain level of confidence.

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a, b, c, and d are distinct real numbers, the terms involving products of three distinct numbers (abc, abd, acd, bcd) are all non-zero. The given equation cannot be factored into linear factors and is irreducible over the real numbers.

The given equation can be simplified using the distributive property of multiplication and combining like terms:

(x - b)(x - c)(x - d) + (x - a)(x - c)(x - d) + (x - a)(x - b)(x - d) + (x - a)(x - b)(x - c)

Expanding each of the terms gives:

(x^3 - (b+c+d)x^2 + (bc+cd+bd)x - bcd) + (x^3 - (a+c+d)x^2 + (ac+cd+ad)x - acd) + (x^3 - (a+b+d)x^2 + (ab+bd+ad)x - abd) + (x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc)

Combining like terms gives:

4x^3 - 2(a+b+c+d)x^2 + 3(ab+ac+ad+bc+bd+cd)x - 6abc - 6abd - 6acd - 6bcd

Since a, b, c, and d are distinct real numbers, the terms involving products of three distinct numbers (abc, abd, acd, bcd) are all non-zero. The given equation cannot be factored into linear factors and is irreducible over the real numbers.

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If a, b and c are distinct real numbers, prove that the equation

(x−a)(x−b)+(x−b)(x−c)+(x−c)(x−a)=0

has real and distinct roots.

Answer:

Each of the four arrangements contains 4 gray tiles.

Step-by-step explanation:

In each of the four arrangements, the black and gray square tiles are arranged in a 2 x 2 grid. Since one of the tiles is black, the remaining three tiles are gray. Therefore, each of the four arrangements contains three gray tiles.

The required central angle in the circle is as follows:

m∠2 = 30 degrees

The central angle of an arc is the central angle subtended by the arc.

Therefore, the measure of an arc is the measure of its central angle.

Hence, let's find the angle m∠2.

Therefore,

arc angle BD = 150 degrees

Therefore,

m∠4 = 150 degrees(central angle to the arc)

Let's find the value of m∠2.

Hence,

m∠4 = m∠3(vertically opposite angles)

m∠2 = 360 - 150 - 150 ÷ 2

m∠2 = 360 - 300 ÷ 2

m∠2 = 60 / 2

m∠2 = 30 degrees

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The surface area of a rectangular prism of dimensions 14 cm, 4.5 cm and 32 cm is given as follows:

1310 cm².

The surface area of a rectangular prism of height h, width w and length l is given by:

S = 2(hw + lw + lh).

This means that the area of each rectangular face of the prism is calculated, and then the surface area is given by the sum of all these areas.

The dimensions for this problem are given as follows:

14 cm, 4.5 cm and 32 cm.

Hence the surface area of the prism is given as follows:

S = 2 x (14 x 4.5 + 14 x 32 + 4.5 x 32)

S = 1310 cm².

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Lawnco should produce 300 bags of grade a, 400 bags of grade b, and 200 bags of grade c fertilizer in order to meet the given nutrient requirements.

Let x, y, and z denote the number of 100-lb bags of grade a, b, and c fertilizer respectively.
Then, we can create the following system of equations based on the given information:
16x + 20y + 24z = 26200 (total nitrogen)
6x + 4y + 3z = 4700 (total phosphate)
7x + 4y + 6z = 6600 (total potassium)
Solving this system of equations, we get:
x = 300 (number of bags of grade a)
y = 400 (number of bags of grade b)
z = 200 (number of bags of grade c)
To find out how many 100-lb bags of each of the three grades of fertilizers Lawnco should produce, we need to set up a system of linear equations using the given information and solve for a, b, and c.
Equation 1 (nitrogen): 16a + 20b + 24c = 26,200
Equation 2 (phosphate): 6a + 4b + 3c = 4,700
Equation 3 (potassium): 7a + 4b + 6c = 6,600
Solving this system of linear equations will give you the number of bags of grade A, B, and C fertilizers Lawnco should produce to use all available nutrients.

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Answer:

x ≈ 62,15°

Step-by-step explanation:

Given:

A right triangle

FG = 8,4

FH = 9,5

Find: x - ?

Use trigonometry:

[tex] \sin(x°) = \frac{fg}{fh} [/tex]

[tex] \sin(x°) = \frac{8.4}{9.5} ≈0.8842[/tex]

[tex]x≈62.15°[/tex]

To find the probability that a plant chosen at random will be between and cm tall, we need to use the normal distribution formula. We know that the mean height of the plant is cm and the standard deviation is cm.

First, we need to standardize the values of and by subtracting the mean and dividing by the standard deviation:

Z1 = ( - ) / = ( - ) /
Z2 = ( - ) / = ( - ) /

Next, we use a standard normal distribution table or calculator to find the area under the curve between these two Z-values. This area represents the probability that a plant chosen at random will have a height between and cm.

Alternatively, we can use the normal distribution function on a calculator or software to find the probability directly. The formula for the normal distribution function is:

P( < X < ) = 1/2[erf(( - )/sqrt(2)) - erf(( - )/sqrt(2))]

where erf is the error function.

Using either method, we can find that the probability that a plant chosen at random will be between and cm tall is approximately %.

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When using the "rule of thirds" when examining an extremity, the bone is divided into thirds. Therefore, the correct option is option C.

First aid is the initial and urgent help provided to anyone who has a little or major disease or injury,[1] with the goal of preserving life, preventing the condition from getting worse, or promoting recovery until medical help arrives. First aid is typically administered by a person with only little medical training. The idea of first aid is expanded to include mental health in mental health first aid. When using the "rule of thirds" when examining an extremity, the bone is divided into thirds.

Therefore, the correct option is option C.

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An advantage of stem-and-leaf plots compared to most frequency distributions is that provide more information about the distribution of the data.

Stem-and-leaf plots offer several advantages over most frequency distributions.

One advantage is that stem-and-leaf plots provide a more detailed representation of the data than frequency distributions.

They allow you to see the individual data values and their magnitudes, which can provide more information about the distribution, such as the spread, central tendency, and outliers.

Additionally, stem-and-leaf plots can be easier to read and interpret than frequency distributions, especially for small data sets.

They can reveal patterns and trends in the data that might not be apparent in a frequency distribution.

Finally, stem-and-leaf plots can be used to compare different data sets or to identify similarities or differences within a single data set.

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The probability that Max will get the job is 17/28 or approximately 0.61. This is calculated by subtracting the sum of the probabilities of the other two companies getting the job from 1.

To calculate the probability that Max's construction company will get the job, we first need to understand that the sum of probabilities for all three companies must equal 1. Let the probability of Max's company getting the job be represented by P(Max).

Since the probabilities of the two competing companies are 1/7 and 1/4, we can write the equation:

P(Max) + 1/7 + 1/4 = 1

To solve for P(Max), we first need to find a common denominator for the fractions. The least common denominator for 7 and 4 is 28. So, we can rewrite the equation as:

P(Max) + 4/28 + 7/28 = 1

Now, we can combine the fractions:

P(Max) + 11/28 = 1

To find P(Max), we subtract 11/28 from 1:

P(Max) = 1 - 11/28

Since 1 is equal to 28/28, the equation becomes:

P(Max) = 28/28 - 11/28

Now, we subtract the fractions:

P(Max) = 17/28

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The perimeter of the rectangle is 24 inches.

We have,

Length = 8 inch

Width = 4 inch

So, Perimeter of rectangle

= 2 (l + w)

= 2 (8 + 4)

= 2 x 12

= 24 inches.

Thus, the required perimeter is 24 inches.

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