Find the distance between the two points rounding to the nearest tenth (if necessary).
(7,-7) and (1, 2)

The proportion is solved and the equivalent value of A = 5 , when R = 3.6

Given data ,

If R is inversely proportional to A, we can write:

R = k / A

where k is a constant of proportionality. To find the value of k, we can use the given information that when A = 1.5, R = 12:

12 = k / 1.5

Multiplying both sides by 1.5, we get:

k = 18

Now we can use this value of k to find R when A = 5:

R = 18 / 5

Simplifying, we get:

R = 3.6

Hence , when A = 5, R is equal to 3.6 and the proportion is solved

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The vector representing the speed and direction of the motorboat is approximately 20.22 mph at an angle of 8.53 degrees with respect to the original direction of the boat.

To find the vector representing the speed and direction of the motorboat, we need to use vector addition. Let the velocity of the boat in still water be Vb and the velocity of the current be Vc. Then, the resulting velocity of the boat relative to the ground is Vr = Vb + Vc.
Since the boat is heading directly across the river, the velocity of the current is perpendicular to the direction of the boat. This means that we can use the Pythagorean theorem to find the magnitude of Vr:
|Vr|^2 = |Vb|^2 + |Vc|^2
|Vr|^2 = (20 mph)^2 + (3 mph)^2
|Vr|^2 = 409
|Vr| ≈ 20.22 mph
To find the direction of Vr, we can use trigonometry. Let θ be the angle between Vr and Vb. Then:
tan(θ) = |Vc| / |Vb|
tan(θ) = 3 / 20
θ ≈ 8.53 degrees

The true speed of the motorboat is simply the magnitude of Vb:  |Vb| = 20 mph

To find the vector representing the speed and direction of the motorboat, we need to consider both the motorboat's speed in still water and the river current's speed.
Step 1: Identify the motorboat's speed in still water (20 mph) and the river current's speed (3 mph).
Step 2: Represent the motorboat's speed as a vector. Since it is heading directly across the river, we can represent it as a horizontal vector: V_motorboat = <20, 0>.
Step 3: Represent the river current's speed as a vector. The current flows perpendicular to the motorboat's direction, so we can represent it as a vertical vector: V_current = <0, 3>.
Step 4: Add the motorboat's vector and the current's vector to find the resultant vector, which represents the true speed and direction of the motorboat: V_resultant = V_motorboat + V_current = <20, 0> + <0, 3> = <20, 3>.
Now we have the vector representing the speed and direction of the motorboat: <20, 3>.
To find the true speed, calculate the magnitude of the resultant vector: True speed = sqrt(20^2 + 3^2) = sqrt(400 + 9) = sqrt(409) ≈ 20.22 mph.
To find the direction, calculate the angle (θ) between the resultant vecor and the x-axis using the tangent function: tan(θ) = (3/20)
θ = arctan(3/20) ≈ 8.53 degrees.
The true speed of the motorboat is approximately 20.22 mph, and its direction is approximately 8.53 degrees from the direct path across the river.

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The 6th term of the geometric sequence for the first sequence is -15625.

The 6th term of the geometric sequence for the second sequence is -762.875

The common ratio of the sequence is found by dividing any term by its preceding term.

For the first sequence:

Common ratio = (-5) / (-1) = 5

To find the 6th term, we can use the formula for the nth term of a geometric sequence:

a_n = a_1 * r^(n-1)

where a_1 is the first term, r is the common ratio, and n is the term we want to find.

For the first sequence, we have:

a_1 = -1

r = 5

n = 6

a_6 = (-1) * 5^(6-1) = -15625

So the 6th term of the first sequence is -15625.

For the second sequence:

Common ratio = (-7) / (-2) = 3.5

Using the same formula, we have:

a_1 = -2

r = 3.5

n = 6

a_6 = (-2) * 3.5^(6-1) = -762.875

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The completed relationship is:

0.001 grams = 0.0000000551 µg

To complete the relationship, we need to convert the units of the left-hand side and simplify the right-hand side.

Starting with the left-hand side:

1 mg = 1 milligram = 0.001 grams

Now, we can substitute this into the relationship:

For the right-hand side:

1100 in binary is equal to[tex]12^3 + 12^2 + 02^1 + 02^0 = 8 + 4 = 12[/tex]

10001000 in binary is equal to [tex]12^7 + 02^6 + 02^5 + 02^4 + 12^3 + 02^2 + 02^1 + 02^0 = 128 + 8 = 136[/tex]

100 in binary is equal to [tex]12^2 + 02^1 + 0*2^0 = 4[/tex]

Now, we can substitute these values into the relationship:

0.001 grams = 12 µg / 136 / 4

Simplifying the right-hand side:

12 µg / 136 / 4 = 12 * (1/1000000) * (1/136) * (1/4) = 0.00000005514706...

Rounding this to a reasonable number of significant digits, we get:

0.0000000551

Therefore, the completed relationship is:

0.001 grams = 0.0000000551 µg

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Answer:

c

Step-by-step explanation:

Answer:

To determine which graph represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8), we can analyze the behavior of the function as x approaches infinity and negative infinity, as well as the location and behavior of any vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts.

First, let's factor the denominator of the rational function:

x^2 - 2x - 8 = (x - 4)(x + 2)

Therefore, the rational function can be written as:

f(x) = (x^2 - 16)/((x - 4)(x + 2))

To find any vertical asymptotes, we need to look for values of x that make the denominator of the rational function equal to zero. Since the denominator is a product of two linear factors, the values that make it equal to zero are x = 4 and x = -2. Therefore, the rational function has vertical asymptotes at x = 4 and x = -2.

To find any horizontal asymptotes, we can look at the behavior of the function as x approaches infinity and negative infinity. As x becomes very large (either positive or negative), the highest degree term in the numerator and denominator of the rational function will dominate the expression. In this case, both the numerator and denominator have a highest degree of x^2, so we can apply the horizontal asymptote rule and divide the leading coefficient of the numerator by the leading coefficient of the denominator. This gives us:

y = 1

Therefore, the rational function has a horizontal asymptote at y = 1.

To find any x-intercepts, we need to look for values of x that make the numerator of the rational function equal to zero. Since the numerator is a difference of two squares, we can factor it as:

x^2 - 16 = (x - 4)(x + 4)

Therefore, the rational function has x-intercepts at x = 4 and x = -4.

To find the y-intercept, we can set x = 0 in the rational function:

f(0) = (-16)/(-8) = 2

Therefore, the rational function has a y-intercept at y = 2.

Based on this information, we can sketch the graph of the rational function as follows:

Out of the provided graphs, only graph (C) matches this description. Therefore, graph (C) represents the rational function f(x) = (x^2 - 16)/(x^2 - 2x - 8).

Answer: C

Step-by-step explanation:

The calculated value of the most reasonable product of 5 10/11 and 3 16/17 is 24

The numbers whose products are to be estimated are given as

5 10/11 and 3 16/17

Express the numbers as decimals

So, we have

5.91 and 3.94

Approximating the numbers, we have

6 and 4

So, the most reasonable product of 5 10/11 and 3 16/17 is

Product = 6 * 4

Evaluate the products of 6 and 4

So, we have

Product = 24

Hence, the most reasonable product of 5 10/11 and 3 16/17 is 24

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Complete question

Which of the following is the most reasonable product of 5 10/11 and 3 16/17

The mean function for {Yt} is 5 + 2t, and the autocovariance function is γk + 2γk(k+1), which implies that {Yt} is stationary.

a. To find the mean function for {Yt}, we take the expected value of Yt:

E(Yt) = E(5 + 2t + Xt)

= 5 + 2t + E(Xt)

Since {Xt} is a zero-mean stationary series, E(Xt) = 0. Therefore, the mean function for {Yt} is 5 + 2t.

b. To find the autocovariance function for {Yt}, we start with the definition:

γYk = Cov(Yt, Yt-k)

= Cov(5 + 2t + Xt, 5 + 2(t-k) + Xt-k)

= Cov(Xt, Xt-k) + 2Cov(t,Xt-k)

Since {Xt} is stationary, its autocovariance function is γk for all k. Thus, Cov(Xt, Xt-k) = γk.

Using the fact that Cov(t, Xt-k) = E(tXt-k) - E(t)E(Xt-k) = 0 (because {Xt} is stationary and t is deterministic), we have:

γYk = γk + 2(0) = γk

Therefore, the autocovariance function for {Yt} is γk, which is the same as the autocovariance function for {Xt}.

c. To determine if {Yt} is stationary, we need to check if its mean and autocovariance functions are constant over time.

As we found in part (a), the mean function for {Yt} is 5 + 2t, which is a linear function of time. Therefore, the mean is not constant over time, and {Yt} is not strictly stationary.

However, the autocovariance function for {Yt} is γk + 2γk(k+1), which does not depend on time. Therefore, {Yt} is weakly stationary, since its autocovariance function is constant over time.

Therefore, the answer is: {Yt} is weakly stationary, since its autocovariance function is constant over time, although its mean function is not constant over time.

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Weight of an apple: quantitative continuous.The most common color of apples in a bag: qualitative.Number of apples in a two-pound bag: quantitative discrete.

1. Weight of an apple: The weight is a measurable quantity with numerical values. Therefore, this data is quantitative. Since weight can take on any value within a range (e.g., 5.2 oz, 5.25 oz), it is continuous. So, the data is quantitative continuous.

2. The most common color of apples in a bag: Color is a non-numerical characteristic that describes the apples. This data is qualitative.

3. Number of apples in a two-pound bag: The number of apples is a countable numerical value. This data is quantitative. Since it can only be a whole number (e.g., 5 apples, 6 apples), it is discrete. So, the data is quantitative discrete.

Your answer:

1. Weight of an apple - Quantitative continuous
2. The most common color of apples in a bag - Qualitative
3. Number of apples in a two-pound bag - Quantitative discrete

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The circumference, in feet of the parachute in discuss as required to be determined is; Choice C; 12pi feet.

It follows from the task content that the circumference, in feet of the parachute in discuss is to be determined.

Since the parachute is said to be circular, it's circumference is given by; C = 2pi × radius

Hence, since the radius is 6, we have that;

C = 2pi × 6 feet

C = 12pi feet.

Consequently, choice C; 12pi feet is correct.

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The finance charge, based on the annual interest rate of 18 % would be $ 9. 07.

You need to find the periodic rate :

=  18 %  / 365

=  0. 04931506849315

Then the average daily balance :

Days 1 - 7 : $ 800 balance

Days 8 - 15 : $ 800 + $ 600 = $ 1400 balance

Days 16 - 20 : $ 1400 - $ 1000 = $ 400 balance

This allows us to find the average daily balance :
= ( ( 800 x 7 ) + ( 1, 400 x 8 ) + ( 400 x 5 ) ) / 20

= $ 940

The finance charge is:

= 940 x 0. 04931506849315 x 20

= $ 9. 07

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Answer:

n=1\2

Step-by-step explanation:

According to the given triangle, the value of ∠AFE = 48 degrees. (option b).

Let's start by looking at the given figure. We have an equilateral triangle ABC, which means all its sides are equal, and all its angles are 60 degrees. We are also given a point D outside the triangle and an angle CAD of 18 degrees. This means that the angle CAD and the angle BAC are supplementary (they add up to 180 degrees), since they both share the side AC.

Now, we can apply the angle bisector theorem to the triangle AEB. This tells us that AE/EB = AD/DB. Since CD = BE and triangle ADE is congruent to triangle BDC, we can say that

AD/DB = AE/EC = (AE + EC)/EC = AC/EC.

Therefore,

=> AE/EC = AC/EC - 1 = (2 cos 18 - 1)/sin 18.

Solving for sin ∠AFE, we get sin

∠AFE = (sin 12)(sin (42 - x))/sin (126 - x)(sin (60 - x)).

Taking the inverse sine of this value, we get

∠AFE = 48 degrees.

Therefore, the answer is option B, 48 degrees.

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Answer:

  97 feet

Step-by-step explanation:

Given a 25-foot gorilla that Kelsey wants to stabilize with ropes at angles of 60°, 45°, and 50°, we want the total length of rope required.

When the geometry is modeled by a right triangle, the sine function relates the height of the gorilla, the rope length, and the angle by ...

  Sin = Opposite/Hypotenuse

Here, the angle is opposite the gorilla height, and the length of rope is the hypotenuse of the right triangle. For each angle, we have ...

  sin(θ) = (25 ft)/(rope length)

  rope length = (25 ft)/sin(θ)

The total rope length required will be the sum of the individual lengths for the different angles:

  total length = (25 ft)/sin(60°) +(25 ft)/sin(45°) +(25 ft)/sin(50°)

  total length = 28.9 ft +35.4 ft +32.6 ft = 96.9 ft

Kelsey needs about 97 feet of rope.

<95141404393>

The value of the product [tex]$abc = 1 * 3 * 1 = 3$[/tex].

To express the number [tex]$345{,}600$[/tex] as a product of powers of prime numbers, we first need to find the prime factorization of the given number.

[tex]$345{,}600 = 2^3 * 3 * 5^3 * 23$[/tex]

Now we need to rewrite the given expression [tex]$6^a5^b4^c$[/tex] in terms of prime numbers:

[tex]$6^a5^b4^c = (2 * 3)^a * 5^b * (2^2)^c = 2^{a+2c} * 3^a * 5^b$[/tex]

By comparing the exponents of the prime factorization of [tex]$345{,}600$[/tex] with the rewritten expression, we can determine the values of [tex]$a$[/tex], [tex]$b$[/tex], and [tex]$c$[/tex]:

[tex]a+2c=3$, \\$a=1$, \\and $b=3$[/tex]

Solving for [tex]$c$[/tex], we get:

[tex]$c = (3 - a) / 2 = (3 - 1) / 2 = 1$[/tex]

Now we have [tex]a=1$, $b=3$, and $c=1$[/tex]. The product [tex]$abc = 1 * 3 * 1 = 3$[/tex].

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Using the substitution partial fraction method to find the indefinite integral, we have: [tex]\mathbf{\dfrac{1}{2}x^2-x+ In(|(x-1)(x+2)|)+C}[/tex]

The method of solving partial fractions using the substitution method is called partial fraction decomposition. The steps in evaluating the indefinite integral are as follows:

Given that:

[tex]\int (\dfrac{x^3-x+3}{x^2+x-2})dx[/tex]

We need to remove the parentheses in the denominator and write the fraction by using the partial fraction decomposition.

[tex]\int \dfrac{x^3-x+3}{x^2+x-2}dx[/tex]

[tex]\int x-1+\dfrac{1}{x-1}+\dfrac{1}{x+2}dx[/tex]

Now, this process is followed by splitting the two integrals into multiple integrals.

[tex]\int xdx + \int-1dx +\int \dfrac{1}{x-1}dx + \int \dfrac{1}{x+2 }dx[/tex]

By using the power rule, the integral of x with respect to x is [tex]\dfrac{1}{2}x^2[/tex]

[tex]\dfrac{1}{2}x^2+C+ \int-1dx +\int \dfrac{1}{x-1}dx + \int \dfrac{1}{x+2 }dx[/tex]

Now, Let's apply the constant rule

[tex]\dfrac{1}{2}x^2+C-x+C +\int \dfrac{1}{x-1}dx + \int \dfrac{1}{x+2 }dx[/tex]

Such that; [tex]u_1 = x - 1[/tex], Then [tex]du_1 = dx[/tex]. So, we can now rewrite it as [tex]u_1 \ and \ du_1[/tex].

[tex]\dfrac{1}{2}x^2+C-x+C +\int \dfrac{1}{u_1}du_1 + \int \dfrac{1}{x+2 }dx[/tex]

Furthermore, taking the integral  of [tex]\dfrac{1}{u_1}[/tex] with respect to [tex]u_1[/tex] is [tex]\mathbf{In (|u_1|)}[/tex]

[tex]\dfrac{1}{2}x^2+C-x+C +In(|u_1|)+C + \int \dfrac{1}{x+2 }dx[/tex]

Now, let [tex]u_2 = x +2[/tex] such that [tex]du_2 = dx[/tex]. So, we can now rewrite it as [tex]u_2 \ and \ du_2[/tex].

[tex]\dfrac{1}{2}x^2+C-x+C +In(|u_1|)+C + \int \dfrac{1}{u_2 }du_2[/tex]

The integral  of [tex]\dfrac{1}{u_2}[/tex] with respect to [tex]u_2[/tex] is [tex]\mathbf{In (|u_2|)}[/tex]

[tex]\dfrac{1}{2}x^2+C-x+C +In(|u_1|)+C + In(|u_2|)+C[/tex]

By simplifying the above process;

[tex]\dfrac{1}{2}x^2-x+ In(|u_1*u_2|)+C[/tex]

Now, using the substitution method to substitute back in for each integration substitution variable, we have:

[tex]\mathbf{\dfrac{1}{2}x^2-x+ In(|(x-1)(x+2)|)+C}[/tex]

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The theoretical probability is 1/6 or approximately 0.1667. The colors are red, blue, green, yellow, purple, and orange. The theoretical probability of landing on any one of these colors with one spin is 1/6 or approximately 0.1667.

This means that if the spinner is spun many times, we would expect the spinner to land on each color approximately 1/6 of the time.  To calculate the theoretical probability, we simply divide the number of outcomes we are interested in by the total number of possible outcomes. In this case, there is one outcome we are interested in (landing on a specific color) and there are six possible outcomes (landing on any one of the six colors). Therefore,

It is important to note that theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. In reality, there may be other factors that can affect the outcomes of the spinner such as the force of the spin or the shape of the spinner. However, assuming that these factors are constant, the theoretical probability can be a useful tool in predicting the likelihood of certain events occurring.

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Yes, the line should be parallel to the triangular faces.

We have,

Bella has drawn a line to represent the parallel cross-section of the triangular prism.

A cross-section is a 2-dimensional shape that is obtained by slicing a 3-dimensional object.

In the case of a triangular prism, if you slice it parallel to one of the rectangular faces, the resulting cross-section will be a rectangle.

The base of a triangular prism is a triangular face.

A "parallel cross-section" is a cross-section taken parallel to the base.

Hence, the parallel cross section should be parallel to the triangular faces.

So, the correct answer is:

Yes, the line should be parallel to the triangular faces.

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The annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car would be $20,000.

You need to know the employee's taxable income to determine the annual tax payment for a worker paying a 40% tax rate who does not contribute financially to an executive automobile.

Assume the employee does not receive any deductions or allowances, and their taxable income is $50,000 per year.

You must first determine their gross income tax, which is the amount of tax they would pay if their whole taxable income were subject to a 40% tax.

Gross Income Tax = Taxable Income x Tax Rate

Gross Income Tax = $50,000 x 0.4 = $20,000

Then, take away any deductions or allowances to which the employee is entitled. The employee's net income tax in this instance is equal to their gross income tax because it is assumed that they are not eligible for any deductions or allowances.

Therefore, the annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car would be $20,000.

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The complete question is -

What is the annual tax payment for an employee at the 40% tax rate who does not make a financial contribution towards an executive car and earns taxable income of $50,000 per year?

The dimensions of the rectangular prism-shaped birdhouse are 742 inches, 4 inches, and 746 inches.

We have,

Let's first convert the volume of the birdhouse from cubic feet to cubic inches, since the dimensions are given in inches.

1 cubic foot = 12 inches x 12 inches x 12 inches = 1728 cubic inches

The volume of the birdhouse in cubic inches is:

= 128 cubic feet x 1728 cubic inches per cubic foot

= 221184 cubic inches

Let's represent the width of the birdhouse in inches as w, and its height

as h.

We know that the depth is 4 inches, and the height is 4 inches greater than the width.

Width = w inches

Depth = 4 inches

Height = w + 4 inches

The volume of the birdhouse is given by the formula:

Volume = Width x Depth x Height

Substituting the given values, we get:

221184 cubic inches = w x 4 inches x (w + 4) inches

Simplifying and rearranging, we get:

w + 4w - 55296 = 0

Using the quadratic formula, we find:

w = (-4 ± √(4² - 4(1)(-55296))) / (2(1))

w = (-4 ± √(2220800)) / 2

w = (-4 ± 1488) / 2

Since the width must be positive, we discard the negative solution and get:

w = (-4 + 1488) / 2

w = 742

Therefore,

The dimensions of the rectangular prism-shaped birdhouse are 742 inches, 4 inches, and 746 inches.

The width of the birdhouse is 742 inches, the depth is 4 inches, and the height is w + 4 = 746 inches.

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Answer:

adding five

Step-by-step explanation:

-8 + 5 = -3

-3 + 5 = 2

2 + 5 = 7

7 + 5 = 12

Answer:

This is the probability density function of X.

Step-by-step explanation:

To find the PDF of X, we need to find the probability of each possible value of X.

When two dice are thrown, the possible outcomes are the integers from 2 to 12, with each outcome having an equal probability of 1/36. Therefore, the PDF of X can be represented as follows:

P(X = 2) = 1/36

P(X = 3) = 2/36

P(X = 4) = 3/36

P(X = 5) = 4/36

P(X = 6) = 5/36

P(X = 7) = 6/36

P(X = 8) = 5/36

P(X = 9) = 4/36

P(X = 10) = 3/36

P(X = 11) = 2/36

P(X = 12) = 1/36

This is the probability density function of X.

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Answer:

it's the natural error that exists between a sample and it's corresponding population

Since f'(x) is always greater than or equal to 2, we know that f(x) is increasing at a rate of at least 2 for any x value between 2 and 6. Using the Mean Value Theorem, we can say that the change in f(x) between x = 2 and x = 6 is at least 8 (2 times the distance between 2 and 6).

We also know that f(2) = 1, so the minimum value of f(6) would be 9 (1 + 8). However, we cannot say for certain that f(6) is exactly 9, as there could be other factors affecting the function that we do not know about.
To find the smallest possible value of f(6), we'll use the given information: f(2) = 1, f '(x) ≥ 2, and 2 ≤ x ≤ 6.

Step 1: Understand the relationship between f '(x) and f(x).
Since f '(x) is the derivative of f(x), it represents the slope of the tangent line to the curve of f(x) at any point x. In this case, f '(x) ≥ 2 means that the function f(x) is increasing with a slope of at least 2 for the interval 2 ≤ x ≤ 6.

Step 2: Determine the smallest possible increase in f(x) from x=2 to x=6.
Since the slope is at least 2, the smallest increase in f(x) occurs when the slope is exactly 2. We can calculate this increase as follows:
Increase in f(x) = (slope) × (change in x) = 2 × (6 - 2) = 2 × 4 = 8

Step 3: Calculate the smallest possible value of f(6).
Using the increase in f(x) from step 2, and the given value of f(2) = 1, we can find the smallest possible value of f(6):
f(6) = f(2) + increase in f(x) = 1 + 8 = 9

So, the smallest possible value of f(6) is 9.

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Answer:

Step-by-step explanation:

Pictograph

Pictograms are a powerful tool for visualizing data and are widely used in a variety of different fields, from marketing and advertising to science and education.

A pictogram is a type of chart that uses symbols instead of words or numbers to portray data. Pictograms are often used in data visualization and are particularly useful for presenting complex information in a simple and easily understandable way. Pictograms can be used to represent a wide range of data, including statistical information, demographic data, and geographical information. They are also commonly used in advertising and marketing, as they are a powerful tool for communicating ideas and concepts quickly and effectively. Pictograms can be created using a variety of different techniques, including hand-drawn illustrations, computer-generated graphics, and photographs. They are typically presented in a grid format, with each symbol representing a single data point. Pictograms can be used to show trends, compare data sets, and highlight key points. They are also a great way to make data more engaging and interactive, as users can explore the data by clicking on individual symbols or groups of symbols.

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a) To find the value of k that makes f(y) a probability density function, we need to ensure that the integral of f(y) over the entire range of y is equal to 1. That is:

∫[0,1] k y(1 − y) dy = 1.

Solving this integral, we get:

k ∫[0,1] y(1 − y) dy = 1

k [(1/2)y^2 - (1/3)y^3] [0,1] = 1

k (1/6) = 1

k = 6.

Therefore, f(y) is a probability density function with k = 6.

b) To find P(0.4 ≤ Y ≤ 1), we need to integrate f(y) over the range [0.4,1]:

P(0.4 ≤ Y ≤ 1) = ∫[0.4,1] f(y) dy

= ∫[0.4,1] 6y(1 − y) dy

= 0.54.

Therefore, P(0.4 ≤ Y ≤ 1) = 0.54.

c) To find P(Y ≤ 0.4|Y ≤ 0.8), we use the formula for conditional probability:

P(Y ≤ 0.4|Y ≤ 0.8) = P(Y ≤ 0.4 and Y ≤ 0.8)/P(Y ≤ 0.8)

= P(Y ≤ 0.4)/P(Y ≤ 0.8)

= [∫[0,0.4] 6y(1 − y) dy]/[∫[0,0.8] 6y(1 − y) dy]

= 0.0225/0.36

= 0.0625.

Therefore, P(Y ≤ 0.4|Y ≤ 0.8) = 0.0625.

a) To find the value of c that makes f(y) a probability density function, we need to ensure that the integral of f(y) over the entire range of y is equal to 1. That is:

∫[0,2] c y dy = 1.

Solving this integral, we get:

c ∫[0,2] y dy = 1

c (1/2) y^2 [0,2] = 1

c = 1/2.

Therefore, f(y) is a probability density function with c = 1/2.

b) To find F(y), we integrate f(y) from 0 to y:

F(y) = ∫[0,y] (1/2) y dy

= (1/4) y^2.

For y < 0 or y > 2, F(y) = 0.

Therefore, the cumulative distribution function F(y) is given by:

F(y) = { 0, y < 0

    (1/4) y^2, 0 ≤ y ≤ 2

    1, y > 2 }

c) To find P(1 ≤ Y ≤ 2), we use the cumulative distribution function:

P(1 ≤ Y ≤ 2) = F(2) - F(1)

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

= 3/4.

Therefore, P(1 ≤ Y ≤ 2) = 3/4.

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Answer:

[tex] {x} = \sqrt{ {5}^{2} + {12}^{2} } = \sqrt{169} = 13 [/tex]

[tex]y = \sqrt{ {3}^{2} + {5}^{2} } = \sqrt{34} [/tex]

x + y = 13 + √34 = 18.83 (to 2 decimal places)

The calculated values of x + y in the triangle is 18.83

From the question, we have the following parameters that can be used in our computation:

The right triangle

Using the above as a guide, we have the following Pythagoras theorem

x^2 = 5^2 + 12^2

y^2 = 5^2 + 3^2

When the above equations are evaluaed

So, we have the following representation

x = 13

y = √34

So, we have

x + y = 13 + √34

Evaluate

x + y = 18.83

Hence, the value of x + y is 18.83

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Answer:

151

Step-by-step explanation:

180-49(complimentary angles sum up to 180

The sample mean of the population is 3/4 and the variance is 3/80. Using the central limit theorem, P([tex]\bar{X}[/tex] > 0.8) can be simplified as 0.003.

The mean of the population can be computed as follows:

µ = ∫x f(x) dx from 0 to 1

= ∫x (3x²) dx from 0 to 1

= 3/4

The variance of the population can be computed as follows:

σ² = ∫(x-µ)² f(x) dx from 0 to 1

= ∫(x-(3/4))² (3x²) dx from 0 to 1

= 3/80

By the Central Limit Theorem, as the sample size n = 80 is large, the distribution of the sample mean [tex]\bar{X}[/tex] can be approximated by a normal distribution with mean µ and variance σ²/n.

Therefore, P([tex]\bar{X}[/tex] > 0.8) can be approximated by P(Z > (0.8-0.75)/(sqrt(3/80)/sqrt(80))), where Z is a standard normal random variable.

Simplifying, we get P([tex]\bar{X}[/tex] > 0.8) ≈ P(Z > 2.73) ≈ 0.003.

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