Find the circumference of the circle shown

The average velocity of the function y = x² + 8x over the interval [5, 8] is 21.

Average velocity is a measure of the rate of change of an object's position over time. It is calculated by dividing the total change in position (displacement) of an object by the time interval over which the change occurred.

In mathematical terms, average velocity is given by the formula: v_avg = Δx / Δt, where Δx is the change in position (displacement) of the object and Δt is the time interval over which the change occurred. The average velocity is a vector quantity, meaning that it has both magnitude and direction.

The average velocity of an object gives a measure of its average speed and direction over a certain time interval. It is a useful concept in physics, especially in the study of mechanics and kinematics, where it is used to describe the motion of objects under different conditions and to calculate the velocity of objects at a specific instant in time.

The average velocity over an interval [a, b] is defined as the total change in position (y) divided by the total change in time (x), i.e., (y(b) - y(a)) / (b - a).

Given the function y = x² + 8x, we have:

y(5) = 5² + 8 × 5 = 25 + 40 = 65

y(8) = 8² + 8 × 8 = 64 + 64 = 128

So, the average velocity over the interval [5, 8] is:

(y(8) - y(5)) / (8 - 5) = (128 - 65) / (8 - 5) = 63 / 3 = 21.

Therefore, the average velocity of the function y = x² + 8x over the interval [5, 8] is 21.

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As per the given joint density function, the marginal density function for y1 is f1(y1) = 1/y1, where y1 > 0.

Joint density functions are used to describe the distribution of two or more variables. They provide the probability of observing certain values for multiple variables at the same time.

For the given joint density function

[tex]= > f(y1, y2) = e^{-(y1 y2)}, y1 > 0, y2 > 0[/tex]

and 0 elsewhere, we are asked to find the marginal density function for y1, which is denoted by f1(y1).

Here the marginal density function of y1 is found by integrating the joint density function over all possible values of y2. It gives us the probability density for y1 alone, without considering the value of y2.

In order to find f1(y1), we integrate f(y1, y2) with respect to y2, from 0 to infinity:

=> f1(y1) = ∫ f(y1, y2)dy2

[tex]= > \int f(y1, y2) = \int {e^{-(y1 y2)},dy2}[/tex]

[tex]= -(1/y1)e^{-(y1 y2)}[/tex]

= −(1/y1) * 0 + (1/y1) = 1/y1

Here it is important to note that the marginal density function must always be non-negative, since it represents a probability density.

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The volume of the triangular prism is 168 cm³ while the surface area of the rectangular and triangular prism are 286 cm² and 270 ft² respectively

An equation is an expression that shows the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on

The surface area of a solid figure is gotten by calculating each area of the surface and then summing.

a) For the rectangular prism:

Surface area = 2(9 * 7) + 2(9 * 5) + 2(7 * 5) = 286 cm²

b) For the triangular prism:

Surface area = 2(0.5 * 12 * 5) + (7 * 12) + (7 * 13) + (5 * 7) = 270 ft²

3) Volume  = (1/3) * base area * height = (1/3) *  (8 * 9) * 7 = 168 cm³

The volume of the triangular prism is 168 cm³

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Answer:

72 sq ft

Step-by-step explanation:

The area of the square pool is 6ft * 6ft = 36 sq.ft.

The perimeter of the pool is 2 * (6ft + 6ft) = 24ft.

The length of the gravel path is equal to the perimeter of the pool, so 24ft.

The width of the gravel path is 3ft, so the total area of the gravel path is 24ft * 3ft = 72 sq.ft.

Since the corners of the path are rounded, we can assume that the loss of area is negligible and assume that the area of the gravel path is 72 sq.ft.

So, the approximate area of the gravel path is 72 sq.ft.

Answer:

The area of the rectangle is  [tex]8\sqrt{6}+\sqrt{18}[/tex]  ft².

Step-by-step explanation:

The formula for the area of a rectangle is length times width

[tex]A=lw[/tex]

We are given

[tex]l=8+\sqrt{3}[/tex]

[tex]w=\sqrt{6}[/tex]

Given this information we can solve for the area.

[tex]A=(8+\sqrt{3})*\sqrt{6}[/tex]

Use the Distributive Property to distribute [tex]\sqrt{6}[/tex] over [tex]8+\sqrt{3}[/tex].

[tex]A=8\sqrt{6}+\sqrt{6} \sqrt{3}[/tex]

We can simplify this to

[tex]A=8\sqrt{6}+ 3\sqrt{2}[/tex]  or  [tex]A=8\sqrt{6}+\sqrt{18}[/tex]

The domain and range of x-intercept (600,0) is Domain: {600}, Range: {0}

The domain and range of y-intercept (0,300) is Domain: {0}, Range: {330}

What is the x-intercept?

The intersection of a function or equation's graph with the x axis is known as the x intercept. This can be compared to a point having a value of 0.

For any given curve, the x-intercept is a coordinate that is plotted on the x-axis. To put it another way, the x-intercept is the value of the x coordinate at the point where the graph crosses the x-axis, or we can say that it is the value of the x coordinate at a point where the value of the y coordinate equals zero.

Let  s= x, y=a

5x+10y=3000

x-intercept:

y=0

5x+10(0) = 3000

5x=3000

=> x = 3000/5 = 600

x-intercept is (600,0)

y-intercept:

x=0

5(0)+10y = 3000

10y=3000

=> y = 3000/10 = 300

y-intercept is (0,300)

Plot these points and join them

graph attached.

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Answer:

Neither

Step-by-step explanation:

Parallel slopes are the same and perpendicular slopes are opposite reciprocals which would be like 2/3 and -3/2.

If the interest is compounded continuously, an investment account with the same initial investment and the same APR will have the highest APY.

APY (Annual Percentage Yield) is the effective annual rate of return after compounding. The higher the APY, the more frequently the interest is compounded.

Compounding yields the highest APY because it effectively provides an infinite number of compounding periods. Compounding daily, monthly, or annually yields lower APYs than compounding continuously.

An investment account is a type of financial product that allows people to save or invest money in the hopes of earning a profit. Individual retirement accounts (IRAs), brokerage accounts, savings accounts, and other types of investment accounts are available.

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For each nth term of the sequence, we can multiply 3 by itself (n-1) times and then multiply that result by 23.The nth term of the sequence b1, b2, b3, … , which is bn = 3^(2n-1) × 23.

A geometric sequence is a sequence of numbers in which each term is found by multiplying the previous term by a constant. In this case, the constant is 3 and the first term is 23. An explicit rule for the nth term of the sequence is given by bn = 3^(n-1) * 23. This rule states that bn is equal to 3 to the power of n minus 1, multiplied by 23. Therefore, for each nth term of the sequence, we can multiply 3 by itself (n-1) times and then multiply that result by 23. This gives us the nth term of the sequence.

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The correct statements are,

⇒ Vertical reflection and translation.

Reflection is a type of transformation that flip a shape in mirror line, so that each point is the same distance from the mirror line as its reflected point.

We have to given that;

Triangles STU and S'T'U' are congruent.

Now, We know that;

Reflection is a type of transformation that flip a shape in mirror line, so that each point is the same distance from the mirror line as its reflected point.

And,  The translated shapes (or the image) appear to be the same size as the original shape, indicating that they are congruent. They've simply shifted in one or more directions.

Hence, The correct statement to Triangles STU and S'T'U' are congruent are,

⇒ Vertical reflection and translation.

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Positive rational numbers cannot be added together.

If real numbers are multiplied, they are closed.

Irrational numbers cannot be divided in any way.

Subtraction causes closed integers.

Positive rational numbers cannot be added together.

a) The set is closed since the product of any two real numbers is a real number.

b) The quotient could occasionally be logical. (18/2) Equals 3, for instance. The set is thus not closed.

c) The set is closed since the difference between any two numbers is also an integer.

d) The set is closed since the product of any two positive rational numbers is a positive rational number.

The complete question is

Select from the drop-down menus to correctly identify whether the given operation is closed or not closed with respect to each set of numbers.

Real numbers are Closed/Not Closed under multiplication.

Irrational numbers are Closed/Not Closed under division.

Integers are Closed/Not Closed under subtraction.

Positive rational numbers are Closed/Not Closed under addition.

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36.74098197 kg in 81 lbs. Likewise, the question how many pounds in 81 kilogram has the answer of 178.57443237 lbs in 81 kg.

81 kg * 2.2046226218 lbs / 1 kg= 178.57443237lbs

The weight of 81 kilograms is equivalent to 178.57443237 pounds (81kg = 178.57443237lbs). It is simple to convert 81 kg to lb. Simply use our calculator above or the method to convert 81 kg to pounds. To convert 81 kg to pounds, multiply the kilogram mass by 2.2046226218. The formula for converting 81 kilograms to pounds is [lb] = 81 * 2.2046226218. Thus, 81 kilos in pounds are 178.57443237 pounds.

One pound is about equivalent to 0.45359237 kilos (kg). The pound to kilogram conversion is accomplished by multiplying the provided pound value by 0.45359237. In addition, one kilogram is about equal to 2.2046226218 pounds.

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1.102 pounds for an approximate result, divide the mass value by 453.6

453.592 grams equals one pound.As a result, you’ll frequently see that abbreviated to 453 grams. However, if you recall from elementary school math lesson, we usually round up when the first figure following a decimal point is five or greater, so you may also use 454 grams.

One pound is about equivalent to 0.45359237 kilos (kg). The provided pound value is multiplied by 0.45359237 pound, unit of avoirdupois weight, equivalent to 16 ounces, 7,000 grains, or 0.45359237 kg, and of troy and apothecaries’ weight, equal to 12 ounces, 5,760 grains, or 0.3732417216 kg. The acronym lb comes from the Roman predecessor of the contemporary pound.

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His mistake is the graph.

The correct graph is given below.

The coordinates in a graph indicate the location of a point with respect to the x-axis and y-axis.

The coordinates in a graph show the relationship between the information plotted on the given x-axis and y-axis.

We have,

y = (-1/2)x + 2

The coordinates on the graph will be:

When x = 0, 1, 2, 3

y = (-1/2)x + 2 = (-1/2) x 0 + 2 = 2

y = (-1/2) x 1 + 2 = -1/2 + 2 = 3/2 = 1.5

y = (-1/2) x 2 + 2 = -1 + 2 = 1

y = (-1/2) x 3 + 2 = -3/2 + 2 = 1/2 = 0.5

And so on....

Now,

The x-intercept.

i.e x = 0,

y = 2

The y-intercept.

i.e y = 0,
0 = (-1/2)x + 2

x = 4

Now,

The graph is given below.

We see there the y-intercept is at y = 2 and the x-intercept is at x = 4.

Thus,

The graph is a mistake.

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The electric field increases linearly with the position x along the x-axis.

The electric field E as a function of the position x can be found by taking the derivative of the electric potential V with respect to x:

E = -dV/dx = -d(-4x^2 + 3x - 8)/dx = 8x - 3

So the electric field E as a function of x is given by the equation:

E = 8x - 3

This means that the electric field increases linearly with the position x along the x-axis.

Therefore, The electric field increases linearly with the position x along the x-axis.

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At age of 25 years, the diameter would be 6.8612mm

At 50 years the diameter would be 5.422mm

What are the logarithms?

A logarithm is an exponent that indicates the power to which a certain number, called the base, must be raised to produce a given number.

d = 2.115Logₑa + 13.669

d = diameter of the pupil

a = number of years

Note : Logₑa = In a (check logarithmic rule)

d = 2.115Ina + 13.669

1. At 25 years,

d = -2.115In25 + 13.669

d = -2.115 × 3.2188 + 13.669

d = -6.807762 + 13.669

d = 6.8612mm

At 25 years, the pupil shrinks by 6.86mm

2. At 50 years,

d = -2.1158In50 + 13.669

d = -2.1158 * 3.912 + 13.669

d = -8.2770 + 13.699

d = 5.422mm

At 50 years, the pupil shirks by 5.422mm

Hence,

At age of 25 years, the diameter would be 6.8612mm

At 50 years the diameter would be 5.422mm

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Answer:

4800 liters fuel to travel 600 kilometers

Answer:

50L

Step-by-step explanation:

Given ,

To Find -

Now,

If Car travels 240 km in 20 L of petrol

Then,

The car travels 1 km in 20/240 L of petrol.

Then,

Petrol needed to travel 600 km

20/240 × 600

20/24 × 60

20/4 × 10

5 × 10

= 50 L

Hence, 50 L of petrol is needed to travel 600 km.

The area at right end point is Rn=7.484 and area at left end points is Ln=7.476

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here f(x) = [tex]\frac{1}{x^3}[/tex] over the interval [1,5] and n=10 then

Δx = [tex]\frac{b -a}{n} = \frac{5-1}{10} = \frac{4}{10} = \frac{2}{5}[/tex]

Δx = 0.4

Then subinterval are, [0,0.4],[0.4,0.8],[0.8,1.2],[1.2,1.6],[1.6,2],[2,2.4],[2.4,2.8],[2.8,3.2],[3.2,3.6],[3.6,4]

Then, area using right end point is

Rn = Δx[f(0.4)+f(0.8)+f(1.2)+f(1.6)+f(2)+f(2.4)+f(2.8)+f(3.2)+f(3.6)+f(4)]

=> Rn = 0.4[[tex]\frac{1}{0.4^3} +\frac{1}{0.8^3} +\frac{1}{1.2^3} +\frac{1}{1.6^3} +\frac{1}{2^3} +\frac{1}{2.4^3} +\frac{1}{2.8^3}+\frac{1}{3.2^3} +\frac{1}{3.6^3}+\frac{1}{4^3}[/tex]]

=> Rn=0.4[18.71]

=> Rn= 7.484

Now area using  Left end point is

=>Ln = Δx[f(0)+f(0.4)+f(0.8)+f(1.2)+f(1.6)+f(2)+f(2.4)+f(2.8)+f(3.2)+f(3.6)]

=> Ln = 0.4[[tex]\frac{1}{0^3}+\frac{1}{0.4^3} +\frac{1}{0.8^3} +\frac{1}{1.2^3} +\frac{1}{1.6^3} +\frac{1}{2^3} +\frac{1}{2.4^3} +\frac{1}{2.8^3}+\frac{1}{3.2^3} +\frac{1}{3.6^3}[/tex]

=> Ln = 0.4[18.69]

=> Ln =7.476

Hence the area at right end point is Rn=7.484 and area at left end points is Ln=7.476

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The mean has increased due to the elimination of the minimum value from the data. It is highly affected by the extreme value while the median is not affected to a very high extent.

According to the question,

Observation 44 is removed from the data.

The arranged data is,

{44,45,46,47,48,56,59,60}

The mean and median will be given, as follows:

[tex]Mean = \frac{59 + 47 + 56+ 46 + 48 + 60 +45 }{7}[/tex]

            = 51.5714

[tex]Mead = \frac{N+1}{2} th term[/tex]

            [tex]= \frac{7 + 1}{2}[/tex]  term

            = 4th term

Median = 47

Hence, it is concluded that the mean has increased due to the elimination of the minimum value from the data. Thus, it is highly affected by the extreme value while the median is not affected to a very high extent.

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The complete question is:

A study examined music performance anxiety. Scores on the Performance Anxiety Inventory (PAI) scale for participants in eight different studies is reproduced in the following table.

59, 47,56,44, 46, 48,60,45

Suppose the PAI score of 44 results from a study involving development-delayed children; consequently, the researchers eliminated this data value from the analysis. What impact does this have on the value of the mean? The median?

Based on the information in the statement, we can establish that the most accurate conclusion would be "we can draw conclusions about cause and effect for students like those in the study" (option B).

According to the information from the study, the conclusion that we can draw must be related to the students who participated in the study because this study only had an impact on these students. We cannot establish a conclusion in which a change occurs to other students because the other students in the school did not identify a change in their daily life.

So, the most appropriate conclusion would be "we can draw conclusions about cause and effect for students like those in the study" because they are the ones who noticed a change in their daily lives after participating in the study.

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The ratio of liters of yellow paint to red paint will make the same shade of orange paint is 10 to 6

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

5 liters of yellow paint with every 3 liters of red paint

This means that

Ratio = 5 litres of yellow : 3 liters of red

Multiply the ratio by 2

So, we have

Ratio = 10 litres of yellow : 6 liters of red

Hence, the ratio is 10 : 6

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The population of a city can be modeled by a radical function, P(x) = 55,000 √x - 1945, where x represents the year and P(x) represents the population in that year.

The year 1945 has been subtracted from x in order to simplify the equation and make it easier to interpret.

In this particular problem, we are asked to find the year in which the population of the city was 275,000. To do this, we need to solve for x in the equation:

55,000 √x - 1945 = 275,000

The first step is to isolate x on one side of the equation. We can do this by adding 1945 to both sides:

55,000 √x = 275,000 + 1945

Next, we need to get rid of the square root symbol. One way to do this is to square both sides of the equation:

x = (275,000 + 1945) / 55,000^2

The square root has been eliminated, but the equation still doesn't give us x in a form that is easy to interpret. To get x in a more useful form, we can take the square root of both sides:

√x = √((275,000 + 1945) / 55,000^2)

So, the year in which the population of the city was 275,000 is approximately x + 1945 = 1976.

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Jack is 100 ft away from the trail guide.

The angle of depression is the angle between the horizontal line and the  observation line of the object from the horizontal line.

Given is that Jack is one side of a 200 foot wide canyon and Jill is on the other side. Jack and Jill can both see the trail guide at angle of depression of 50 degrees.

From the figure, we can write -

tan (50) = OB/OC

OB = OC tan(50)                                   ......... { 1 }

and

tan (50) = OB/OA

OB = OA tan (50)                                  ......... { 2 }

Equation { 1 } and { 2 } -

OC tan(50) = OA tan(50)

OC = OA                                                   ...... { 3 }

Now -

AC = AO + OC

2OA = AC

2OA = 200

OA = 100 ft

Therefore, jack is 100 ft away from the trail guide.

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The value of segment BC = 9.8 and the value of BD = 7 in the given right triangle ABC.

In a right-angled triangle, the square of the hypotenuse side is equal to the sum of the squares of the other two sides, according to Pythagoras's Theorem. These triangle's three sides are known as the Perpendicular, Base, and Hypotenuse. Due to its position opposite the 90° angle, the hypotenuse in this case is the longest side. When the positive integer sides of a right triangle (let's say sides a, b, and c) are squared, the result is an equation known as a Pythagorean triple.

Given that the length of the segment CA = 14.

As BD is the bisector of line CA we have:

CD = 7 and

DA = 7

Given that angle C = 45 degrees.

Then,

tan C = opposite side/ adjacent side

tan (45) = BD / 7

1 (7) = BD

BD = 7

Using the Pythagorean Theorem we have:

BC^2 = BD^2 + CD^2

Substituting the values of BD = 7 and CD = 7 we have:

BC^2 = 7^2 + 7^2

BC^2 = 49 + 49

BC = 9.8

Hence, the value of segment BC = 9.8 and the value of BD = 7 in the given right triangles ABC.

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The radius of the circle is 3√5 cm.

What is Pythagoras theorem?

The hypotenuse's square is equal to the sum of the squares of the other two sides if a triangle has a straight angle (90 degrees), according to the Pythagoras theorem. Keep in mind that BC2 = AB2 + AC2 in the triangle ABC signifies this. Base AB, height AC, and hypotenuse BC are all used in this equation. The longest side of a right-angled triangle is its hypotenuse, it should be emphasized.

Here given in the question that one side of the wall 5 feet

So as it is a rectangular bedroom it can be split into 2 right-angled triangles.

Let O be the circle's centre and r its radius. Its chords are AB and CD, which have lengths of 6 cm and 12 cm, respectively.

Draw OM LAB which cuts CD at N AB||CD and ON ICD OM LAB AM = MB (*.* Perpendicular drawn from centre of circle bisects the chord).

AM = MB = 1/2AB

AM = 1/2 * 6 =3cm

Distance between chords AB and CDMN = 3cm.

Let ON = x cm.

In right angled AOCN, by Pythagoras theorem

OC² = ON² + CN2

(OC)² = (x)² +(6)²

(OC)² = x²+36.....(i)

(OC)² = (3)²+36

=9+36

= 45

OC = √45

OC = 3√/5cm.

Hence, The radius of the circle is 3√5 cm.

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From the Histogram shown below , the number of values in the data set that are greater than 5.5 but less than 8.5 are 17 .

A Histogram is defined as a graphical representation of data that displays the frequency distribution of a set of continuous or discrete data. It shows how data is distributed across a range of values, by dividing the data into a set of bins and then plotting the number of data points that fall into each bin as bars.

From the Histogram we see that between 5.5 and 8.5 there are 3 bars, that means the three heights are 7, 5, and 5.

So , to find the number of values ,  we add these three heights:

that means ⇒ 7 + 5 + 5 ⇒ 17.

Therefore , there are 17 values in the data set that are greater than 5.5 but less than 8.5  .

The given question is incomplete , the complete question is

Given the following histogram for a set of data, how many values in the data set are greater than 5.5 but less than 8.5 ?

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Kate need 3/4 cup of flour. She only ha two measuring cup, a 1/3 and a 1/8. The cup should Kate use is 1/8 .

The cup of flour she need is 3/4 that is 75% of the use.

There are two cups available 1/3 and 1/8

In percentages 1/3 can be 33.33%

And in percentages 1/8 can be 12.5%

The amount to be filled is 3/4 that is 75% so the cup that will be used for this is 1/8 that is 12.5% so that,

= 12.5 x 6

= 75

So we used 1/8 cup for 6 times we can get the 3/4 cup of flour.

A fraction represents a portion of a total. This entire may refer to a place or a group of places. The Latin word "fractio," which meaning "to break," is the source of the English term "fraction." The distribution of food and supplies as well as the lack of a metal currency were among the mathematical issues that the Egyptians utilized fractions to solve because they were the first civilization to understand fractions.

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Answer:

3(3+4n)

Step-by-step explanation:

3x3=9

3x4=12

so, gcf is 3

The total meter reading of Dan electricity usage from July to October including the meter reading as at July is 33,289 units

Quantity of electricity Dan used from July to October = 942 units

Meter reading in July= 32, 347 units

The total meter reading = Quantity of electricity Dan used from July to October + Meter reading in July

= 942 units + 32, 347 units

= 33,289 units

Therefore, Dan used a total of 33,289 units of electricity from July to October including the previous reading.

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