whats the radius of the circle (x+8)^2 + (y-2)^2=81

r=___

Answer:

7.

Step-by-step explanation:

To find what is equivalent to it:

[tex]9\frac{3}{2}[/tex]

You have to simplify it into an improper fraction:

21 / 3

= 7

Answer:

the correct answer is 7 according to me

Answer:

Step-by-step explanation:

Answer:

$80.00

Step-by-step explanation:

Price Paid: $48

Discount Received: 40%

Percentage Paid: 100% - 40% = 60%

$48 / 60% =  $48/0.6 = $80

Answer:

80.00

Step-by-step explanation:

I don't know why, but you don't need the explanation so I have none.

The arithmetic series can be written in summation notation as:

∑_{i=1}⁶ 5i = 5 + 10 + 15 + 20 + 25 + 30 = 105.

What is arithmetic series?

The sum of a series in which each term is computed from the preceding one by adding and subtracting constants is known as an arithmetic series. Alternately, we could say that an arithmetic progression is a set of numbers where, for every pair of successive terms, the second number is obtained by adding a fixed amount to the first.

The common difference in this arithmetic series is 10 - 5 = 5. Therefore, the nth term of the series is given by a_n = 5n.

The first term of the series is 5, which corresponds to n = 1. The last term is 30, which corresponds to n = 6.

The sum of an arithmetic series can be calculated using the formula: S_n = (n/2)(a_1 + a_n), where S_n is the sum of the first n terms, a_1 is the first term, and a_n is the nth term.

Substituting the values of a_1, a_n, and n, we get:

S_6 = (6/2)(5 + 30) = 3(35) = 105

Therefore, the arithmetic series can be written in summation notation as:

∑_{i=1}⁶ 5i = 5 + 10 + 15 + 20 + 25 + 30 = 105.

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Answer: u = -15/32 ( check if you can simplify, I didn't)

- subtraction/negative

x multiplication

/ division/fraction bar

Step-by-step explanation:

-8/3u -1/2 = 3/4

         +1/2   +1/2

-8/3u = 3/4 + 1/2

                     x2

            3/4 + 2/4

-8/3u = 5/4

x-3/8     x-3/8

u = -15/32

The biased statistical measure among the given options is mean.

Statistical bias is anything that leads to a systematic difference between the true parameters of a population and the statistics used to estimate those parameters.

The mean of a sampling distribution is biased when the sample is drawn from a non-normal population or when the sample size is small. In such cases, the mean tends to be pulled towards the extremes of the distribution, resulting in a biased estimate of the population mean.

In contrast, the median is a robust measure that is less affected by extreme values, and is therefore less likely to be biased. The variance and proportion are also unbiased measures of the sampling distribution.

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(a) A/x + Bx + [tex]C/(1 + x) + Dx^2 + Ex/(1 - x) + Fx^2[/tex]

(b)A/x + [tex]B/x^2 + C/x^3[/tex] + D/(x - 2) + [tex]E/(x^2[/tex]+ 2x + 4) + F/(x + 2) + [tex]G/(x^2[/tex] - 2x + 4)

(a) To perform partial fraction decomposition on the function:

[tex](x^4 (x^3 + x)) / (x^2 - x^4)[/tex]

we first factor the denominator:

[tex]x^2 - x^4 = x^2(1 - x^2) = x^2(1 + x)(1 - x)[/tex]

The partial fraction decomposition of the function is:

[tex](x^4 (x^3 + x)) / (x^2 - x^4) = A/x + Bx + C/(1 + x) + Dx^2 + Ex/(1 - x) + Fx^2[/tex]

where A, B, C, D, E, and F are constants to be determined.

(b) To perform partial fraction decomposition on the function:

[tex]4 / (x^6 - 8x^3)[/tex]

we first factor the denominator:

[tex]x^6 - 8x^3 = x^3(x^3 - 8) = x^3(x - 2)(x^2 + 2x + 4)(x + 2)(x^2 - 2x + 4)[/tex]

The partial fraction decomposition of the function is:

[tex]4 / (x^6 - 8x^3)[/tex] = [tex]A/x + B/x^2 + C/x^3[/tex] + D/(x - 2) + [tex]E/(x^2 + 2x + 4) + F/(x + 2)[/tex] + [tex]G/(x^2 - 2x + 4)[/tex]

where A, B, C, D, E, F, and G are constants to be determined.

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The area of the shaded portion as shown in the diagram is 118.74 ft².

Area is the region bounded by a plane shape.

To calculate the area of the shaded portion, we use the formula below

Formula:

Where:

From the diagram,

Given:

Substitute these values into equation 1

Hence, the area of the shaded portion is 118.74 ft².

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y = 3x + 6 is the equation of the line with gradient 3 and passing through the point (-1,3).

The equation of a straight line with gradient m and passing through the point (x₁, y₁) is given by:

y - y₁ = m(x - x₁)

In this case, we have a gradient of m = 3 and a point (x₁, y₁) = (-1, 3). Plugging in these values, we get:

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

Simplifying this equation, we get:

y - 3 = 3x + 3

Adding 3 to both sides, we get:

y = 3x + 6

Therefore, the equation of the line with gradient 3 and passing through the point (-1,3) is y = 3x + 6.

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The point estimate for least-squares regression line x=1 is approximately 3.85, and the point estimate for x=7 is approximately 24.02.

In a linear regression, the coefficient of determination (R-squared) is a statistical indicator that shows how much of the variance in the dependent variable is explained by the independent variable (s). Higher values suggest a better fit of the regression line to the data, and it has a value between 0 and 1. R-squared is derived by dividing the regression's sum of squares (SSR) by the sum of all squares (SST).

The least-squares regression line is given by the formula:

b1 = Σ[(xi - x)(yi - y)] / Σ[(xi - x)²]

The means of x and y:

x = (−2 + 2 + 5 + 7 + 10) / 5 = 4.4

y = (2 + 9 + 13 + 19 + 24) / 5 = 13.4

Substituting the value we have:

Σ[(xi - x)(yi - y)] = (-2-4.4)(2-13.4) + (2-4.4)(9-13.4) + (5-4.4)(13-13.4) + (7-4.4)(19-13.4) + (10-4.4)(24-13.4) = 400.8

Σ[(xi - x)²] = (-2-4.4)²+ (2-4.4)² + (5-4.4)² + (7-4.4)² + (10-4.4)² = 86.8

Thus, the value of:

b1 = Σ[(xi - x)(yi - y)] / Σ[(xi - x)²] = 400.8 / 86.8 ≈ 4.617

Now, for y-intercept we have:

b0 = y - b1x = 13.4 - 4.617(4.4) ≈ -0.767

y = -0.767 + 4.617x

Now, the point estimate is:

For x=1, y = -0.767 + 4.617(1) ≈ 3.85

For x=7, y = -0.767 + 4.617(7) ≈ 24.02

Therefore, the point estimate for x=1 is approximately 3.85, and the point estimate for x=7 is approximately 24.02.

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The time clocks necessarily follow two formats which are 12-hour format and 24-hour format.

12-hour:

half past three in the afternoon

9 minutes to midnight

quarter after 00:00 H (this should be "quarter after 12:00 am")

quarter to 19:00 H

half past 13:00 H

5 to 23:00 H

24-hour:

03:02 before 08:40

15:30

23:51

00:15

18:45

13:30

22:55

The most typical time format in the United States, Canada, and some other English-speaking nations is the 12-hour clock. It employs the digits 1 through 12 to represent the hours, and "am" or "pm" to denote morning or afternoon/evening, respectively.

The majority of non-American nations, including most of Europe and the rest of the globe, use the 24-hour clock as their primary time system. The hours are denoted by the digits 0-23, with no distinction made between morning and afternoon/evening.

Decimal time: In some nations, including France, the day is split into ten hours, each hour into one hundred minutes, and each minute into one hundred seconds.

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Step-by-step explanation:

i dony know the answer of this question

Answer:

please see the attached I put the answer

Step-by-step explanation:

you answer were correct but you change the sign one of them will be positive.

If the point Q(10,0) lies on the circle, then the other points that lie on the circle C are (a) (5 , 5√3) and (b) (6,4).

We know that, if "circle C" is centered at the origin, its equation is of the form ⇒ x² + y² = r², where r is = radius of circle,

We know that point Q(10,0) lies on circle C, so we substitute its coordinates into the equation of the circle to find the value of r:

⇒ 10² + 0² = r²,

⇒ r² = 100,

⇒ r = 10,

So, the equation of circle C is : x² + y² = 100,

Now, we check each of the given points to see which ones lie on the circle:

Option (a) : (5 , 5√3)

⇒ 5² + (5√3)² = 100,

⇒ 25 + 75 = 100,

The point (5 , 5√3)lie on circle.

Option (b) : (6,4)

⇒ 6² + 4² = 100,

⇒ 36 + 16 = 100,

The point (6,4) lies on circle.

Option (c) : (4,5√3)

⇒ 4² + (5√3)² = 100,

⇒ 16 + 75 = 91 ≠ 100,

This point (4,5√3) does not lie on circle.

Option (d) : (3 , √34)

⇒ 3² + (√34)² = 100,

⇒ 43 ≠ 100

This point (3 , √34) does not lie on circle.

Therefore, (a) (5 , 5√3) and (b) (6,4) are the point that "lies-exactly" on the circle.

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The given question is incomplete, the complete question is

Circle C is centered at the origin. If Q(10,0) lies on circle C, which of the following points also lies on circle C?

(a) (5 , 5√3)

(b) (6,4)

(c) (4,5√3)

(d) (3 , √34)

Answer:

Yes, the students' reasoning is correct. The percent of a part compared to the whole is calculated by dividing the part by the whole and multiplying the result by 100. Since the part is much less than the whole, the result of the division will be a very small number, and when multiplied by 100, it will be an even smaller number, which is the percentage.

Step-by-step explanation:

1. Divide the part by the whole to calculate the fraction that the part is of the whole.

2. Multiply the result by 100 to convert the fraction to a percentage.

3. Since the part is much less than the whole, the result of the division will be a very small number, and when multiplied by 100, it will be an even smaller number, which is the percentage.

A game that uses exponent rules and scientific notation on paper are given below as exponent explorer.

We have,

A game that incorporates exponent rules and scientific notation:

Title: Exponent Explorer

Objective: The goal of the game is to become an "Exponent Explorer" by solving a series of challenges related to real-world phenomena that involve exponent rules and scientific notation.

Materials: Pencils, paper, and a list of challenges.

Instructions:

The game is played in rounds, with each round consisting of a challenge.

Before starting the game, explain the exponent rules and scientific notation to the players, and provide examples of how they are used in real-world phenomena.

Start the first round by presenting a challenge to the players. Each challenge should involve a real-world phenomenon, such as the size of the universe, the speed of light, or the population of a city. The challenge should require players to use exponent rules and/or scientific notation to solve a problem related to the phenomenon.

Players work individually or in teams to solve the challenge. They must show their work and write their answer in scientific notation.

After a set amount of time, reveal the correct answer and award points to the players or teams that solved the challenge correctly.

The number of points awarded can vary based on the difficulty of the challenge.

Play additional rounds with new challenges, and continue awarding points.

The player or team with the most points at the end of the game is the winner and becomes an "Exponent Explorer."

Example Challenge:

Challenge:

The population of New York City is approximately 8,336,817.

Write this number in scientific notation, and then determine how many times larger it is than the population of San Francisco, which is approximately 883,305.

Solution:

8,336,817 = 8.336817 x 10^6

To find how many times larger New York City's population is than San Francisco's, divide the two numbers:

8,336,817 / 883,305

= 9.44

New York City's population is approximately 9.44 times larger than San Francisco's population.

Thus,

A game that uses exponent rules and scientific notation on paper is given above.

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A) The exponential function that models the mass show many grams remain from a 480-gram sample after t minutes is G(t) = 22(1/2)^(t/5.5)

B) The grams that remain after 2 hours is 51.42 grams.

A substance's half-life is the amount of time needed for half of a radioactive substance to decay. It is a word that is used in nuclear chemistry to describe how quickly unstable atoms undergo radioactive decay into other nuclear species by emitting particles or the amount of time needed for the rate of disintegrations per second of radioactive material in order to reduce by half of its initial value.

Given:

The half-life of goo is 22 minutes/4 = 33.75 minutes

b) A formula for the amount remaining could be ...

G(t) = 22(1/2)^(t/5.5)

c) After 2 hours, the amount remaining is ...

G(120) = 135(1/2)^(47/33.75) ≈ 51.42 . . . grams

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A). After t minutes, the exponential function used to describe the mass remaining from a 480-gram sample is [tex]G(t) = 22(\frac{1}{2} )^{(t/5.5)}[/tex]

B). 51.42 grammes are still there after two hours.

The amount of time required for a radioactive substance to decay by half is known as its half-life. It is a term used in nuclear chemistry to describe how quickly unstable atoms transform into different nuclear species by emitting particles or how long it takes for the rate of radioactive material's disintegrations per second to fall by half of its initial value.

Given:

The half-life of goo is 22 minutes/4 = 33.75 minutes

b) The amount left could be calculated using the method...

[tex]G(t) = 22(\frac{1}{2} )^{(t/5.5)}[/tex]

c) The amount left after two hours is...

[tex]G(120)=135(\frac{1}{2} )^{(\frac{47}{33.5} )}[/tex] ≈ 51.42 . . . grams

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The penny will fall from distance of 232 feet during the 8th second only, as it falls 960 feet in the first 8 seconds, but 728 feet in the first 8 seconds.

The distance the penny falls each second is increasing by 32 feet (16 feet + 32 feet = 48 feet, 48 feet + 32 feet = 80 feet, and so on). Therefore, during the 8th second, the distance it will fall is:

16 + 48 + 80 + ... + (8 - 1) * 32 feet

Using the formula for the sum of an arithmetic sequence, we get:

(8 / 2) * (16 + (8 - 1) * 32) feet

= 4 * (16 + 7 * 32) feet

= 4 * 240 feet

= 960 feet

So the penny will fall 960 feet during the first 8 seconds. However, we need to subtract the distance it fell during the first 7 seconds to find the distance it fell during the 8th second only. The total distance it fell during the first 7 seconds is:

16 + 48 + 80 + ... + (7 - 1) * 32 feet

= (7 / 2) * (16 + (7 - 1) * 32) feet

= 3.5 * (16 + 6 * 32) feet

= 3.5 * 208 feet

= 728 feet

Therefore, the penny will fall 960 - 728 = 232 feet during the 8th second only.

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Answer: 700π/3 OR 233.33π

Step-by-step explanation:

Volume of Cylinder = [tex]\pi r^2h[/tex]

Volume of Hemisphere = [tex]\frac{2}{3}\pi r^3[/tex]

[tex]V_{c}[/tex]  = 5² x 6 x π

[tex]V_{c}[/tex]  = 25 x 6 x π

[tex]V_{c}[/tex]  = 150π

[tex]V_{h}[/tex] = (2 x 5³ x π) / 3

[tex]V_{h}[/tex] = (2 x 125 x π) / 3

[tex]V_{h}[/tex] = 250π/3

[tex]V_{T}[/tex] = [tex]V_{c}[/tex] + [tex]V_{h}[/tex]

[tex]V_{T}[/tex] = 150π + 250π/3

[tex]V_{T}[/tex] = 700π/3 OR 233.33π

Answer:

1 3/5

Step-by-step explanation:

[tex](2\frac{2}{3} )(\frac{3}{5})= (2\frac{10}{15})( \frac{9}{15} )= (\frac{40}{15} )( \frac{9}{15} )= \frac{360}{225}[/tex]

Divide by 45 to simplify

[tex]\frac{8}{5} = 1\frac{3}{5}[/tex]

The calculated t-statistic (3.24) is greater than the critical value (1.699), reject the null hypothesis

Since the calculated t-statistic (3.24) is greater than the critical value (1.699), we reject the null hypothesis.
In conclusion, the dentist's claim is supported.

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Answer:

Step-by-step explanation:

it would be shaped like a but because you cut it into 6cm

Answer:

False

Step-by-step explanation:

The answer is False

The given statement is false, as the letters are used as keys in the dictionary.

The dictionary used in counting letters examples to keep track of letter frequencies uses the letters as the key and the count as the value. This means that for each letter encountered in a string, the value corresponding to the key (letter) in the dictionary is incremented by 1.

For example, if the string is "hello", the dictionary would initially be empty, and for each letter encountered in the string, the value for the corresponding key in the dictionary would be incremented by 1, resulting in a dictionary like {"h": 1, "e": 1, "l": 2, "o": 1}.

Therefore, the correct answer is false, as the letters are used as keys in the dictionary.

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The appropriate null hypothesis for this study is that there is no significant difference between the mean weight of U.S. adults before and after the Thanksgiving holiday period.

The null hypothesis is a statement that there is no significant difference between two groups being compared. In this case, the two groups are the weight of U.S. adults before and after the Thanksgiving holiday period. The null hypothesis assumes that there is no significant difference between these two groups.

To test this hypothesis, the researchers will collect data on the weight of U.S. adults before and after Thanksgiving and compare the mean weight of the two groups. If the mean weight after Thanksgiving is significantly different from the mean weight before Thanksgiving, the null hypothesis will be rejected, and the researchers can conclude that there is strong evidence that the weight of U.S. adults changes between Thanksgiving and January.

Therefore, the appropriate null hypothesis for this study is that there is no significant difference between the mean weight of U.S. adults before and after the Thanksgiving holiday period.

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A two-digit number's tens digit is three times larger than its units digit. Reversing the digits results in a new number that is 54 less than the original. The reversal of the original number is 651, which equals 156.

Let's name the units digit "u" and the tens digit of the original number "t."

From the first statement, we know that:

t = 2u + 3

From the second statement, we know that when the digits are reversed, the new number is 54 less than the original number. Using an equation, we can visualize this:

10u + t = 10t + u - 54

Now we can substitute the first equation (t = 2u + 3) into the second equation:

10u + (2u + 3) = 10(2u + 3) + u - 54

Simplifying this equation, we get:

12u + 33 = 21u - 24

Subtracting 12u from both sides, we get:

33 = 9u - 24

Adding 24 to both sides, we get:

57 = 9u

Dividing both sides by 9, we get:

u = 6

Now we can use the first equation (t = 2u + 3) to find the value of t:

t = 2u + 3 = 2(6) + 3 = 15

Therefore, the original number is 156 (with tens digit 1 and units digit 6) and the reversed number is 651. And indeed, 651 is 54 less than 156.

The complete question is:-

The tens digit of a 2-digit number is 3 more than twice the units digit. if the digits are reversed, the new number is 54 less than the original number. Find the number.

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So the angle measure for ∠L is 22.5° when two parallel lines cut by a transversal.

In geometry, a transversal is a line that intersects two or more other lines in a plane. When a transversal intersects two parallel lines, it creates a series of corresponding angles, alternate angles, and interior angles, which have specific relationships and properties.

Here,

Since lines a and b are parallel, we know that alternate interior angles are congruent. Therefore, we have:

∠F = ∠L + 180° - 135°

Simplifying this expression, we get:

∠F = ∠L + 45°

Substituting the value of ∠F as 67.5°, we can solve for ∠L:

67.5° = ∠L + 45°

22.5° = ∠L

The relationship between ∠F and ∠L is that they are supplementary angles, meaning they add up to 180°. This is because they are on opposite sides of the transversal and both intersect with the same parallel lines a and b.

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Complete question:

Two parallel lines cut by a transversal. For our safety, it is important that drivers recognize a stop sign immediately. Stop signs are made with exact side and angle measurements to form a regular octagon. They are red so they are easy to see. The three lines drawn below are straight lines. Lines a and b are parallel. a) The stop sign is a regular octagon, so the measure of ∠F must be 67.5°. What is the angle measure for ∠L? Describe the relationship between ∠F and ∠L. (1 point)

(a) 22.5 = 22 is the stem and 8 is the leaf., 23.8 = 23 is the stem and 8 is the leaf.(c) 23.2 = 23 is the stem and 2 is the leaf.(d) 22.8 = 22 is the stem and 8 is the leaf.(e) 10.1 = 10 is the stem and 1 is the leaf.(f) 23.5 = 23 is the stem and 5 is the leaf.(g) 24.0 = 24 is the stem and 0is the leaf.(h) 23.2 = 23 is the stem and 2 is the leaf.(g) 24.2 = 24 is the stem and 2 is the leaf.(i) 24.3 = 24 is the stem and 3 is the leaf.(j) 23.3 = 23 is the stem and 3 is the leaf.(k) 23.4 = 23 is the stem and 4 is the leaf.(l) 23.0 = 23 is the stem and 0 is the leaf.(m)23.5 = 23 is the stem and 5 is the leaf.

And (n) 22.8 = 22 is the stem and 8 is the leaf.

(b) Sample Mean = 22.50 and Sample median = 47.4

(c) A box and scatter chart or chart (also called a box chart) is a chart that shows data.

(d) Outliers affect the mean value of the data but have little effect on the median or mode of a given set of data.

(a) Given data set is:

22.5, 23.8, 23.2, 22.8, 10.1, 23.5, 24.0, 23.2, 24.2, 24.3 ,,23.3, 23.4 ,23.0, 23.5, 22.8.

We know some of the facts about stem and leaf diagram:

1) The stem is the first digit or digits;

2) The leaf is the final digit of a value;

3) Each stem can consist of any number of digits; but.

4) Each leaf can have only a single digit.

based on the given information:

(a) 22.5 = 22 is the stem and 8 is the leaf.

(b) 23.8 = 23 is the stem and 8 is the leaf.

(c) 23.2 = 23 is the stem and 2 is the leaf.

(d) 22.8 = 22 is the stem and 8 is the leaf.

(e) 10.1 = 10 is the stem and 1 is the leaf.

(f) 23.5 = 23 is the stem and 5 is the leaf.

(g) 24.0 = 24 is the stem and 0is the leaf.

(h) 23.2 = 23 is the stem and 2 is the leaf.

(g) 24.2 = 24 is the stem and 2 is the leaf.

(i) 24.3 = 24 is the stem and 3 is the leaf.

(j) 23.3 = 23 is the stem and 3 is the leaf.

(k) 23.4 = 23 is the stem and 4 is the leaf.

(l) 23.0 = 23 is the stem and 0 is the leaf.

(m)23.5 = 23 is the stem and 5 is the leaf.

(n) 22.8 = 22 is the stem and 8 is the leaf.

(b) Sample Mean = 22.5+ 23.8 + 23.2+ 22.8+ 10.1+ 23.5+ 24.0+ 23.2+ 24.2+ 24.3 + 23.3+ 23.4 +23.0 + 23.5+ 22.8 ÷ 15

                            = 22.50

    Sample Median = 23.2 + 24.2 ÷ 2

                               = 47.4

(c) The boxplot is a graphical representation of the data distribution based on five numerical points ("minimum", Q1 [Q1], median, Q3 [Q3], and "Maximum"). It can give you information about its benefits and benefits.

(d) The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

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The dimensions of the package of maximum volume that can be sent are approximately 7.8 inches in radius and 21.3 inches in height.

Let's assume that the package has a height 'h' and a radius 'r'. The perimeter of the cross-section of a cylinder is the circumference of the circle plus the height multiplied by two, which can be written as:

Perimeter = 2πr + 2h

Given that the maximum combined length and circumference is 114 inches, we can write:

2πr + 2h + 2r = 114

Simplifying this equation, we get:

πr + h = 57 - r ----(1)

The volume of the cylinder is given by:

[tex]V = πr^2h[/tex]

Now, we can use equation (1) to eliminate 'h' from the volume equation:

[tex]V = πr^2(57 - r - πr)[/tex]

Expanding and simplifying this equation, we get:

[tex]V = πr^3 - π^2r^3/3 + 57πr^2 - 57π^2r[/tex]

To find the dimensions of the package that maximize the volume, we need to find the value of 'r' that maximizes the volume. We can do this by taking the derivative of the volume equation with respect to 'r' and setting it equal to zero:

[tex]dV/dr = 3πr^2 - π^2r^2 + 114πr - 57π^2 = 0[/tex]

Solving for 'r', we get:

r ≈ 7.8 inches

Substituting this value of 'r' back into equation (1), we can find the value of 'h':

h ≈ 21.3 inches

Therefore, the dimensions of the package of maximum volume that can be sent are approximately 7.8 inches in radius and 21.3 inches in height.

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For the given sample space we have the probabilities as follows: 1. P(sum=11) = 6/64 = 3/32 2. P (sum < 6) = 10/64 3. P ( sum is greater than or equal to 7) = 49/64 4.  P(sum is a multiple of 3 or a multiple of 4) = 33/64.

Probability refers to potential. A random event's occurrence is the subject of this area of mathematics. The range of the value is 0 to 1. Mathematics has included probability to forecast the likelihood of certain events. The degree to which something is likely to happen is basically what probability means. You will understand the potential outcomes for a random experiment using this fundamental theory of probability, which is also applied to the probability distribution.

For the sample space of rolling two 8 sided die we have:

1. P (sum = 11):

The samples that give the sum equal to 11 are:

(3, 8), (4, 7), (5, 6), (6, 5), (7, 4), and (8, 3)

Thus,

P(sum=11) = 6/64 = 3/32

2. P (sum < 6):

(1, 1), (1, 2), (1, 3), (1, 4)

(2, 1), (2, 2), (2, 3)

(3, 1), (3, 2)

(4, 1)

Thus, P (sum < 6) = 10/64

3. P ( sum is greater than or equal to 7):

(1, 6), (1, 7), (1, 8) = 3

(2, 5), (2, 6) (2, 7), (2, 8) = 4

(3, 4), (3, 5) (3, 6) (3, 7) (3, 8) = 5  

(4, 3) (4, 4) (4, 5) (4, 6), (4, 7) (4, 8) = 6

(5, 2) (5, 3) (5, 4) (5, 5) (5, 6) (5, 7) (5, 8) = 7

(6, 1) upto (6, 8) = 8

(7, 1) upto (7, 8) = 8

(8, 1) upto (8, 8) = 8

Total = 3 + 4 + 5 + 6 + 7 + 8 + 8 + 8 = 49

Thus, P ( sum is greater than or equal to 7) = 49/64

4. P (sum is a multiple of 3 or a multiple of 4):

Multiple of 3: 3, 6, 9, 12, ........

(1, 2) (1, 5) (1, 8) (2, 1) (2, 4) (2, 7) (3, 1) (3, 3) (3, 6) (4, 2) (4, 5) (4, 8) (5, 1) (5, 4) (5, 7), (6, 3), (6, 6) (7, 2) (7, 5) (7, 8) (8, 1) (8, 4), (8, 7) = 23

Multiple of 4: 4, 8, 12, 16, .........

(1, 3) (1, 7) (2, 2) (2, 6) (3, 1) (3, 5) (4, 4) (4, 8), (5, 3), (5, 7), (6, 2), (6, 6) (7, 1) (7, 5) (8, 4) (8, 8) = 16

Common terms = 6

Thus, P(sum is a multiple of 3 or a multiple of 4) = 23 + 16 - 6 / 64 = 33/64

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a. ln 7 = y in exponential equation is e^y = 7

b. e² = x in logarithm function is log x base e = 2

What is exponential and logarithmic function?

An exponential function is a mathematical function used to calculate the exponential growth or decay of a given set of data.

The logarithmic function is an inverse function to exponentiation. The logarithmic function is defined as. For x > 0 , a > 0, and a ≠1, y= loga x if and only if x = a^y

Therefore,

ln 7 = y

using inverse of ln

7 = e^y

and;

e² = x

using logarithm function as inverse

log x base e = 2.

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