Copy the axes below. a) By completing the tables of values to help you, plot the lines y - 1/4 x + 6 and y = -2x + 3/2 on your axes b) Use your diagram to find the solution to the siimultaneous equations y = 1/4x + 6 and y = -2x +3/2

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 following is a conditional statement: If an is true, then b is true. A conditional statement is a statement that can be written in the form “If P then Q,” where P and Q are sentences.

If p, then q, or "p implies q," is the conditional of q by p, and it is symbolised by p q. If both p and q are true, then it is false; otherwise, it is true. Contrapositive: "If q then p" is the opposite of a conditional statement of the form "If p then q."

Has a truth value of true if both a and b are true.

In addition to b's truth value, an is untrue.

if an is true and b is false, and has a truth value of false.

Then, with relation to the following claims:

1. p -> q

This sentence is true because p and q are both true.

2. p -> q

This is a false assertion because p is true and q is false..

3. p->q

Regardless of the truth value of q, this assertion is true because p is false.

4. ∼p -> q

P is the inverse of p. P is false if and only if p is true. And regardless of the truth value of q, this assertion is accurate.

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The complete question is

What is the truth value for the following conditional statement?

1.)

p: true

q: true

p → q

2.)

p: true

q: false

p → q

3.)

p: false

q: false

p → q

4.)

p: true

q: true

∼p → q

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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Answer:

Step-by-step explanation:

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

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:

$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 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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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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Step-by-step explanation:

i dony know the answer of this question

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

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.

a

Answer:

10% of 30

=10/100×30

=3

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)

(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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Answer:

Step-by-step explanation:

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

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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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

The volume of a hemisphere is 66.9cm³

Volume of cube is 768cm³

Composite volume is 834.9cm³

The volume of a cube is the amount of space enclosed within the boundaries of a cube. A cube is a three-dimensional object with six congruent square faces, all of which meet at right angles.  Therefore, the formula for calculating the volume of a cube is:

Volume = side length x side length x side length = side³

Given:

Sides of cube

a=8cm

b=8cm

c=12cm

Radius of hemisphere=4cm

The volume of a hemisphere = (1/3)πr³cubic centimeters.

=1/3×π×4³

=66.9cm³

Volume of cube=a×b×c

=8×8×12

=768cm³

Composite volume=The volume of a hemisphere + Volume of cube

=66.9+768

=834.9cm³

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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 rate of the function is 4 dollars per gallon of gas. 4 is the cost per gallon of gas, and 8 is the fixed cost of the car wash.

A gallon of gas = 20 pounds of CO₂!

20 pοunds οf carbοn diοxide are prοduced when 6.3 pοunds οf petrοl are burned.

The twο οxygen atοms accοunt fοr a large pοrtiοn οf the weight οf carbοn diοxide (CO₂) (the O₂). Atοms οf carbοn and hydrοgen are jοined tο fοrm petrοl mοlecules. The carbοn and hydrοgen in the gas mοlecules in petrοl burn apart. H₂O, οr water, is created when twο hydrοgen atοms unite with οne οxygen atοm.

Each petrοl carbοn atοm jοins twο οxygen atοms that are already present in the atmοsphere. It prοduces CO₂.

To find the rate of the function, we need to calculate the slope of the line that represents the total cost of Owen's stop at the gas station as a function of the number of gallons of gas. We can use the slope formula:

slope = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line. We can use the coordinates (0, 8) and (7, 36) to calculate the slope:

slope = (36 - 8) / (7 - 0) = 28 / 7 = 4

Therefore, the rate of the function is 4 dollars per gallon of gas.

To find the cost per gallon of gas, we can use the y-intercept of the line, which represents the fixed cost of the car wash. From the graph, we can see that the y-intercept is 8 dollars. Therefore, the gas costs (y - 8) dollars per gallon.

We can write this equation as:

y = 4z + 8

where y is the total cost of Owen's stop at the gas station, z is the number of gallons of gas, 4 is the cost per gallon of gas, and 8 is the fixed cost of the car wash.

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The complete question is:

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)

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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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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See the graphed function image below.

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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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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(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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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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