Find the quotient and remainder if \( f(x) \) is divided by \( p(x) \). \[ f(x)=3 x^{4}+2 x^{3}-x^{2}-x-6 ; \quad p(x)=x^{2}+1 \] quotient \( \quad 3 x^{4}+2 x^{3}-x^{2}-x-6 \) remainder

Answer:

Width: w = 12.5 feet

Length: L = 32.5 feet

Step-by-step explanation:

Let's use two variables to represent the dimensions of the rectangular garden:

Let w be the width of the garden (in feet)

Let L be the length of the garden (in feet)

The problem tells us that the length of the fence is 90 feet, so we can write the equation:

2L + 2w = 90

We also know that the length of the fence (L) is 5 feet less than 3 times the width (w). We can write this as another equation:

L = 3w - 5

Now we have two equations with two variables. We can solve this system of equations using substitution or elimination.

Let's use substitution:

Substitute the expression for L in terms of w from the second equation into the first equation:

2(3w - 5) + 2w = 90

Simplify and solve for w:

6w - 10 + 2w = 90

8w = 100

w = 12.5

Now we can use this value of w to find L:

L = 3w - 5 = 3(12.5) - 5 = 32.5

Therefore, the dimensions of the rectangular garden are as follows:

Width: w = 12.5 feet

Length: L = 32.5 feet

Given equation 2x − 3​ − 1 = 0.

Now, bring -1 to the right side of the equation:

2x − 3​ = 1

Now, add 3 to both sides:

2x = 4

x = 2

Now, we need to check whether this is the correct solution or not. Putting the value of x in the given equation:

2(2) − 3​ − 1 = 0

⇒ 1 − 1 = 0

⇒ 0 = 0

Hence, x = 2 is the solution to the given equation.

(a) The solution set of the equation 2x − 3​ − 1 = 0 is {2}.

Given equation 3x² + 2​ = 5x² − 6​.

Now, bring 5x² − 6 to the left side of the equation:

-2x² + 2 = 0

Now, bring 2 to the right side of the equation:

-2x² = -2

⇒ x² = 1

⇒ x = ±1

Now, we need to check whether these are the correct solutions or not. Putting the value of x in the given equation:

3x² + 2​ = 5x² − 6​

For x = 1,

3(1)² + 2 = 5(1)² − 6

⇒ 3 + 2 = 5 − 6

⇒ -1 = -1

For x = -1,

3(-1)² + 2 = 5(-1)² − 6

⇒ 3 + 2 = 5 − 6

⇒ -1 = -1

Hence, x = 1 and -1 are the solutions to the given equation.

(b) The solution set of the equation 3x² + 2​ = 5x² − 6​ is {-1,1}.

Given equation 6x + 7​ = x + 2.

Now, bring x to the left side of the equation:

6x − x + 7​ = 2

Now, add 7 to both sides:

5x = -5

⇒ x = -1

Now, we need to check whether this is the correct solution or not. Putting the value of x in the given equation:

6(-1) + 7​ = (-1) + 2

⇒ -6 + 7​ = -1 + 2

⇒ 1 = 1

Hence, x = -1 is the solution to the given equation.

(c) The solution set of the equation 6x + 7​ = x + 2 is {-1}.

Given equation x + 1 = x² + 9.

Now, bring x² to the left side of the equation:

x² − x + 1 = 9

Now, bring 9 to the right side of the equation:

x² − x − 8 = 0

Now, factorizing the given equation:

(x − 4)(x + 2) = 0

So, the solutions are x = 4 and x = -2.

Now, we need to check whether these are the correct solutions or not. Putting the value of x in the given equation:

x + 1 = x² + 9

For x = 4,

4 + 1 = 4² + 9

⇒ 5 = 25

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(a) 0, 1/2, 3/8.

(b) The conditional probability distribution of X is 1/3.

(c) the expected value of Y is 0.

(d) Yes, X and Y are independent.

What is Joint distribution?

Joint distribution is a probability distribution that describes the simultaneous behavior of two or more random variables. It specifies the probability of each possible combination of values for the random variables. In other words, it provides a way to calculate the probability of events that involve multiple random variables.

(a) To find the probabilities P(X < 0.5, Y < 1.5), P(X > 0.25, Y < 4.5), and P(X ≤ 0.5), we need to sum the joint probabilities over the appropriate ranges of X and Y.

P(X < 0.5, Y < 1.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) = 0

P(X > 0.25, Y < 4.5) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) + p(2,-2) + p(2,0) + p(2,1) + p(2,2) = 1/2

P(X ≤ 0.5) = p(-1,-2) + p(-1,0) + p(-1,1) + p(0,-2) + p(0,0) + p(0,1) + p(0,2) + p(1,-2) + p(1,0) + p(1,1) = 3/8

(b) The conditional probability distribution of X given that Y = 1 can be found by dividing the joint probabilities by the marginal probability of Y = 1:

P(X = -1 | Y = 1) = p(-1,1) / ∑ p(-1,y) = 1/3

P(X = 0 | Y = 1) = p(0,1) / ∑ p(0,y) = 1/3

P(X = 1 | Y = 1) = p(1,1) / ∑ p(1,y) = 1/3

Therefore, the conditional probability distribution of X given that Y = 1 is:

X | P(X|Y=1)

--- | --------

-1 | 1/3

0 | 1/3

1 | 1/3

(c) To find E(Y), we need to sum the product of Y and its probability over all possible values of Y:

E(Y) = ∑ y p(x,y)

E(Y) = (-2)(1/8) + (-1)(1/4) + (0)(1/8) + (1)(1/4) + (2)(1/8) = 0

(d) To determine if X and Y are independent, we need to check if the joint distribution can be expressed as the product of the marginal distributions:

p(x,y) = p(x) * p(y)

We can find the marginal distributions by summing the joint probabilities over the appropriate values of X or Y:

p(x) = ∑ p(x,y)

p(-1) = p(-1,-2) + p(-1,0) + p(-1,1) = 1/4

p(0) = p(-1,0) + p(0,-2) + p(0,0) + p(0,1) = 1/2

p(1) = p(1,-2) + p(1,0) + p(1,1) + p(1,2) = 1/4

p(y) = ∑ p(x,y)

p(-2) = p(-1,-2) + p(1,-2)

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The two triangles are similar using the AA criteria of similarity.

When two lines cross at a point, vertical angles are created. They are always on an equal footing. In other words, four angles are created anytime two lines cross or meet. It is evident that two opposed angles are equal and are referred to as vertical angles. They are also known as "Vertically opposed angles" due to the fact that they are perpendicular to one another.

In the given figure it is given that, AB is parallel to DF.

Thus, angle ABC = angle CDF

Also, the angle ACB and DCF form vertically opposite angles, and are equal.

Thus, the two triangles are similar using the AA criteria of similarity.

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The most likely data sets to have a negative association and a correlation coefficient between 0 and -1 are: number of minutes spent exercising; number of calories burned, which is option C.

The correlation coefficient, a numerical measure of the strength and direction of the relationship between two variables, is used to describe the association between two data sets. It ranges from -1 to +1, where a correlation coefficient of -1 indicates a negative correlation and +1 indicates a positive correlation.

Number of minutes spent exercising and the number of calories burned while exercising have a negative association. That is, as the number of minutes spent exercising increases, the number of calories burned decreases.

Therefore, the most likely data sets to have a negative association and a correlation coefficient between 0 and -1 are "number of minutes spent exercising; number of calories burned."

Hence, the correct answer is C.

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Answer: Average Annual Temperature in the US; Annual sweater sales by an American retailer

Step-by-step explanation:

The inequalities for each pollen count are given as follows:

The four inequality symbols, along with their meaning, are presented as follows:

A medium pollen count is greater than 4 and less than or equal to 8, hence:

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I will have $2500 at the end of 5 years

Interest Rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed. It is the percentage of principal charged by the lender for the use of its money.It is also percentage a lender charges on the amount of money borrowed.

Where, The principal is the amount of money loaned.

The formula = Principal * Rate * Time/100

Where Principal = $10000

Rate = 5%

Time = 5 years

Simple Interest = 10000 * 5 * 5/100

Simple Interest = 250000/100

Simple Interest = $2500

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The value of x in the given right triangle is 22.55 units.

Simply put, trigonometric functions—also referred to as circular functions—are the functions of a triangle's angle. This means that these trig functions provide the connection between the angles and sides of a triangle. The ratio of the lengths of the adjacent and opposing sides is known as the tangent function. It should be noted that the ratio of sine and cosine to the tan may also be used to express the tan.

For the given triangle the given sides are opposite and adjacent to the given angle.

The trigonometric function that relates the two sides are:

tan (64) = x/11

2.05(11) = x

x = 22.55

Hence, the value of x in the given right triangle is 22.55 units.

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Jenny has 7/15 of her original box of stickers left after giving some to her brother and after her dog ate three of them.

Jenny had an original box of 15 stickers. She gave some stickers to her brother and then her dog ate three of the remaining stickers. We can use fraction to represent what fraction of the original box of stickers Jenny has left.

Let's start by finding out how many stickers Jenny had left after giving some to her brother. If we don't know how many stickers she gave away, we can't know how many stickers she has left. So let's say Jenny gave away 5 stickers to her brother.

Jenny had 15 stickers - 5 stickers = 10 stickers left.

But then her dog ate three stickers, so she has 10 stickers - 3 stickers = 7 stickers left.

To find the fraction of the original box of stickers that Jenny has left, we need to divide the number of stickers she has left by the original number of stickers:

7 stickers ÷ 15 stickers = 7/15.

Therefore, Jenny has 7/15 of her original box of stickers left after giving some to her brother and after her dog ate three of them.

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George eats a total of 25/12 satsumas. This is an improper fraction, meaning the numerator is greater than the denominator, but it is the correct answer to the problem.

George starts with 2 satsumas, which he then splits into 6 equal parts, resulting in 12 parts in total. He eats one of these parts, which is 1/12 of a satsuma.

To find out how many satsumas he eats in total, we need to add the 2 whole satsumas he ate at the beginning to the fraction of a satsuma he ate later:

2 + 1/12

To add these together, we need to find a common denominator. The smallest common multiple of 12 and 1 is 12, so we can convert 2 to twelfths by multiplying it by 12/12:

2 + 1/12 = 24/12 + 1/12

Now we can add the two fractions together:

24/12 + 1/12 = 25/12

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The amount of flour in each box is given as follows:

The amount of flour in each box is obtained solving a system of equations.

The variables for the system of equations are given as follows:

2 boxes combined weighed 155 pounds of flour, hence:

x + y = 155

y = 155 - x.

20 pounds of flour was moved from the first box to the second. Now the first box has 12/19 of what is in the 2nd box, hence the ratio is:

(x - 20)/(y + 20) = 12/19.

Replacing the first equation into the second, the value of x is obtained as follows:

(x - 20)/(155 - x + 20) = 12/19

(x - 20)/(175 - x) = 12/19

19(x - 20) = 12(175 - x)

31x = 2480

x = 2480/31

x = 80 pounds.

Then the value of y is obtained as follows:

y = 155 - x

y = 155 - 80

y = 75 pounds.

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Half of the intercepted arc is equals to the inscribed angle. Therefore, the  measure of the arc is 170 degrees.

The arc of a circle is said to be the part or segment of the circumference of a circle.

The degree of an arc is equals to the measure of the central angle that creates the arc.

Therefore, half of the intercepted arc is equals to the inscribed angle. In other words, the inscribed angle theorem states that the angle inscribed inside a circle is always half the measure of the central angle.

Hence,

∠JKL = 1 / 2 arc angle

Therefore,

85 = 1 / 2 x

cross multiply

x = 85(2)

x = 170 degrees

Therefore,

arc angle = 170 degrees

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The area of the shaded segment in the given circle with radius 24cm and central angle 60° is approximately 301.5929 cm². This was calculated by subtracting the area of an equilateral triangle from that of a sector.

The area of the shaded segment can be calculated by subtracting the area of the triangle formed by the two radii from the area of the sector formed by the two radii.

First, let’s find the area of the sector. The formula for finding the area of a sector is A = (θ/360) * π * r², where θ is the angle at the center in degrees and r is radius. Substituting θ = 60° and r = 24cm, we get:

A = (60/360) * π * 24² A = 1/6 * π * 576 A = 96π

Next, let’s find the area of triangle. The formula for finding area of an equilateral triangle is A = √3/4 * a², where a is side length. Since all sides are equal to radius in this case, substituting a = 24cm, we get:

A = √3/4 * 24² A = √3/4 * 576 A = 144√3

Now we can find area of shaded segment by subtracting area of triangle from that of sector:

Area of shaded segment = Area of sector - Area of triangle = (96π) - (144√3) = 96π - 144√3 cm²

So, the area of shaded segment is approximately 301.5929 cm².

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The missing figure is in the image below

The given inequality will be satisfied by any value of x that is less than -1. Here are five potential remedies. x = -2, x = -2.5, x = -3, x = -1.2,x = -1.8. We can see that each of these values fulfils the given inequality.

When x > Y and a > 0, the result is (x/a) > (y/a), and when x Y and a > 0, the result is (x/a) (y/a). On the other hand, if the inequality sign is reversed, the division of both sides of an inequality by a negative integer results in an equivalent inequality.

We can solve the inequality 3(x+4)<9 as follows:

3(x+4) < 9 (given inequality)

3x + 12 < 9 (distributing the 3)

3x < -3 (subtracting 12 from both sides)

x < -1 (dividing both sides by 3 and changing the direction of the inequality)

So, any value of x that is less than -1 will satisfy the given inequality. Here are five possible solutions:

x = -2

x = -2.5

x = -3

x = -1.2

x = -1.8

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In a right angled triangle ABC, if angle A

= 30 degree and AB = 8 cm then the length of side BC is equals to the 4.62 cm.

As we see in figure, ∆ABC is an right angled triangle with side length of AB

= 8 cm and measure of angle A = 30°. Measure of angle B is 90°. So, measure of angle C = 180° - 90° - 30°

= 69°. So, AB is the opposite side for the angle C, AC is the opposite side for the angle B and BC opposite side for the angle A. We have to determine the length of side BC. Using the Trigonometric functions, tan x = height / base length

In ∆ABC, tan A = BC/AB

=> tan(30°) = BC/8

=> 1/√3 = BC/8 ( tan(30°) = 1/√3)

=> BC = 8/√3 = 4.62

Hence, required length of side BC is 4.62 cm.

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Complete question:

In triangle ABC, angle A=30 degrees and AB =8cm. Find the length of side BC? see the above figure.

a) cos  φ= -4/13`

b) cos φ  -2√13/13`c)

c) inverse tangent function will not give us the angle we are looking for because our angle is in the third quadrant

d) tanφ=32​ => φ ≈ -57.99°

e) coterminal angles to -57.99° in radians is:`(-319.93 + 360n)π/180`, where `n` is an integer.

a) The formula for the tangent of an angle in the third quadrant is, `tan(π + φ) = tan φ` and, hence, we have:`tan(π + φ) = 3/2`Using the fundamental identity for the tangent, we get:`tan(π + φ) = -tan φ``tan φ = -3/2`Then, using the Pythagorean identity `sin^2 φ + cos^2 φ = 1` to solve for `cos φ` in the third quadrant where `cos φ < 0`, we get:`cos φ = -√(1 - sin^2 φ) = -√(1 - (tan^2 φ)/(1 + tan^2 φ)) = -√(1 - (9/13)) = -4/13`b) Since `tan φ = 3/2`, we can construct a right triangle with legs of length `3` and `2` and hypotenuse of length `√(3^2 + 2^2) = √13`.Since the angle is in the third quadrant, the cosine of the angle is negative. Thus:`cos φ = -2/√13 = (-2/√13) * (√13/√13) = -2√13/13`c) The inverse tangent function is only able to give you the value of the angle in the first or fourth quadrant. Therefore, using the inverse tangent function will not give us the angle we are looking for because our angle is in the third quadrant.d) `tanφ=32​ => φ ≈ -57.99°`e) All coterminal angles to -57.99° in radians are given by:`θ = -57.99° + 360n, n ∈ ℤ`Thus, we can convert to radians using the formula `π/180°`:`θ = (-57.99° + 360n)π/180°`Simplifying:`θ = (-319.93 + 360n)π/180`Therefore, the expression for all coterminal angles to -57.99° in radians is:`(-319.93 + 360n)π/180`, where `n` is an integer.

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The domian of the graph is (-11, 11]

The domain of a graph is the set of all possible input values (or independent variable values) for which the function is defined and produces a valid output (or dependent variable value).

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

The graph

As stated above:

The domain of a graph is the set of input values of the graph

Using the above as a guide, we have the following:

When this is represented as an interval, we have

(-11, 11]

Hence, the solution is (-11, 11]

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Answer: 72

Step-by-step explanation: Each angle in a pentagon is 108 degrees. Since there is a line making the angle supplementary just subtract 180 from 108 and the answer is 72.

Answer:

The answer is (w-9)(w+9).

Answer:

(w - 9)(w + 9)

Step-by-step explanation:

w² - 81 ← is a difference of squares and factors in general as

a² - b² = (a - b)(a + b)

then

w² - 81

= w² - 9²

= (w - 9)(w + 9)

The measure of angle b is which is a supplement of angle a is 162 degrees.

If angles a and b are supplementary, that means they add up to 180 degrees.

Given that;

Since angle a and angle b are supplementary, So, we can set up the equation:

a + b = 180

We know that angle a measures 18 degrees, so we can substitute this value into the equation:

18 + b = 180

Solving for b, we can subtract 18 from both sides:

b = 180 - 18

b = 162 degrees

Therefore, angle b measure 162 degrees.

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Answer:

13.5w + 5.4.

Step-by-step explanation:

Using the distributive property, we would write it out like
-6(0.6w-3w-0.9)-0.9w = -3.6w + 18w + 5.4 - 0.9w
Then we would combine like terms by adding the coefficients of the w terms:
-3.6w + 18w - 0.9w = 13.5w
Lastly we can add the constant terms.
13.5w + 5.4 = 13.5w + 5.4

The distance between Mrs. Meyer and Dalton would be = 8.5m

The shape that is being formed between the three individuals is the shape of a triangle.

Distance can be defined as the length that is covered by a moving object.

The distance between Mrs Meyer and Leslie =a= 8m(opposite)

The distance between Dalton and Leslie =b = 3 m (adjacent)

Therefore, the hypotenuse = ?

Using the Pythagorean theorem;

c² = a² + b²

C ² = 8²+3²

C = 64 + 9

c² = 73

C = √ 73

C = 8.5m

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The slope of the line that best represents the relationship between the number of cupcakes and the total cost is 3.

To find the slope of a line that represents the relationship between two variables, we can use the formula: slope = (change in y) / (change in x).

Let's choose the first and last points from the table:

x1 = 0, y1 = 0

x2 = 3, y2 = 9

Using the slope formula:

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

= (9 - 0) / (3 - 0)

= 3

Therefore, the slope of the line that best represents the relationship between the number of cupcakes and the total cost is 3.

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She logs 18,000 feet of walking each shift based on perimeter of the rectangle field.

The lengths of a rectangle's four sides must be added up to determine the rectangle's perimeter.

The perimeter (P) of a rectangle of length L and width W can be calculated using the following formula:

P = 2L + 2W

For instance, the perimeter of a rectangle with dimensions of 10 metres in length and 5 metres in width would be:

[tex]P = 2(10) plus 2(5) equals 20 plus 10 metres.[/tex]

Hence, the rectangle's perimeter is 30 metres.

The fencing has a length and a width and is rectangular in design.

Size: 5000 feet

width is 4000 feet.

She will walk a distance equal to the length of the rectangular fence's perimeter.

Perimeter = 2 (length + width) = 2 (5000 + 4,000) = 2 (9000) = 18,000 ft.

She logs 18,000 feet of walking each shift.

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The total area of the shape is 90cm²

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. The area of a triangle is expressed as ;

A = 1/2 bh, where b is the base and h is the height if that triangle.

The base of the a triangle in the shape is 6cm and the height is 5cm.

A = 1/2 bh

A = 1/2 × 6 × 5

A = 3× 5

A = 15cm²

Since the size of the triangles are equal ,it means that all the triangles will have thesame area.

Therefore the area of six triangles =

6× 15 = 90cm²

Therefore the area of the shape is 90cm²

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In respοnse tο the questiοn, we may say that  In 100 years, there will be functiοn arοund 125 grammes left.

Mathematicians research numbers, their variants, equatiοns, assοciated structures, fοrms, and pοssible cοnfiguratiοns οf these. The wοrd "functiοn" describes the cοnnectiοn between a grοup οf inputs, each οf which has a cοrrespοnding οutput. A functiοn is a cοnnectiοn between inputs and οutputs where each input results in a single, distinct οutcοme. Each functiοn has a dοmain, cοdοmain, οr scοpe assigned tο it. Functiοns are usually denοted by the letter f. (x). An x is entered. On functiοns, οne-tο-οne capabilities, sο multiple capabilities, in capabilities, and οn functiοns are the fοur main categοries οf accessible functiοns.

Setting t = 0 in the prοvided functiοn will reveal the sample's οriginal quantity:

A(0) = 647

[tex](1/2)^{(0/30)}[/tex] = 647

As a result, there are 647 grammes οf starting material in the sample.

In οrder tο calculate the amοunt left after 100 years, we must enter t = 100 intο the supplied functiοn:

[tex]A(100) = 647(1/2)^{(100/30) }= 647(1/2)^{(10/3)}[/tex] ≈ 125.24

Thus, there will be arοund 125 grammes left after 100 years (rοunded tο the nearest gram).

There are 647 grammes οf starting material in the sample.

In 100 years, there will be arοund 125 grammes left.

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The initial amount of the sample is 647 grams and the amount remaining after 100 years is 69 grams.

what is functiοn?

Mathematicians research numbers, their variants, equatiοns, assοciated structures, fοrms, and pοssible cοnfiguratiοns οf these. The wοrd "functiοn" describes the cοnnectiοn between a grοup οf inputs, each οf which has a cοrrespοnding οutput.

The given exponential function is:

A(t) = 647(1/2)^(t/30)

where t is the time in years.

To find the initial amount of the sample, we need to evaluate A(0):

A(0) = 647(1/2)^(0/30) = 647(1) = 647

Therefore, the initial amount of the sample is 647 grams.

To find the amount remaining after 100 years, we need to evaluate A(100):

A(100) = 647(1/2)^(100/30) ≈ 69.35

Rounding this to the nearest gram gives the amount remaining after 100 years as 69 grams.

Therefore, the initial amount of the sample is 647 grams and the amount remaining after 100 years is 69 grams.

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The line with an equation (2/5)x + (1/10)y = 2 have an x intercept at (5, 0) and y intercept at (0, 20)

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

The slope intercept form of a linear equation is:

y = mx + b

where m is the rate of change (slope) and b is the y intercept.

Given the equation:

(2/5)x + (1/10)y = 2

The x intercept is at y = 0, hence:

(2/5)x + (1/10)(0) = 2

(2/5)x = 2

x = 5

The x intercept is (5, 0)

The y intercept is at x = 0, hence:

(2/5)(0) + (1/10)(y) = 2

(1/10)y = 2

y = 20

The y intercept is (0, 20)

The line with an equation (2/5)x + (1/10)y = 2 have an x intercept at (5, 0) and y intercept at (0, 20)

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