Determine the Domain of the relation

{(x, y): y = x/x-8}

Answer:

y=+3

Step-by-step explanation:

the equation of a line is y=mx+c

where m is the slope and c is the y-intercept.

First, let's find what m is, the slope of the line...

The slope of a line is a measure of how fast the line "goes up" or "goes down". A large slope means the line goes up or down really fast (a very steep line). Small slopes means the line isn't very steep. A slope of zero means the line has no steepness at all; it is perfectly horizontal.

For lines like these, the slope is always defined as "the change in y over the change in x" or, in equation form:

So what we need now are the two points you gave that the line passes through. Let's call the first point you gave, (0,3), point #1, so the x and y numbers given will be called x1 and y1. Or, x1=0 and y1=3.

Also, let's call the second point you gave, (-5,3), point #2, so the x and y numbers here will be called x2 and y2. Or, x2=-5 and y2=3.

Now, just plug the numbers into the formula for m above, like this:

m=[tex]\frac{3-3}{-5-0}[/tex]

or

m=[tex]\frac{0}{-5}[/tex]

or

m=0

So, we have the first piece to finding the equation of this line, and we can fill it into y=mx+c like this:

y=0x+c

Now, what about c, the y-intercept?

To find c, think about what your (x,y) points mean:

(0,3). When x of the line is 0, y of the line must be 3.

(-5,3). When x of the line is -5, y of the line must be 3.

Because you said the line passes through each one of these two points, right?

Now, look at our line's equation so far: y=0x+c. c is what we want, the 0 is already set and x and y are just two "free variables" sitting there. We can plug anything we want in for x and y here, but we want the equation for the line that specfically passes through the two points (0,3) and (-5,3).

So, why not plug in for x and y from one of our (x,y) points that we know the line passes through? This will allow us to solve for c for the particular line that passes through the two points you gave!.

You can use either (x,y) point you want..the answer will be the same:

(0,3). y=mx+c or 3=0x 0+c, or solving for c: c=3-(0)(0). c=3.

(-5,3). y=mx+c or 3=0x -5+c, or solving for c: =3-(0)(-5). c=3.

See! In both cases we got the same value for c. And this completes our problem.

The equation of the line that passes through the points

(0,3) and (-5,3)  

is  

y=+3

Hope this helps :)

y = 3

First find the gradient of the line:

(Y²-Y¹)/(X²-X¹)

(3-3)/(-5-0)

0/-5 = 0

Your gradient is 0

Now find the y intercept. This is whrre where the line crosses the y axis, when X = 0

(0,3)

Your y intercept is 3.

Now put It into the formula y = mx + c

m = 0

c = 3

y = 0x + 3

y = 0 + 3

y = 3

Answer:

24 cubic inches

Step-by-step explanation:

The formulas for the volume of a cylinder and cone are ...

volume of a cylinder = πr^2·h

volume of a cone = (1/3)πr^2·h

For the same radius and height, the cone has 1/3 the volume of the cylinder. Your cone's volume is ...

(1/3)·(72 cubic inches) = 24 cubic inches

Final answer:

To find the volume of a cone with the same height and radius as a cylinder with a volume of 72 cubic inches, divide the cylinder's volume by 3. The resulting volume of the cone is 24 cubic inches.

Explanation:

Calculating the Volume of a Cone

The student's question involves finding the volume of a cone that has the same height and radius as a given cylinder with a known volume. Since the volume of the cylinder is given as 72 cubic inches, we can use this information to find the volume of the cone. The formula for the volume of a cylinder is V = πr²h, where r is the radius and h is the height. For a cone with the same height and radius, the formula to calculate the volume is V = ⅓πr²h. This formula for cone volume is derived from the fact that it's one-third of the volume of a cylinder with the same height and radius.

Given that the cylinder's volume is 72 cubic inches, the volume of the cone would be V = ⅓ × 72 cubic inches, which simplifies to V = 24 cubic inches. This means the volume of the cone is 24 cubic inches.

To summarize the process:

Use the given volume of the cylinder to reference the volume of the cone.

Apply the formula for the cone's volume: V = ⅓πr²h.

Since the volume of the cylinder is three times that of the cone, divide the cylinder's volume by 3 to get the cone's volume.

Answer:

[tex]tan(D)=\frac{12}{5}[/tex]

Step-by-step explanation:

we know that

In the right triangle DEF

The tangent of angle D is equal to the opposite side angle D divided by the adjacent side angle D

so

[tex]tan(D)=\frac{EF}{ED}[/tex]

substitute the values

[tex]tan(D)=\frac{12}{5}[/tex]

Answer:

  24 terms

Step-by-step explanation:

The sum of an arithmetic sequence is the average of the first and last terms, multiplied by the number of terms. The last term is given by ...

  an = a1 + (n-1)d

We have a sequence with first term a1 = 2 and common difference d = 2. So the last term is ...

  an = 2+ 2(n -1) = 2n

Then the average of first and last terms times the number of terms is ...

  Sn = 600 = n(2 + 2n)/2 = n(n+1) . . . . . . close to n²

We can solve the quadratic in n, or we can estimate the value of n as the integer just below the square root of 600.

  √600 ≈ 24.5

so we believe n = 24.

_____

Check

  S24 = 24·25 = 600 . . . . . . as required.

The number of terms in the arithmetic sequence (2,4,6,8,...) that would give a sum of 600 is approximately 35. This conclusion is reached by substituting the values into the formula for the sum of an arithmetic sequence and solving the quadratic equation derived from it.

The subject question pertains to an arithmetic sequence and seeks to find the number of terms that would give a sum of 600. Considering the arithmetic sequence defined as (2,4,6,8....), the difference (d) between consecutive terms is 2, and the first term (a) is 2 itself. The formula for the sum (S) of an arithmetic sequence is S=n/2 * (2a + (n-1)d) where n is the number of terms. Therefore, we want to solve the equation for n providing that the sum S is 600:

600 = n/2 * (2 * 2 + (n - 1) * 2) simplifies to 600 = n * (2 + n - 1) which leads to the quadratic equation n^2 + n - 300 = 0. Solving this quadratic equation using the quadratic formula [tex]\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}[/tex], we end up having  [tex]n = -1 \pm \sqrt{1 + 1200}[/tex]or  [tex]n = -1 \pm \sqrt{1201}[/tex]

Therefore, we have two potential solutions: n = -1 + sqrt(1201) and n = -1 - sqrt(1201). However, since the number of terms cannot be negative or fraction in a sequence, the only valid solution here is n = -1 + sqrt(1201) which equals approximately 35. This means that 35 terms of this arithmetic sequence would give a sum of 600.

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Answer:

a) each hour the number is multiplied by 16. Such a geometric sequence is described by an exponential growth function.

b) n = 16^t . . . . . n is number of bacteria; t is hours

c) about 7.923×10^28

d) the rule would be multiplied by 100/16=6.25, so becomes n=6.25·16^t

Step-by-step explanation:

a) We presume the table represents exponential growth because it was formulated with that purpose in mind. ("Why?" is always a tricky question.) It represents exponential growth because each table entry is 16 times the one before. A constant multiplier like that is characteristic of exponential growth.

___

b) In part (a) we noted the common ratio is 16. Since the first term (for t=1) is also 16, we can write the equation as ...

n = 16^t

where n is the number of bacteria and t is time in hours

___

c) Evaluating the formula for t=24, we get ...

n = 16^24 ≈ 7.923×10^28

n = 79,228,162,514,264,337,593,543,950,336

This is the estimate of the number present after 24 hours based on the formula.

___

d) If the starting number is different, the rule is multiplied by a factor that gives the appropriate starting value. The above answer assumes your "starting value" is 100 after 1 hour. If the starting value at 0 hours is 100, then the rule of part (b) gets multiplied by 100, not 6.25

n = 100·16^t . . . . . starting with 100 at t=0

n = 6.25·16^t . . . . starting with 100 at t=1

_____

Comment on exponential bacteria growth

The number of bacteria estimated in part (c) have an approximate volume of 102 cubic kilometers, roughly the volume of Lake Nicaragua.

Answer:

Step-by-step explanation:

The square has 4 sides with length 4

The right triangle has the right side equal to 4yd + 4yd(from the square) = 8yd

Using the Pythagorean Theorem, we find that the left side of the triangle has length = 10yd

The area of the whole thing is the area of the square + the area of the triangle

The formula for the area of a square with sides l is [tex]A_s = l^2[/tex]

The area of the triangle is trickier, but you can imagine tracing a line in the left side and the upper side to form a rectangle, and the area of that is A = [tex]l * L[/tex], the area of the triangle will be half the area of the rectangle so it'll be [tex]A = \frac{6 * 8}{2} = 24 (yd)^{2}[/tex]

The total area will be;

Area of the square + Area of the triangle = [tex]16 + 24 = 40 yd^{2}[/tex]

5/6=f/15

15•5=6•f

75=6•f

75/6=f

12.5=f

The flag pole is 12.5 ft tall

¶

. |

__& ____|

Answer:

  The flagpole is 12.5 ft tall.

Step-by-step explanation:

The ratio of object height to shadow length is ...

  (teen height) : (teen shadow) = (5 ft) : (6 ft) = 5 : 6

so the height of an object is 5/6 the length of its shadow. Then the height of the flagpole is ...

  (5/6) · (15 ft) = 12.5 ft

_____

A suitable scale drawing can show you the answer.

Answer:

the answer is .75 as a decimal or 3/4 as a fraction.

Step-by-step explanation:

Lara and three girl friends share three sandwiches. They can share .75

HEY!

SOLUTION :

[tex]a(x + b) = c \\ \\ = ax + ab = c \\ \\ = ax = c - ab \\ = x = \frac{c - ab}{a} [/tex]

Thanks!

Answer:

Step-by-step explanation:

you have to change it into a x and keep change flip and

1/5 / 3/7

1/5 * 7/3 then x

To divide two fractions, you should flip the second fraction to get its reciprocal, then multiply the first fraction with this reciprocal. For instance, in dividing 3/5 by 4/7, you flip 4/7 to get 7/4, then multiply 3/5 by 7/4 to obtain 21/20.

To divide two fractions, you will need to follow a few simple steps. Firstly, you need to flip or find the reciprocal of the second fraction (the one that you are dividing by). Secondly, you will multiply the first fraction (the one being divided) by the reciprocal of the second fraction. This process can also be remembered as 'copy, flip, multiply'.

For example, if we have to divide 3/5 by 4/7, first we flip 4/7 to get 7/4. Then we multiply 3/5 by 7/4. The result is 21/20.

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The domain of the function c(t) = 0.09t is 0 < t < 444, so that he spend no more than $40 per month on her texting bill.

An equation is an expression that shows the relationship between two or more numbers and variables.

Let c(t) represents the total phone bill based on the number of texts (t). Since:

c(t) = 0.09t and c ≤ $40, hence:

40 = 0.09t

t = 444

The domain of the function c(t) = 0.09t is 0 < t < 444, so that he spend no more than $40 per month on her texting bill.

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Answer:

The height of the soccer ball is above 35 ft when t is more than 0.947 seconds and less than 2.178 seconds after it is kicked.

Step-by-step explanation:

You are right, it doesn't make sense. The question seems to ask for the specific times at which height is 35 feet, but the "answer" wording suggests an interval is of interest.

_____

This is a good question to ask your teacher about.

Since only the first term has ax for coefficients the GCF cant be -3ax^5.

The GCF would be -3a

Factor -3a out of all the terms:

-3a(2x^7)-3a(-4a^5)-3a(a^4)

Now factor -3a out of the equation for the final answer:

-3a(2x^7-4a^5+a^4)

Answer:

196,900,000

Step-by-step explanation:

Multiply it out, or have your calculator or spreadsheet show you. (Some places, this *is* standard form.)

1.969×10^8 = 1.969×100,000,000 = 196,900,000

Answer:

3 km/h

Step-by-step explanation:

Let c represent the speed of the current in km/h, and t the time in hours after the raft leaves point A. 2 hours 40 minutes is 8/3 hours.

Distance = Speed × Time

The watercraft meet at a point 27 km upstream from B, so we have two equations involving these variables:

27 = 44 - ct

27 = (12-c)(t -8/3)

Equating these two versions of "27", we have ...

44 -ct = 12t -32 -ct +(8/3)c

76 = 12t + 8/3·c . . . . . add 32+ct

9t +2c = 57 . . . . multiply by 3/4

t = (57 -2c)/9 . . . .solve for t

Now, we can substitute this into the first of the above equations, after rearranging it to ...

ct = 17

c(57 -2c)/9 = 17

2c^2 -57c +153 = 0 . . . . multiply by 9, subtract the left side

c = (57-√((-57)^2 -4(2)(153)))/(2·2) = (57 -√2025)/4 = 12/4 . . . . quadratic formula

c = 3

The speed of the current is 3 km/h.

Answer:

The student assumed he was asked to factor

(a^2 - b^2)

Which is equal to

= (a+b)*(a-b)    (His response)

But, he was asked to simplify

(a^2 + b^2)

Which has imaginary roots, and can be factored as:

(b +ai)(b-ai)

Answer:

3^2+7*4~  Addition should be last

Step-by-step explanation:

Parenthesis        ----- Addition is second to last in order of operations

Exponents                    

Multiplication

Division

Addition

Subtraction

Given the problem 3² + 7 × 4, the last operation that should be performed is addition due to the rules dictated by the order of operations (PEMDAS).

In the given problem, 3² + 7 × 4, the operation that should be performed last is addition. This is due to the rule of the order of operations, often remembered by the acronym PEMDAS which stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Therefore, in this expression, you first perform the exponentiation (3²), then multiplication (7x4), and add the results last.

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2.52 × 10 = 25.2

2.52 = 240 × 0.008

The total thickness of a stack of 10 textbooks, where each book has 240 sheets of 0.008 cm thickness and two covers of 0.3 cm thick, is 25.2 cm.

The question requires us to find the total thickness of a stack of 10 textbooks. Each book has 240 sheets and two covers. The thickness of each sheet is 0.008 cm and each cover is 0.3 cm thick.

First, we calculate the thickness of one book. The thickness of sheets in one book is 240 multiplied by 0.008 cm (sheet thickness), which equals 1.92 cm. The thickness of the covers of one book is 2 multiplied by 0.3 cm (cover thickness), which equals 0.6 cm. Hence the total thickness of one book is 1.92 cm (sheets) plus 0.6 cm (covers), which equals 2.52 cm.

Therefore, the thickness of a stack of 10 books would be 10 multiplied by 2.52 cm (single book thickness), which equals 25.2 cm

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35 round to the nearest 10 equal 40

37 round =40

Answer:

7.  combination; 25C6 = 177100

8.  combination; 15C5 = 3003

Step-by-step explanation:

7. The order of the committee member selections is not important. (It would be if specific people filled specific positions on the committee.) Hence, the number is a number of combinations of 25 people taken 6 at a time.

__

8. The order of players is not important, as it might be if specific choices filled specific positions on the team. (The problem statement gives no indication that is the case. It is only by our knowledge of basketball teams that we entertain the possibility that order might be important.) Hence, the number is a number of combinations of 15 people taken 5 at a time.

_____

Of course, nCk = n!/(k!(n-k)!)

Answer:

[tex](x+2)^2+(y-1)^2=25[/tex]

Step-by-step explanation:

The equation of a circle with center at (h,k) and radius r, is given by

[tex](x-h)^2+(y-k)^2=r^2[/tex]

Since the given circle contains the point (-5,-3) and it is centered at (-2,1), we can determine the radius of the circle using the distance formula;

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

This implies that;

[tex]r=\sqrt{(-2--5)^2+(1--3)^2}[/tex]

[tex]r=\sqrt{(-3)^2+(4)^2}[/tex]

[tex]r=\sqrt{9+16}[/tex]

[tex]r=\sqrt{25}[/tex]

[tex]r=5[/tex]

We now substitute (h,k)=(-2,1) and r=5 into the formula to obtain;

[tex](x+2)^2+(y-1)^2=5^2[/tex]

[tex](x+2)^2+(y-1)^2=25[/tex]

Answer:

  A ≈ 67°

Step-by-step explanation:

Since you have all three sides of the right triangle, you can use any of the inverse trig functions to find the angle. SOH CAH TOA reminds you ...

  Sin(A) = Opposite/Hypotenuse = 12/13

  A = arcsin(12/13) ≈ 67°

__

  Cos(A) = Adjacent/Hypotenuse = 5/13

  A = arccos(5/13) ≈ 67°

__

  Tan(A) = Opposite/Adjacent = 12/5 = 2.4

  A = arctan(2.4) ≈ 67°

_____

Comment on calculator use

When you use your calculator for these inverse functions, make sure it is in "degrees" mode (not "radians"). Your calculator keys may be labeled with a "-1" superscript to indicate the inverse function. You can use the rounding function of your calculator, or you can round the number yourself (probably easier).

  sin⁻¹ = arcsin

  cos⁻¹ = arccos

  tan⁻¹ = arctan

A Google search box is also capable of showing you the inverse trig function value. (see attachment)

Answer:

Step-by-step explanation:

It's a right triangle, because:

[tex]5^2+12^2=13^2\\25+144=169\\169=169[/tex]

Pythagorean teorem.

Use sine:

[tex]sine=\dfrac{opposite}{hypotenuse}[/tex]

We have:

[tex]opposite=12\\hypotenuse=13[/tex]

[tex]\sin A=\dfrac{12}{13}\approx0.9231\Rightarrow67^o[/tex]

To turn a number into scientific notation, move the decimal point so that there is only 1 number to the left of the decimal point.

Count the number of spaces you moved the decimal point.

If you move the decimal point to the right, the exponent is negative, if you move it to the left the exponent is positive.

0.000075

You need to move the decimal point 5 places to the right to get 7.5

Because it was moved to the right, the exponent is negative 5

The answer would be 7.5 x 10^-5 which is B.

The scientific notation of 0.000075 is 7.5 x [tex]10^{-5[/tex]. This is because in converting to scientific notation, the goal is to have a number >=1 but <10 multiplied by 10 raised to an exponent. In this case, the decimal was moved 5 places to the right to result in a number <10.

In mathematics, scientific notation is used to express very large or very small numbers in a simpler way by using powers of ten. In this case, you're trying to represent the number 0.000075. To do this accurately in scientific notation, we would write it as 7.5 x [tex]10^{-5[/tex]. This is because the exponent of the 10 represents how many places the decimal point was moved to convert the number to a new form where a number greater than or equal to 1 but less than 10 is multiplied by 10 raised to an exponent.

In the case of option D - 75 x [tex]10^{-6[/tex], it's suggesting we moved the decimal place 6 places to the right to a number that's >= 10, which is incorrect. The decimal is moved 5 places to the right to get a number that's less than 10. Thus, the correct answer is B - 7.5 x [tex]10^{-5[/tex].

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Answer:

  1/8, 7/8

Step-by-step explanation:

1/2 is 4/8.

3/8 less than 4/8 is 1/8.

3/8 more than 4/8 is 7/8.

Answer:

[tex]y=2\cdot4^x\\y=4\cdot\left(\frac{1}{2}\right)^x[/tex]

Step-by-step explanation:

We're given both the y-intercept and a point on the graphs of both functions, so our work her is largely just substituting x and y values in the general form of the equation for an exponential function.

For the first one, we can start by using the point (0, 2) to solve for a:

[tex]y=a\cdot b^x\\2=a\cdot b^0\\2=a[/tex]

Next, we can use that a-value and the second point to solve for b:

[tex]y=2\cdot b^x\\128=2\cdot b^3\\64=b^3\\4=b[/tex]

This gives us the equation [tex]y=2\cdot4^x[/tex] for the first function.

We can repeat the same process for the second function. Solving for a:

[tex]y=a\cdot b^x\\4=a\cdot b^0\\4=a[/tex]

And then for b, using the point (2, 1):

[tex]1=4\cdot b^2\\\frac{1}{4}=b^2\\\frac{1}{2}=b[/tex]

This gives us the general equation [tex]y=4\cdot\left(\frac{1}{2}\right)^x[/tex]

To write an exponential function in y = a · bx form that represents the given situations, substitute the given points into the equation to find the values of a and b.

To write an exponential function in the form y = a · bx that represents the given situations:

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Answer:

  (-√10)/10

Step-by-step explanation:

The trig identity that relates the sine and the tangent is ...

  sin(x) = ±tan(x)/√(1 + tan(x)²)

For your given value of tan(x), this evaluates to ...

  sin(x) = ±(1/3)/(√(1 + (1/3)²) = ±1/√10 = (±√10)/10

The given value of the tangent function is positive, so the sine and cosine functions have the same sign (both are negative in this case).

So, your sine function is ...

  sin(x) = (-√10)/10

_____

You can derive the above relationship using SOH CAH TOA.

  Tan = Opposite/Adjacent

You are given the value of the tangent function as 1/3, so it is convenient to assign Opposite = 1 and Adjacent = 3. Then Hypotenuse is found using the Pythagorean theorem as ...

  Hypotenuse = √(Opposite² + Adjacent²) = √(1² + 3²) = √10

Now, the sine is ...

  Sin = Opposite/Hypotenuse = 1/√10

Rationalizing the denominator, this is ...

  Sin = (√10)/10

__

Now, you need to take into account the quadrant of the angle. Tangent is positive and Cosine is negative in the 3rd quadrant, where the Sine is also negative. Then your sine value is ...

  sin(x) = (-√10)/10

For both problems, expand the left hand side and match up the coefficients.

a. [tex](5x-2)^2=25x^2-20x+4=ax^2-bx+c[/tex]

[tex]\implies a=25,b=20,c=4[/tex]

[tex]\implies a+c^2=25+4^2=41[/tex]

b. [tex](3x+5)^2=9x^2+30x+25=ax^2+bx+c[/tex]

[tex]\implies a=9,b=30,c=25[/tex]

[tex]\implies a+b+c=9+30+25=59[/tex]

Answer:

  Table B

Step-by-step explanation:

The table represents a linear function if the ratio of change in y (∆y) to change in x (∆x) is a constant.

A — first two points: ∆y/∆x = (1-2)/(3-0) = -1/3

  second two points: ∆y/∆x = (6-1)/(4-3) = 5 ≠ -1/3

__

B — first two points: ∆y/∆x = (2-(-3))/(4-(-1)) = 5/5 = 1

  second two points: ∆y/∆x = (4-2)/(6-4) = 2/2 = 1, the same as for the first points. This is the table that answers the question.

__

C — first two points: ∆y/∆x = (0-(-2))/(0-(-3)) = 2/3

  second two points: ∆y/∆x = (4-0)/(2-0) = 4/2 = 2 ≠ 2/3

__

D — first two points: ∆y/∆x = (-2-(-7))/(0-5) = 5/-5 = -1

  second two points: ∆y/∆x = (2-(-2))/(2-0) = 4/2 = 2 ≠ -1

Table 3, with the ordered pairs (-3, -2), (0, 0), and (2, 4), appears to represent a linear function due to the consistent changes in 'y' as 'x' increases.

Let's analyze each table to determine which one appears to represent a linear function:

Table 1:

In this table, as 'x' increases by 3 units (from 0 to 3) and then by 1 unit (from 3 to 4), 'y' changes from 2 to 1 and then to 6. The changes in 'y' are not consistent, so this table does not appear to represent a linear function.

Table 2:

In this table, as 'x' increases by 5 units (from -1 to 4) and then by 2 units (from 4 to 6), 'y' changes from -3 to 2 and then to 4. The changes in 'y' are not consistent, so this table does not appear to represent a linear function.

Table 3:

In this table, as 'x' increases by 3 units (from -3 to 0) and then by 2 units (from 0 to 2), 'y' changes from -2 to 0 and then to 4. The changes in 'y' are consistent, indicating a linear relationship between 'x' and 'y.'

Table 4:

In this table, as 'x' increases by 5 units (from 0 to 5) and then by 2 units (from 2 to 0), 'y' changes from -2 to 2 and then to -7. The changes in 'y' are not consistent, so this table does not appear to represent a linear function.

Based on the consistent changes in 'y' as 'x' increases in Table 3, it appears to represent a linear function. So, Table 3 is the one that shows a set of ordered pairs that appears to lie on the graph of a linear function.

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The solution is : the area, in square feet, of the mat is 169π/4 ft^2.

Area is defined as the total space taken up by a flat (2-D) surface or shape of an object. The space enclosed by the boundary of a plane figure is called its area. The area of a figure is the number of unit squares that cover the surface of a closed figure. Area is measured in square units like cm² and m². Area of a shape is a two dimensional quantity.

here, we have,

we know that,

Circumference = π * d

Diameter = 2r

Area of a circle = πr²

From our first equation, we can find our value for our diameter:

We do this by dividing 13π by π to get 13.

so, we get,

C = πd

or, 13π = πd

or, d = 13π/π

or, d = 13

Using our second equation, we can find out the value for our radius:

d = 2r

so, r = 13/2

Now we have a value for our radius, we can use our third equation to find out our area:

A = πr²

so, A = π(13/2)²

or, A = 169π/4

Hence, The solution is : the area, in square feet, of the mat is 169π/4 ft^2.

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The area of the circular mat with a circumference of 13π feet is 42.25π square feet, calculated by first determining the radius from the circumference and then using the area formula.

To find the area of the circular mat when given the circumference, we can start by recalling the relationship between the circumference (C) and the radius (r) of a circle. The formula for circumference is C = 2πr. From this, we can solve for the radius:

C = 2πr
13π = 2πr
r = 13π / 2π
r = 6.5 feet

Now we have the radius, so we can find the area (A) of the circle using the formula A = πr²:

A = π(6.5)²
A = π×42.25
A = 42.25π square feet

Thus, the area of the circular mat is 42.25π square feet.

Answer:

y = -2(x+2)(x-1)/((x+3)(x-6)) = (-2x^2 -2x +4)/(x^2 -3x -18)

Step-by-step explanation:

A polynomial function will have a zero at x=a if it has a factor of (x-a). For the rational function to have zeros at x=-2 and x=1, the numerator factors must include (x+2) and (x-1).

For the function to have vertical asymptotes at x=-3 and x=6, the denominator of the rational function must have zeros there. That is, the denominator must have factors (x+3) and (x-6). Then the function with the required zeros and vertical asymtotes must look like ...

f(x) = (x+2)(x-1)/((x+3)(x-6))

This function will have a horizontal asymptote at x=1 because the numerator and denominator degrees are the same. In order for the horizontal asymptote to be -2, we must multiply this function by -2.

The rational function may be ...

y = -2(x +2)(x -1)/((x +3)(x -6))

If you want the factors multiplied out, this becomes

y = (-2x^2 -2x +4)/(x^2 -3x -18)

Answer:

y = -2(x+2)(x-1)/((x+3)(x-6)) = (-2x^2 -2x +4)/(x^2 -3x -18)

Step-by-step explanation:

A polynomial function will have a zero at x=a if it has a factor of (x-a). For the rational function to have zeros at x=-2 and x=1, the numerator factors must include (x+2) and (x-1).

For the function to have vertical asymptotes at x=-3 and x=6, the denominator of the rational function must have zeros there. That is, the denominator must have factors (x+3) and (x-6). Then the function with the required zeros and vertical asymptotes must look like ...

f(x) = (x+2)(x-1)/((x+3)(x-6))

This function will have a horizontal asymptote at x=1 because the numerator and denominator degrees are the same. In order for the horizontal asymptote to be -2, we must multiply this function by -2.

The rational function may be ...

y = -2(x +2)(x -1)/((x +3)(x -6))

If you want the factors multiplied out, this becomes

y = (-2x^2 -2x +4)/(x^2 -3x -18)