Difference between synthetic division and synthetic substitution

The equation of a hyperbola is  [tex]x^{2}[/tex]/9 - [tex]y^{2}[/tex]/4 = 1

What are the steps to Hyperbola equation ?

To find the equation of hyperbola, the following steps must be taken. You need to identify;

The general equation of a hyperbola can be expressed as

[tex]x^{2}[/tex]/[tex]a^{2}[/tex] - [tex]y^{2}[/tex]/[tex]b^{2}[/tex] = 1

From the graph, we have the following parameters

a = 3

[tex]a^{2}[/tex] = [tex]3^{2}[/tex] = 9

b = 2

[tex]b^{2}[/tex] = [tex]2^{2}[/tex] = 4

The equation of a hyperbola can be expressed as

[tex]x^{2}[/tex]/9 - [tex]y^{2}[/tex]/4 = 1

The vertices = V(+/-a,0) = (+/-3,0)

The center = C(0,0)

The focus = F(+/-C, 0)

Where [tex]C^{2}[/tex] = [tex]a^{2}[/tex]  + [tex]b^{2}[/tex]

C = [tex]\sqrt{9 + 4}[/tex]

C = [tex]\sqrt{13}[/tex]

Focus = F(+/- [tex]\sqrt{13}[/tex], 0)

The general equation for asymptote = +/- b/a X

= +/-2/3X

Therefore, the equation of a hyperbola can be expressed as

[tex]x^{2}[/tex]/9 - [tex]y^{2}[/tex]/4 = 1

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Answer:

a=10%, b=25/100, c=1/2

Step-by-step explanation:

a: If the denominator is exactly 100, then the numerator represents a percentage.

b: when you change a fraction to have a greater denominator, an easy way to figure out how to change the numerator to match it and make the value of the rewritten fraction equal to the old version, is to divide the new value of the denominator by the old value (in this case being 100/4=25), and put the answer where the numerator goes.

c: When simplifying a fraction with a lesser numerator than denominator, find the greatest common factor (a factor is a whole number a number can be divided by without resulting in a negative or a decimal number) and divide both the numerator and denominator by it.

The number of number of four-legged and three-legged stools in stock is 38 and 16 respectively.

let

x + y = 54

4x + 3y = 200

From equation (1)

x = 54 - y

4x + 3y = 200

4(54 - y) + 3y = 200

216 - 4y + 3y = 200

216 - y = 200

- y = 200 - 216

-y = -16

y = 16

x + y = 54

x + 16 = 54

x = 54 - 16

x = 38

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Answer:

The surface area (SA) of the given triangular prism is 120m.

Step-by-step explanation:

Greetings !

B/c, SA=bh+(b₁+b₂+b₃)l

Answer:

SA = 120 m2

Step-by-step explanation:

one 9×5 rectangle = 45 m2
one 4×9 rectangle = 36 m2
one 9×3 rectangle = 27 m2

two (4×3)/2 triangles = 6 + 6 = 12 m2

Add the areas:

surface area = 45 + 36 + 27 +12 = 120 square meters

Hope this helps

The pricing scheme of the function is (a) The first two hours cost $2, between two hours and six hours cost $2 per hour, and all hours after that cost $1.

The points are given as:

The constant interval implies that the first two hours cost $2

Next, we calculate the slope/rate of the other points.

This is done using

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

So, we have:

m1 = (10 - 2)/(6 - 2) = 2

m2 = (11 - 10)/(7 - 6) = 1

This means that:

Hence, the pricing scheme of the function is (a)

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Answer:

A minimum of 21250 packets per day.

Step-by-step explanation:

If Monday produced 15,000 packets, and Tuesday was closed they produced 15,000 packets in 2 days.

The factory is open 6 days a week.

They need 100,000 packets.

They have 15,000 packets.

6 subtract 2 is 4.

100,000 subtract 15,000 is 85,000.

They need 85,000 packets in 4 days.

That is a minimum of 21250 packets per day.

The slope of the median of the triangle whose vertices are P(-6,1),Q(-2,5) R(8,1) and passes through R is -1/6.

Given that vertices of triangle is P(-6,1),Q(-2,5),R(8,1) and there is a median which passes through R.

Median of a triangle is a line segment which originates from a vertex of triangle and divides the opposite side in two equal parts. let median touches the line at L point.

When line passes through R then it must be a perpendicular on PQ. The coordinates of L is {(-2-6)/2,(5+1)/2}=(-4,3)

Now because median passes through R,it will be LR and we know that the formula of calculating slope of line from two points is [tex](y_{2} -y_{1} )/(x_{2} -x_{1} )[/tex] where [tex](x_{1} ,y_{2} ),(x_{2} ,y_{2} )[/tex] are the points of the end points of line.

Slope=(1-3)/(8+4)

=-2/12

=-1/6

Hence the slope of the median of the triangle whose vertices are P(-6,1),Q(-2,5) R(8,1) and passes through R is -1/6.

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Answer:

4 ft

Step-by-step explanation:

The area of each tile is tile is 1 square ft, so the total area of the floor is 16 square ft.

Therefore, the side length is 4 ft.

Answer:

Step-by-step explanation:

area=length×width

=(2x+3)(3x-5)

Area=6x²-10x+9x-15

=6x²-x-15

73 division 84, 649 would give 8. 64 × 10^-4

In order to determine the value of the division, we have

= [tex]\frac{73}{84649}[/tex]

Using the calculator, put the values in

We would have

= 8. 64 × 10^-4

Thus, 73 division 84, 649 would give 8. 64 × 10^-4

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Answer:

  -7, 15

Step-by-step explanation:

The positive difference between E and G is 11.

  G - E = 11

  G = 11 + E = 11 + 4

  G = 15 . . . . . . . . . . . . . when G is right of E

__

  E - G = 11

  E - 11 = G = 4 - 11

  G = -7 . . . . . . . . . . . . when G is left of E

Possible coordinates for G are -7 and 15.

Answer: c. the mean is greater than the median

Explanation:

To find out the correct answer, we first must know the true meanings and the method of calculation of the terms, "mean", "mode" and "median".

Mean: The mean is a type of average. It is the sum of all the values in a set of data, such as numbers of measurements, divided by the numbers of values on the list.

[tex]\bar{x} = \dfrac{\sum x}{n}[/tex]                                      where, [tex]\bar{x}=[/tex] sample mean
                                                               [tex]\sum x =[/tex] sum of each value in sample
                                                               [tex]n =[/tex] number of values in the sample

Example: in {6, 3, 9, 6, 6, 5, 9, 3} the mean is 5.875 (6+3+9+6+6+5+9+3/8).
                           
                                         

Mode: The mode is the number which appears most often in a set of numbers.
Example: in {6, 3, 9, 6, 6, 5, 9, 3} the Mode is 6.

Median: The median is the "middle" of a sorted list of numbers in ascending order (small to big). To find the Median, place the numbers in value order and find the middle.
Example: in {6, 3, 9, 6, 6, 5, 9, 3} the median is 6. (in 3,3,5,6,6,9,9, the middle number is 6 )

with an even amount of numbers things are slightly different.
In that case we find the middle pair of numbers, and then find the value that is half way between them. This is easily done by adding them together and dividing by two. (basically averaging the middle two numbers incase there is an even set of numbers)
Example: 3, 13, 7, 5, 21, 23, 23, 40, 23, 14, 12, 56, 23, 29

When we put those numbers in order we have:

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

There are now fourteen numbers and so we don't have just one middle number, we have a pair of middle numbers.

3, 5, 7, 12, 13, 14, 21, 23, 23, 23, 23, 29, 40, 56

In this example the middle numbers are 21 and 23.

To find the value halfway between them, add them together and divide by 2: 21 + 23 = 44, then 44 ÷ 2 = 22

So the Median in this example is 22.
(Note that 22 was not in the list of numbers ... but that is OK because half the numbers in the list are less, and half the numbers are greater.)

Now that you understand the terms clearly, let's find out the mean, mode and median from your given set of numbers.
                                           5, 15, 25, 30, 3

Mean: (5+15+25+30+3)/5  is, 15.6
Mode: None since none of the numbers repeated more than once.
Median: 3,5,15,25,30  , so the middle number is 15 .

Since there is no mode, option d. can never be the answer. And clearly, 15.6 is greater than 15 so the answer should be c. the mean is greater than the median.

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Using translation concepts, it is found that g(x) is positive on the following interval:

(3, ∞)

A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.

Function [tex]g(x) = \sqrt[3]{x - 3}[/tex] is a shift right of 3 units of [tex]f(x) = \sqrt[3]{x}[/tex], which is positive on the interval (0, ∞), hence g(x) is positive on the interval (3, ∞).

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(a). All four sides are equal and the angles are right-angled (90 degrees).

(b). A quadrilateral has no pairs of parallel sides is a Kite and irregular trapezium.

(c). A triangle having one pair of equal sides and one pair of equal angles is an isosceles Triangle.

One of the first areas of mathematics is geometry, along with arithmetic. It is focused on spatial characteristics that relate to the proximity, size, shape, and relative placement of objects.

(a). Two features that make the shape square are all four sides equal and the angles are right-angled (90 degrees).

(b). A quadrilateral has no pairs of parallel sides is a Kite and irregular trapezium.

(c). A triangle having one pair of equal sides and one pair of equal angles is an isosceles Triangle.

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Considering the similarity of triangles, the corresponding sides are:

[tex]\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}[/tex].

If two triangles have the exact same angles, they are called similar, and the corresponding sides, that is, the sides adjacent to the same angles, have proportional lengths.

In this problem, the corresponding sides have equal ratios given as follows:

[tex]\frac{DE}{AB} = \frac{EF}{BC} = \frac{DF}{AC}[/tex].

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The values of (2/3 + 6/3), ( 4/6 + 9/6 ) and ( 6/6 + 6/6 ) are 8/3, 13/6 and 2 respectively.

Given that;

1) 2/3 + 6/3

We combine the numerators over the common denominators

= ( 2 + 6 ) / 3

= 8/3

2) 4/6 + 9/6

We combine the numerators over the common denominators

= ( 4 + 9 ) / 6

= 13/6

3) 6/6 + 6/6

We combine the numerators over the common denominators

= ( 6 + 6 ) / 6
= 12/6

= 2

The values of (2/3 + 6/3), ( 4/6 + 9/6 ) and ( 6/6 + 6/6 ) are 8/3, 13/6 and 2 respectively.

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Answer: Option (3)

Step-by-step explanation:

Since the parabola has a vertical axis of symmetry, we can eliminate Options (1) and (4).

To see whether options (2) or option (3) is correct, we can substitute in [tex]x=-1[/tex] and see if we get y=3.

Doing this with option (2), we get y = -3, so it is wrong.

Doing this with option (3), we get y = 3, as required.

So, option (3) is correct.

The limit of

[tex]$\lim _{x \rightarrow 0}\left(\frac{x \cos (\theta)-\theta \cos (x)}{x-\theta}\right)=1$$[/tex].

In Mathematics, a limit exists described as a value that a function approaches the output for the provided input values. Limits exist necessary in calculus and mathematical analysis and exist utilized to determine integrals, derivatives, and continuity.

Given:

[tex]$\lim _{x \rightarrow 0}\left(\frac{x \cos (\theta)-\theta \cos (x)}{x-\theta}\right)$$[/tex]

Substitute the value x = 0, then we get

[tex]$=\frac{0 \cdot \cos (\theta)-\theta \cos (0)}{0-\theta}$$[/tex]

Simplify the above equation, we get

[tex]$\frac{0 \cdot \cos (\theta)-\theta \cos (0)}{0-\theta}= 1$[/tex]

Therefore, the limit of

[tex]$\lim _{x \rightarrow 0}\left(\frac{x \cos (\theta)-\theta \cos (x)}{x-\theta}\right)=1$$[/tex]

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The area of the square is given by the  quadratic expression:

[tex]A = 9x^2 - 6xy + y^2[/tex]

If we have a square whose side measures S, the area of said square is given by the equation:

[tex]A = S^2[/tex]

In this case, we know that the side length of our square is given by:

[tex]S = 3x - y[/tex]

Replacing that in the area equation we get:

[tex]A = (3x - y)^2[/tex]

Now we just need to expand that product, we will get:

[tex]A = (3x - y)^2 = (3x)*(3x) + (3x)*(-y) + (-y)*(3x) + (-y)*(-y) \\\\A = 9x^2 - 6xy + y^2[/tex]

Then we can see that the correct option is the last one.

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Using an exponential function, it is found that:

The exponential function for population growth is given as follows:

[tex]P(t) = P(0)e^{kt}[/tex]

In which:

For Country A, we have that k = 0.016. The doubling time is t for which P(t) = 2P(0), hence:

[tex]P(t) = P(0)e^{kt}[/tex]

[tex]2P(0) = P(0)e^{0.016t}[/tex]

[tex]e^{0.016t} = 2[/tex]

[tex]\ln{e^{0.016t}} = \ln{2}[/tex]

[tex]0.016t = \ln{2}[/tex]

[tex]t = \frac{\ln{2}}{0.016}[/tex]

t = 43 years.

For Country B, P(36) = 2P(0), hence we have to solve for k to find the growth rate.

[tex]P(t) = P(0)e^{kt}[/tex]

[tex]2P(0) = P(0)e^{36k}[/tex]

[tex]e^{36k} = 2[/tex]

[tex]\ln{e^{36k}} = \ln{2}[/tex]

[tex]36k = \ln{2}[/tex]

[tex]k = \frac{\ln{2}}{36}[/tex]

k = 0.019.

For Country B, the growth rate is of 1.9% per year.

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Answer:

x = 5 ,  x = - 6

Step-by-step explanation:

First we factor using factorisation method :

We need 2 numbers that multiply to give +1 and add to give -30 :

To find these numbers we write out all the factors of 30 :

1, 30

2 , 15

3 , 10

5 , 6    ----> These must be the numbers as -5+6 = 1

Now we split +x into -5x and +6x :

x² -5x + 6x -30 = 0

Factor the first 2 terms together and the last two terms :

x(x-5) + 6(x-5) = 0

Keep 1 bracket and terms outside the brackets collect them into one:

(x-5)(x+6) = 0

We know that 1 of these brackets must be equal to zero as anything multiplied by zero = 0

Either x - 5 =0      OR x + 6 = 0

So x = 5                x = - 6

These are our final answers

Hope this helped and have a good day

Answer:

(x-5) (x+6)

x=5

X=-6

Step-by-step explanation:

x²+x-30=0

factorizing

x²-5x+6x-30=0

x(x-5)+6(x-5) =0

(x-5)(x+6)=0

now solve individually

x-5=0

send 5 to the other side

x=5

x+6=0

send 6 to the other side

x=-6

hope you understand the procedure

Answer: [tex]a=\frac{b}{zm-1}[/tex]

Step-by-step explanation:

Let's first multiply both sides by m to get rid of the denominator.

[tex]za=\frac{a+b}{m}\\zam=a+b[/tex]

Now, we have an a on both sides of the equation. It will be easier to solve if there is only one, so let's divide both sides by a.

[tex]\frac{zam}{a}=\frac{a+b}{a}\\zm=\frac{a}{a}+\frac{b}{a}\\zm=1+\frac{b}{a}[/tex]

The 1 on the right can be moved to the left to isolate the fraction.

[tex]zm-1=\frac{b}{a}[/tex]

We want to get a by itself, so let's multiply it on both sides to remove it from the denominator.

[tex]a(zm-1)=b[/tex]

Finally, we can divide zm-1 from both sides to get a by itself.

[tex]a=\frac{b}{zm-1}[/tex]

Answer:

Multiples of 4:4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, …

Multiples of 6:6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …

Answer: The least common denominator of 1/4 and 1/6 is 12.

Prime factorisations of 8= 2 × 2 × 2.

Prime factorisations of 12= 2 × 2 × 3.

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Hi! ⋇

                 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

   To find the least common denominator of [tex]\sf{\frac{1}{2}}[/tex] and [tex]\sf{\frac{5}{6}}[/tex], we should first find the least common multiple of 2  and 6, which is 6:

 Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16... ∞

 Multiples of 6: 6, 12, 18, 24... ∞

(all common multiples are in bold-face; all multiples of 6 are in bold-face because all multiples of 6 are also multiples of 2)

· In both sets, 6 is the least common multiple, and it's also the least common denominator.

Hope this made sense to you !!

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Answer:

$592.92

Step-by-step explanation:

Bills:

47 ones = 47*1 = $47

22 fives = 22*5 = $110

9 tens = 9*10 = $90

17 twenties = 17*20 = $340

Total Bills: 47+110+90+340 = $587

Coins:

67 pennies = 67*1 = $0.67

12 nickels = 12*5 = $0.60

9 dimes = 9*10 = $0.90

15 quaters = 15*25 = $3.75

Total Coins: 0.67+0.6+0.9+3.75 = $5.92

Total: $587+$5.92 = $592.92

The outcome P-Value is 0.0179 when Zstat = +2.10

According to the statement

we have given the value of Z-Stat which is +2.10

And we have to find the P value when H0 is rejected only in the upper tail

Z-stat is defined as the Z-score indicates how much a given value differs from the standard deviation. The Z-score, or standard score, is the number of standard deviations a given data point lies above or below mean.

And

we know that the relationship between Z-Stat and the p value is the P-Value is calculated by converting your statistic (such as mean / average) into a Z-Score.

So, the outcome p value is 0.0179

So, The outcome P-Value is 0.0179 when Zstat = +2.10

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Answer:

50<x<60

Step-by-step explanation:

The triangle can only be formed with these two side lengths if the third side is bigger than (Subtract them)

55 - 5

...and smaller than (Add them)

55 + 5

55-5 < x < 55+5

50 < x < 60

Answer:

First answer is 1200

Second answer is 900 for each new employee

Step-by-step explanation:

First answer:

2200+1300+800+940+560+x = 7000

5800+x = 7000 subtract 5800 from both sides and you get your answer

x = 1200

Second answer:

x = 900.  Each of the two employees made 900 a month or 1800 a month for both of them.

To find the average we take the total salaries and divide by the number of people to find the average salary.  In this case, we know the average and we know all of the salaries, but two.  We can figure this out.

(7000 + 2x)/8 = 1100  multiple both sides by 8 to clear the fraction/

7000 +2x = 8800  Subtract both sides by 7000

2x = 1800  Divide both sides by 2

x = 900

The number of bats are expected to be in the colony after 11 years if the colony continues to grow at the same rate is 69,632

Let

exponential function;

y = ax^t

When t = 3

272 = ax³

when t = 5

1088 = ax^5

272/1088 = ax^3 / ax^5

1/4 = x^3-5

1/4 = x^-3

1/4 = 1/x²

1 × x² = 4 × 1

x² = 4

x = 2

Substitute x = 2 into

272 = ax³

272 = a × 2³

272 = a × 8

272 = 8a

a = 272/8

a = 34

Number of bat expected at the colony after 11 years

y = ax^t

= 34 × 2^11

= 34 × 2048

y = 69,632 bats

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The Piecewise -Defined function and how it is graphed is given below..

Given the function in the attached image,

See the attached Graph for better understanding.

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Applying the Pythagorean theorem, the missing side of the triangle is: C. √193 km.

To find the missing side (x) in the right triangle, based on the Pythagorean theorem, we would have the following equation:

x = √(12² + 7²)

x = √(144 + 49)

x = √193

The missing side of the triangle is: C. √193 km

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Answer:

C.)  193√ km

Step-by-step explanation:

I got it right on founders edtell