Mathematics High School

## Answers

**Answer 1**

The table can be used to determine the missing **value** of x by examining the pattern in the data. One way to do this is to look for a relationship between the values in the table.

For example, we can see that the values in the second column are all twice the values in the first column. Similarly, the values in the third column are all three times the values in the first column.

Using this pattern, we can determine the missing value of x by **multiplying** the value in the first column by four. This is because the value in the fourth column should be four times the value in the first column, based on the pattern observed in the other columns.

So, to find the missing value of x, we can multiply the value in the third row of the first column, which is 7, by 4 to get 28. Therefore, the missing **value** of x is 28.

In general, when using a table to determine missing values, it's important to look for patterns and relationships in the **data** to help identify the missing value.

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A 2-column table with 5 rows. Column 1 is labeled Decimal with entries 0.1 6 repeating, 0.3 repeating, 0.5, x, 0.8 3 repeating. Column 2 is labeled Fraction with entries one-sixth, two-sixths, three-sixths, four-sixths, five-sixths.

Which best shows how the table can be used to determine the missing value x?

A) 2(two-sixths) equals four-sixths, so 2(0.5) = 1, and x = 1.

B) 4(1-sixth) equals four-sixths, so 4(0.16) = 0.64, and x = 0.64.

C) Three-sixths plus one-sixth equals four-sixths, so 0.5 + 0.16 = 0.6, and x = 0.6.

D) Five-sixths plus one-sixth equals four-sixths, so 0.83 + 0.16 = 0.99, and x = 0.99.

## Related Questions

Khalid spent Rs 156 out of Rs1200. What percent did he spend?

### Answers

**Khalid **spent 13% of the total **amount **he had, which is Rs 156 out of Rs 1200.

To find out the percentage that **Khalid **spent, we need to divide the amount he spent by the total amount he had and then multiply the result by 100.

**Percentage **spent = (Amount spent / Total amount) x 100%

In this case, Khalid spent Rs 156 out of Rs 1200.

Percentage spent = (156 / 1200) x 100%

Percentage spent = 0.13 x 100%

Percentage spent = 13%

Therefore, Khalid spent 13% of the total **amount **he had.

Let's say Khalid spent x percent of the total amount he had, then we can write the equation as:

x% of Rs 1200 = Rs 156

To solve for x, we can first convert x% to a decimal by dividing by 100:

x/100 * Rs 1200 = Rs 156

Next, The **equation** we can solve for x by isolating it on one side:

x/100 = Rs 156 / Rs 1200

x/100 = 0.13

x = 0.13 * 100

x = 13

Therefore, Khalid spent 13% of the total amount he had, which is Rs 156 out of Rs 1200.

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excluded value of (3ab/4xy)/(-21a^2b/10x^2y)

### Answers

Simplifying the **expression** (3ab/4xy) ÷ (-21a²b/10x²y) = -5x/14a

What is an expression?

A** mathematical expression** is a relationship that does not contain an equality sign.

Since we have the expression (3ab/4xy)/(-21a^2b/10x^2y), we desire to simplify it.

We proceed as follows

(3ab/4xy) ÷ (-21a²b/10x²y) = 3ab/4xy × 10x²y/-21a²b

Expanding out the factors, we have

(3ab/4xy) ÷ (-21a²b/10x²y) = 3ab/4xy × 10x²y/-21a²b

= 3 × a × b/(4 × x × y) × 10 × x × x × y/(-21 × a × a × b)

= 3 × a × b/(2 × 2 × x × y) × 2 × 5 × x × x × y/(-3 × 7 × a × a × b)

Canelling out the** common factors,** we have that

= 3 × 2 × a × b × x × 5 × y × x/(3 × 2 × 2 × a × b × x × y × -7 × a)

Cancelling out 3 × 2 × a × b × x × y, we have that

= 5 × x/(2 × -7 × a)

= 5x/-14a

= -5x/14a

So, (3ab/4xy) ÷ (-21a²b/10x²y) = -5x/14a

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The amount of money Sarah has saved (S) can be modeled by the function equation:

S(w) = 8w + 20, where 'w' represents the number of weeks since she started saving. What does the rate of change mean for this situation?

A. Sarah has saved a total of $8

B. Sarah started with a savings of $8

C. Sarah saves $8 per week

D. Sarah saves $20 a week

### Answers

**Answer:**

The correct answer is C.

Sarah saves $8 per week.

The rate of change in this situation describes how much money Sarah saves each week, which is $8.

The solution to this equation is that Sarah saves $8 per week. This can be found by substituting "w" with the number of weeks since she started saving and solving for S.

write the complex number in standard form. √ – 0.04

### Answers

**Answer:**

0.9798 + 0i

**Step-by-step explanation:**

Assuming that the square root symbol applies only to the number 0.04 and not the entire expression, we can write the complex number in standard form as:

√(1 - 0.04) = √0.96

Since √0.96 cannot be simplified further without using decimals, we can convert it to a decimal approximation using a calculator or by long division. One possible decimal approximation is:

√0.96 ≈ 0.9798

Therefore, the complex number in standard form is:

0.9798 + 0i

Where 0i represents the imaginary part, which is zero in this case since the complex number is a real number.

The concept of zero:

1. What is the history of this concept?

a. Who was using it?

b. Where was it developed?

c. Why was this important during this time in history?

d. How did this change the way people did math?

2. How have teachers/family/friends taught you this topic in the past?

a. Any tips or tricks that you could share with the class?

3. How is this concept used in everyday life?

a. How do adults use it?

b. How do teenagers use it?

### Answers

The **concept of zero **was used by Indian mathematicians.

It was **developed** in India

What is the history of this concept?

1. History of the concept of zero:

a. The **concept of zero **was used by Indian mathematicians as early as the 5th century AD, including Aryabhata and Brahmagupta.

b. It was **developed** in India, and then spread to the Middle East and Europe.

c. The concept of zero was important during this time in **history **because it revolutionized mathematics by allowing for more complex calculations and the development of algebra. It also had implications for astronomy and navigation.

d. The use of zero changed the way people did math by allowing for the development of place value notation and the ability to represent numbers using a positional system. This made arithmetic and higher level mathematics much easier and more efficient.

2. Teaching of the concept of zero:

a. Teachers, family and friends typically teach the concept of zero in early elementary school through the use of counting, number lines and place value charts.

b. A tip for teaching the concept of zero is to use visual aids such as drawings, diagrams or objects to help illustrate the concept. It can also be helpful to use real-life examples to demonstrate the importance and usefulness of zero.

3. Everyday use of the concept of zero:

a. Adults use the concept of zero in everyday life when dealing with money, measurements, and other numerical calculations. For example, when making change, measuring ingredients for a recipe, or calculating interest on a loan.

b. **Teenagers** use the concept of zero in many of the same ways as adults, but also in areas such as computer programming, where zero is often used as a placeholder or to represent an absence of data. It is also used in video games, where scores often start at zero and increase as the player progresses.

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Find the length of the third side. If necessary, round to the nearest tenth.4 5

### Answers

**Answer:**

6.4

**Step-by-step explanation:**

use Pythagorean theorem

Solve questions 1 and 2 please

### Answers

The **geometric **mean of the **numbers **are 9.8 and 10.3

How to determine the geometric mean

Question 1

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

8 and 12

The geometric **mean **is calculated as

Mean = √8 * 12

Evaluate the **expression**

Mean = 9.8

Question 2

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

8.5 and 12.4

The **geometric **mean is calculated as

Mean = √8.5 * 12.4

**Evaluate **the expression

Mean = 10.3

Hence, the **geometric **means are 9.8 and 10.3

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There are 1152 learners at school. 11%are athletes and 3/5 of the learners will attend the sport meeting to cheer the athletes. The distance from the school to Athlone stadium is 16,7km. For how many learners must the bus service cater?

### Answers

The number of **leaners **that the **bus service** must cater for can be found to be 818 learners

How to find the learners ?

First, find the number of **athletes **that the bus service should cater for :

= 1, 152 x 11 %

= 126. 72

= 127 athletes

The number of **learners** that the **bus service **should cater for to attend the sport meeting is :

= 3 / 5 x 1152

= 691 learners

The total number of learners is :

= 691 + 127

= 818 learners

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Khufu is building a pyramid of height 135

cubits and with a square base of side

length 212 cubits. He knows that the

formula for the volume of a pyramid of

height ŉ and base length b ishb².

What is the volume of the rock needed to

build it (assuming it is solid)?

How much rock will have been used when

the as-yet-unfinished pyramid reaches

half the height of the final pyramid?

You may assume that the top of the

unfinished pyramid is a flat platform

parallel to the base.

### Answers

The **volume** of the rock needed to build the pyramid is; 6,067,440 cubits³.

The **volume** of the rock used at half the intended height is; 3,033,720 cubits³.

What volume of rock is used for the pyramid?

As evident in the task content; the **volume** of the pyramid is given by;

V = hb²

Therefore; V = 135 × 212²

V = 6,067,440 cubits³.

Hence, the quantity of block in **volume** measures which will be needed to build the pyramid is; 6,067,440 cubits³.

Also, at **half** the **height** of the pyramid, the **volume** of rock used as required is;

V = 6,067,440 / 2

V = 3,033,720 cubits³.

Consequently, the quantity of block in **volume** measures which will be needed to build for the half-finished pyramid is; 3,033,720 cubits³.

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The function f(x) = 3x 10 represents the number of animals adopted at one shelter, where x is the number of months. the function g(x) = x 5 represents the number of animals adopted at another shelter, where x is the number of months. part a: what is (f g)(x)? show all necessary steps. (2 points) part b: evaluate (f g)(6). show your work. (2 points) part c: explain what your answer represents in terms of the scenario. (2 points)

### Answers

The answer to the **function **(f g)(x) = 3x - 5, (f g)(6) = 13 and (f g)(6) = 13 contradicts this situation and could be a sign of a computation error.

a) To find (f g)(x), we need to first find the composition of f and g, which is f(g(x)). **Substituting **g(x) into f(x), we get:

f(g(x)) = 3(g(x)) + 10

= 3(x - 5) + 10

(substituting g(x) = x - 5)

= 3x - 15 + 10

= 3x - 5

Therefore, (f g)(x) = 3x - 5.

b) To evaluate (f g)(6), we simply substitute x = 6 into the **expression** we found in part a:

(f g)(6) = 3(6) - 5

= 18 - 5

= 13

Therefore, (f g)(6) = 13.

c) (f g)(6) represents the number of animals adopted in total by both shelters after 6 months. The function (f g)(x) takes the number of months as **input**, and outputs the total number of animals adopted by both shelters up to that point. In this case, after 6 months, Shelter 1 has adopted 3(6) + 10 = 28 animals, while Shelter 2 has adopted (6) - 5 = 1 animal. Therefore, the total number of animals adopted by both shelters after 6 months is 28 + 1 = 29. The result we obtained in part b, (f g)(6) = 13, is not consistent with this scenario and may indicate a **calculation **error.

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What is the area of the polygon?

A=

square units

### Answers

**The area of the polygon = 21 square units**

**Area of an object.**

To calculate the **area** of an **object**, we need to identify the **shapes** that are **embedded** in the **image.**

From the image, I can get 2 **triangles**. Using the formula for finding the area of a triangle, we can find the area of the given polygon.

A = 1/2 x b x h

where b = base

h = height

triangle A has base of 6 units and height of 4units

triangle B has base of 6 units and height of 3 units

Area of A = 1/2 x 6 x 4 = 12 square unit

Area of B = 1/2 x 6 x 3 = 9 square unit

Total area will be area of triangle A + area of triangle B

= 12 + 9 = 21 square unit.

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Determine the voltage across resistorR2

given: R1=30Ω,R2=75Ω,andI1=0.5A

Someone help me please

### Answers

**Answer:**

15 V

**Step-by-step explanation:**

You want the **voltage** across **parallel resistor R2** when the **current** through **30 Ω parallel resistor R1** is **0.5 A**.

Ohm's Law

The voltage across resistor R1 is the product of its resistance and the current through it:

E = IR

E = (0.5A)(30Ω) = 15 V

Parallel circuits

All of the branches of a parallel circuit have the same voltage across them. **The voltage across R2 is 15 V**.

￼please help me with this!! I need this ASAP thank you!

### Answers

**Answer:**

I'm not sure but I think it's d)8

The water temperature

in the swimming pool

began at 79 degrees,

and it is rising 0. 8

degrees each hour. If

the water temperature is

84 degrees, then how

many hours have passed?

### Answers

**Answer:**

80

**Step-by-step explanation:**

Katie and her friend Alma went to their county's annual fair. Katie spent $39 on admission and 6 rides. Alma spent $51 for admission and 9 rides. Use this information to write a system of equations

### Answers

From the **given **conditions **system** of **equations** is

x + 6y &= 39

x + 9y &= 51

**What is expression ?**

Expressions can be evaluated or simplified using mathematical rules and operations to obtain a single value or a simpler form. Expressions are used in various mathematical contexts, including algebra, calculus, and geometry, to represent mathematical relationships and solve mathematical problems.

Let x be the cost of admission and y be the cost of a ride at the fair.

Katie's expenses can be represented by the equation:

39 = x + 6y

Alma's expenses can be represented by the equation:

51 = x + 9y

Thus, the system of equations is:

x + 6y &= 39

x + 9y &= 51

This system of equations can be used to find the values of x and y that satisfy both equations and give the cost of admission and rides at the fair.

Therefore, from the **given **conditions **system** of **equations** is

x + 6y &= 39

x + 9y &= 51

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a transverse wave has a wave function y = (0.2 m) sinn 2 π (2 m−1 ) x − (2 s−1 )t o where x and y are in meters and t is in seconds. this wave is

### Answers

The wave function for a** transverse wave** is given by y = (0.2 m) sin(2π(2 m-1)x - (2 s-1)t) where x and y are in meters and t is in seconds. This wave is a **sinusoidal wave. **

This can be seen from the **wave function **y = (0.2 m) sin (2π (2 m^-1) x - (2 s^-1) t). The **sine function** indicates that the wave is a sinusoidal wave. Additionally, the wave is a transverse wave, as indicated in the question, which means that the displacement of the particles in the medium is perpendicular to the direction of the wave's propagation. The amplitude of the wave is 0.2 m, the wavelength is 0.5 m (as indicated by the coefficient of x in the sine function), and the frequency is 2 s^-1 (as indicated by the coefficient of t in the sine function).

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Helppppp

The Medal of Honor, the highest award given to a member of the U.S. military, was awarded to 882 men who served between 1865 and 1918. That’s 855 more than b, the number of Buffalo Soldiers who received the Medal of Honor during this time period. Write and solve an equation to find b.

### Answers

By answering the above question, we may state that Thus, there **equation **were 27 Buffalo Soldiers awarded the Medal of Honor during this time.

What is equation?

A **mathematical **equation is a process that links two statements and indicates equality using the **equals sign **(=). A mathematical proposition that proves the equality of two **formulae **is known as an equation in algebra. For instance, the equal sign separates the numbers 3x + 5 and 14 in the equation 3x + 5 = 14. A mathematical formula may be used to understand the connection that exists between the two phrases that have been written on opposite **sides **of a letter. Most of the time, the logos and the particular software match. like in 2x - 4 = 2, for example.

the formula

882 = b + 855

The overall number of Medal of Honor winners (882) is calculated by adding the number of Buffalo Soldiers who got the award (b) to the number of other recipients (855).

By removing 855 from both sides of the equation, we may concentrate the solution for b on just one side:

882 - 855 = b

27 = b

Thus, there were 27 Buffalo Soldiers awarded the Medal of Honor during this time.

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how to do by solving this equation? Please help!

### Answers

The **equation **of the line of fit is given as follows:

y = x.

How to define a linear function?

The **slope-intercept** definition of a linear function is given as follows:

y = mx + b.

In which;

The slope m represents the rate of change.The intercept b represents the value of y when x = 0.

**Two points **on the graph of the function are given as follows:

(50, 50) and (75,75).

When x increases by 25, y also increases by 25, hence the **slope m** is given as follows:

m = 25/25 = 1.

Hence:

y = x + b.

When x = 50, y = 50, hence the **intercept b **is given as follows:

50 = 50 + b

b = 0.

Meaning that the **equation **is of:

y = 50x.

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state whether a dilation with the given scale factor is a reduction or an enlargement.

1.k= 3 2.k= 1/3 3.k= 5/4 4.k= 0.93

### Answers

**Answer: 1. Enlargement 2. Reduction 3. Enlargement 4. Reduction**

**Step-by-step explanation: If the scale factor is greater that one the dilation is an enlargement, but if the scale factor is greater than zero but less than one it is a reduction.**

Ann and Ben translate documents from German into English.

A set of documents that would take Ann 9 days would take Ben 11 days.

Ann starts to translate the documents.

After 3 days Ann and Ben both work on translating the documents.

It will take an extra 3 days and h hours to complete the work.

Work out h, giving your answer in the form h=.

(h can be left as a top heavy fraction or given in decimal form)

+

Total marks: 4

### Answers

The **time** **taken** by both Ann and Ben to complete the work will be 3 days and 3/10 hours, that means it will take 3 days and 7.2 hours to complete the work.

**Number** of days Ann takes to complete the work = 9 days

**Work** **done** in 1 day = 1/9

Work done in 3 days = 3/9

Works **remaining** after 3 days = 1- 3/9 = 6/9

Number of days Ben takes to complete the work = 11

Work done by Ben in 1 day = 1/11

**Total** **work** done by Ann and Ben in 1 day = 1/9 + 1/11 = 20/99

Remaining work = 3/9

Number of days to complete remaining work = 6/9 ÷ 20/99

= 6×99/9×20 = 594/180

= 33/10 = 3 [tex]\frac{3}{10}[/tex]

So it will take 3 days and 3/10 hours to **complete** the work

That means 3 days [tex]\frac{3}{10}[/tex]× 24 = 3 days and 7.2 hours to complete the work.

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Hey Can Someone Help me with this geometry? Thanks!

### Answers

**Answer: **

**1 - **x = 112°

**2 -** x = 23°

**3 - **x = 97°

**4 - **x = 78°

Robert picked 65 apples at a local orchard 26 of the apples were green what percent were NOT green

### Answers

**Answer:**

1

**Step-by-step explanation:**

40% percent were not

Which relationship have the same constant of proportionality between y and x as the equation 3y=27d

### Answers

A relationship have the same **constant** of proportionality between y and x as the **equation** 3y = 27x include the following: A, B, and C.

What is a proportional relationship?

In Mathematics, a **proportional relationship** is a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical expression:

y = kx

Where:

y represents the x-variable.x represents the y-variable.k is the constant of **proportionality**.

Next, we would determine the required **equation** as follows:

3y = 27x

y = 27x/3

y = 9x

By multiplying both sides of the **equation** by 2, we have:

y = 9x

2y = 18x

Constant of **proportionality**, k = 9/1 = 9.

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2) A coin is flipped 60 times, resulting in 12 heads and 48 tails. What is the experimental probability of

the coin landing on heads?

### Answers

the **experimental probability** of the coin landing on heads is 1/5 or 0.2, which can also be expressed as a percentage of 20%.

The experimental probability of an event is the ratio of the number of times the event occurs to the total number of **trials**.

In this case, the event is getting a head when flipping the coin, and it occurred 12 times out of 60 trials. So the experimental probability of getting a head is:

Experimental probability of getting a head = number of **heads** / total number of trials

Experimental probability of getting a head = 12 / 60

**Simplifying**, we get:

Experimental probability of getting a head = 1 / 5

So the experimental probability of the coin landing on heads is 1/5 or 0.2, which can also be expressed as a **percentage** of 20%.

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how to solve for examples of nonlevel annuities for which the payments for a pattern that is neither geometric nor arithmetic

### Answers

The formula that is used to **solve for examples** of **nonlevel annuities** for which the payments for a pattern that is neither geometric nor arithmetic is explained below .

In order to solve for the present value or future value of a non-level annuity with payments that do not follow a **geometric or arithmetic** pattern, We use the **formula **for the present value or future value of an annuity that has varying payments:

⇒ PV = P₁/(1+r)¹ + P₂/(1+r)² + ... + Pₙ/(1+r)ⁿ

and

⇒ FV = P₁×(1+r)ⁿ⁻¹/(1+r)¹ + P₂×(1+r)ⁿ⁻²/(1+r)² + ... + Pₙ/(1+r)ⁿ

Where : PV = **present value** of the annuity

FV = **future value **of the annuity

P₁, P₂, ..., Pₙ = the payments for **each period**

r = the **interest rate **per period

n = the number of periods .

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3. Decide whether each equation is true for all, one, or no values of x

a. 6x4 = -4+6x

b. 4x6 = 4x+3

c. -2x + 4 = -3x+4

### Answers

**Equatiοns **a and c are true fοr οne value οf x where equatiοn b is cannοt be determined.

What is equatiοn ?

An equatiοn is a **mathematical statement **that asserts the equality οf twο expressiοns. In οther wοrds, an equatiοn is a statement that twο quantities are equal tο each οther. Equatiοns are written using an equals sign (=) between twο **expressiοns**.

a. 6x⁴ = -4 + 6x

Tο determine if this equatiοn is true fοr all, οne, οr nο values οf x, we can simplify it:

6x⁴ = -4 + 6x

Rearranging the terms, we get:

6x⁴ - 6x + 4 =

This is a fοurth-degree pοlynοmial equatiοn. We can try tο factοr it οr use numerical methοds tο find its sοlutiοns. Hοwever, we can alsο οbserve that the degree οf the **pοlynοmial **is even, sο it must have at least οne real rοοt, which means that the equatiοn is true fοr at least οne value οf x.

b. 4x⁶ = 4x + 3

Again, we can simplify the equatiοn:

4x⁶ = 4x + 3

Rearranging the terms, we get:

4x⁶ - 4x - 3 = 0

This is a sixth-degree pοlynοmial equatiοn, which is mοre difficult tο sοlve than a fοurth-degree equatiοn. We can use numerical methοds οr try tο factοr it, but it is nοt clear if it has any real rοοts. Therefοre, we cannοt determine if the equatiοn is true fοr any value οf x.

c. -2x + 4 = -3x + 4

Simplifying the equatiοn:

-2x + 4 = -3x + 4

Adding 3x tο bοth sides, we get:

x + 4 = 4

Subtracting 4 frοm bοth sides, we get:

x = 0

Therefοre, the equatiοn is true fοr οne value οf x, namely x = 0.

Hence, equatiοns a and c are true fοr οne value οf x where equatiοn b is cannοt be determined .

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URGENT PLEASE HELP ME!!! 50 POINTS!!!

### Answers

**Answer:****5/13= sinΘ**** since tanΘ = 5/12 and tanΘ= Perp/base****perp = 5 ****base= 12****use pythagorean theorem**

H2= P2 + B2

H=13 by putting values.

now sinΘ= Perp/hypo

so sinΘ= 5/13

**Step-by-step explanation:**

complete this proof using the two column proof method

Given: Prove: 2 ≅ 3

(Hint: 7 statements with reasons will complete this proof)

### Answers

When writing a **two-column** proof, keep these things in mind:

Number each step.Start with the given information.Statements with the same **reason** can be combined into one step.

What is the proof about?

A two-column **geometric** proof consists of a list of assertions and the justifications for each one's truth. The statements are listed in a left-hand **column**, while the justifications for the statements are listed in a right-hand column.

The **data** is often summarized in a two-column format. The first **column** contains assertions that are true. In the second column, a justification for each statement's truth is listed.

An overview was given as the information is incomplete.

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Graph the hyperbola 4x2 – y2 – 6y = 25.

Be sure you have identified (by giving the coordinates) each of these:

1. Center

2. Vertices

3. Foci

4. Asymptotes

Show your work. Thanks!

### Answers

To graph the **hyperbola**, we can draw the **asymptotes **passing through the centre, as well as the centre, vertices, and foci. The hyperbola, which will approach but never touch the asymptotes, can then be sketched.

To graph the **hyperbola **4x2 - y2 - 6y = 25, we must first convert the equation to standard form by completing the **square**.

4x² – y² – 6y = 25

4x² – (y^2 + 6y) = 25

4x² – (y² + 6y + 9) = 25 + 4(9) (9)

4x² – (y + 3)² = 61

The **equation **is now in standard form:

(4x²/61) – ((y + 3)²/61) = 1

1. Center: The **hyperbola's **centre is (0, -3).

2. Vertices are located along the transverse axis, which in this case is the x-axis. The distance between the centre and each vertex is given by a, which is the **square root **of the positive term's denominator in the standard form **equation**. As a result, a = [tex]\sqrt{(61/4)[/tex]= [tex]\sqrt{(15.25)} = 3.9[/tex](rounded to one decimal place). As a result, the vertices are (-3.9, -3) and (3.9, -3).

3. Foci: The distance between the centre and each focus is given by c, which is the square root of the sum of the squares of a and b, where b is the square root of the negative term in the standard form **equation**. As a result, [tex]b = \sqrt{(61/-61)} = -1[/tex], and [tex]c = \sqrt{(15.25 - 1)} = \sqrt{(14.25)} = 3.8.[/tex](rounded to one decimal place). As a result, the foci are at (-3.8, -3) and (3.8, -3).

4. Asymptotes: The asymptotes' slopes are given by plus or minus b/a. As a result, the asymptote slopes are plus or minus (-1/3.9) = plus or minus 0.26. (rounded to two decimal places). The asymptotes are lines with these slopes that pass through the centre of the **hyperbola**. The asymptotes' **equations **are y = -3 +/- 0.26x.

To graph the **hyperbola**, we can draw the asymptotes passing through the centre and plot the centre, vertices, and foci. The hyperbola can then be sketched, which will approach but never touch the asymptotes.

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f (x) = x^2 - 4/ x - 2 if x is not equal to 2 and f (x) = 1 if x = 2. Let f be the function defined earlier. Which of the following statements about f are true? (a) f has a limit at x = 2 (b) f is continuous at x = 2 (c) f is differentiable at x = 2

### Answers

**Given:**

**f(x≤2)=4x+2**

**f(x>2)=3x+4**

**Function is continuous:**

**lim(x→2) =10 from either piece.**

**Each piecewise function is differentiable alone, but we see the f(x) is not differentiable because the derivatives are not the same:**

**(Depending on your calculus experience, you can use the definition of a derivative or the power rule to find the derivative)**

**For x≤2**

**f’=lim(h→0) {[4(x+h)+2]-[4x+2]}/h def. of derivative **

**f’=lim(h→0) (4x+4h+2-4x-2)/h expand and cancel terms**

**f’=lim(h→0) (4h/h) simplify**

**f’=4**

**or:**

**f’=4 power rule**

**For x>2**

**f’= lim(h→0){[3(x+h)+4]-[3x+4]}/h def. of derivative**

**f’=lim(h→0)(3x+3h+4-3x-4)/h expand and cancel terms**

**f’=lim(h→0)3h/h simplify**

**f’=3**

**or**

**f’=3 via power rule**

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