Mathematics College

## Answers

**Answer 1**

We can then draw a **smooth **curve through these three **points **to graph the **function**, as shown below.

What is equation?

In mathematics, an **equation **is a statement that asserts that two mathematical expressions are equal to each other. It usually contains one or more variables, which are placeholders for values that can vary. Equations are used to describe mathematical relationships and to solve problems. They can take many forms, such as **linear **equations, quadratic equations, polynomial equations, exponential equations, and so on. The specific form of an equation depends on the type of problem being solved and the mathematical relationships involved. Solving an equation typically involves finding the values of the variables that make the equation true. This is done by manipulating the equation using algebraic operations such as **addition**, subtraction, multiplication, division, and exponentiation. The goal is to isolate the variable on one side of the equation, so that its value can be determined.

Here,

To graph the quadratic function f(x) = -2x² - 4x + 1 using completing the square method, we need to follow these steps:

Step 1: Group the x² and x terms together and factor out the coefficient of x², which is -2 in this case:

f(x) = -2(x² + 2x) + 1

Step 2: To complete the square, we need to add and subtract the square of half of the coefficient of x, which is (2/2)² = 1:

f(x) = -2(x² + 2x + 1 - 1) + 1

Step 3: Simplify the expression inside the parentheses by factoring it as a perfect square:

f(x) = -2[(x + 1)² - 1] + 1

Step 4: Distribute the -2 and simplify:

f(x) = -2(x + 1)² + 3

Now, we can see that the function is in vertex form, where the vertex is at the point (-1, 3), and the coefficient of the squared term is negative, which means the parabola opens downward.

To graph the function, we can start by plotting the vertex at (-1, 3). Then, we can find two more points on the parabola by plugging in x-values to the equation and solving for y. For example, when x = -2, we get:

f(-2) = -2(-2 + 1)² + 3 = -5

So one point on the graph is (-2, -5). Similarly, when x = 0, we get:

f(0) = -2(0 + 1)² + 3 = 1

So another point on the graph is (0, 1).

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## Related Questions

Liam is setting up folding chairs for a meeting. If he arranges the chairs in 5 rows of the same length, he has 5 chairs left over. If he arranges the chairs in 3 rows of that same length, he has 23 left over. How many chairs does Liam have?

QUICK!!!

### Answers

Liam has 50 chairs:

5x9+5=50

3x9+23=50

**Let the row length = L**

**This implies that, since the number of total chairs is the same in each case, we have **

**5L + 3 = 3L + 21 subtract 3L, 3 from both sides**

**2L = 18 divide both sides by 2**

**L = 9 [ 9 chairs in a row ]**

**So......the total number of chairs = 5(9) + 3 = ****48**

**Step-by-step explanation**

help me please! I’ll give brainliest

### Answers

Answer: 5+2*3

Step-by-step explanation:

the formula multiplies 2 by 3 and then adds 5 to the result

need help on this giving 25 points plus brainliest.

### Answers

A **right angle** triangle also has two sides that are perpendicular to each other, which means that they meet at a right angle.

What is a right angle triangle?

A **right angle **triangle is a type of triangle that has one angle which measures exactly 90 degrees. This angle is referred to as the "right angle". The other two angles of a right angle triangle are acute angles, which means that they measure less than 90 degrees.

1) sin 30 = x/10

x = 10 sin 30

x = 5

2) sin 30 = x/30

x = 30 sin 30

x = 15

3) sin 30 = 12/x

x = 12/sin 30

= 6

4) sin 30 = 28/x

x = 28/sin 30

x = 14

5) Tan 30 = 4/x

x = 4/Tan30

= 7

6) Tan 30 = 11√3/x

x = 11√3/Tan 30

x = 11√3/1/√3

x = 11/√3 * √3/1

x = 11

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Which graph represents two functions that are increasing on all points across the domain that is common to both functions?

### Answers

Two **functions** that are increasing on all points across the **domain** that is common to both functions is represented by graph 1.

What are increasing functions?

Calculus uses a function's **derivative** to determine whether a function is increasing or decreasing at various intervals in a specific domain. Given a given function, y = F(x), the function is known as an **increasing function** if the value of y increases as the value of x increases, and the function is known as a decreasing function if the value of y decreases as the value of x increases.

From the given graphs we observe the following:

Graph 1:

For the domain of the two graphs the two functions increase continuously.

Graph 2:

For the domain of the two graphs the function in blue increases, but the function in the red decreases.

Graph 3:

For the domain of the two graphs the function in blue increases, but the function in the red decreases.

Graph 4:

For the domain of the two graphs the function in blue increases, but the function in the red decreases.

Hence, two functions that are increasing on all points across the domain that is common to both functions is represented by graph 1.

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The graph represents a relation where x represents the independent variable and y represents the dependent variable. A coordinate plane with points at negative 5 comma negative 4, negative 2 comma 0, 0 comma negative 1, 1 comma 3, and 4 comma 1. What is the domain of the relation? {−5, 0, 3, 4} {−4, −1, 0, 1, 3} {−5, −2, 0, 1, 4} {−5, −4, −2, −1, 0, 1, 3, 4}

### Answers

The **domain **of the **relation **will be {-5,-2,0,1,4} i.e. **C**.

What is a **domain **of a relation or function?

**Domain**

The domain of a function is the set of all potential values that qualify as **inputs **to a function. It may alternatively be described as the full **set **of values available for **independent **variables. The domain may be discovered if the fraction's **denominator **is not zero and the digit under the square root bracket is positive. (In the case of a fractional function).

**Range**

The range of a function is the set of all the outputs of a function, or, after replacing the domain, the whole set of all values imaginable as **dependent variable **outcomes.

Now,

as domain is the input/**independent variables **for a **function **or relation

and here given independent variable is** x** with values

-5,-2,0,1,4.

Then the **domain **will be set of these values.

Hence,

The **domain **of the **relation **will be **{-5,-2,0,1,4}** .

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Two students created paintings for their art class. The canvas size used for each is shown.

8 ft

12 ft

2 ft

3 ft

Which statement is true about the ratios comparing the width and length of each painting?

O The paintings' width to length ratios are equal because the ratio 12 to 8 is equal to the ratio 2 to 3.

O The paintings' width to length ratios are different because the ratio 3 to 2 is not equal to 12 to 8.

O The paintings' width to length ratios are different because the ratio 8 to 12 is not equal to 2 to 3.

O The paintings' width to length ratios are equal because the ratio 8 to 12 is equal to the ratio 2 to 3.

### Answers

The correct statement is that the painting **width** to **length** **ratio** are equal because the **ratio** 8to 12!is equal to the **ratio** 2 to 3

What is ratio?

A **ratio** is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two **ratios** are set equal to each other. For example, if there is 1 boy and 3 girls you could write the **ratio** as: 1 : 3 (for every one boy there are 3 girls) 1 / 4 are boys and 3 / 4 are girls.

The first canvass has a ratio **width** to **length** 8:12

= 2:3 = 2/3

The second canvass has a **ratio** **width** to **length** 2/3

Therefore the **ratio** of **width** to **length** of the paintings are equal

8/12 = 2/3

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In 2008, a country used 27% or 135 million tons of all the grain grown that year to produce ethanol. How much grain was grown in the country in 2008?

### Answers

From the **equation **formed below, we found that the total** amount **of grains that were grown in that country in 2008 is **500 **million tons.

**What is an equation?**

A mathematical **equation **is a formula that uses the equals sign (=) to represent the equality of two expressions. The expressions on each side of the equals sign are referred to as the "left-hand side" and "right-hand side," respectively, of the **equation**. Typically, we consider an **equation's **right side to be zero. As we can balance this by deducting the right-side expression from both sides' expressions, this won't reduce the generality.

Let the total **amount **of grains grown in the year 2008 in that country be **x**.

Then it is said that 27% of **x **was used for producing ethanol.

This **amount **is equal to 135 million tons

So we can write the above relation as the following **equation**.

27% of **x **= 135 million tons

0.27 * **x **= 135

From this **equation**, we can find the value of x.

**x **= 135 / 0.27 = **500 **million tons.

Therefore from the above **equation**, we found that the total** amount **of grains that were grown in that country in 2008 is **500 **million tons.

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Drag each tile to the table to multiply (6x - y) (2x - y + 2).

### Answers

Answers in the attachment

And the math below

6x * 2x = 12x^2

6x * -y = - 6xy

6x * 2 = 12x

-y * 2x = - 2xy

-y * -y = y^2

-y * 2 = -2y

The function y = 40x describes the distance Shu Ling has traveled in miles after x hours. Use the graph to estimate how many miles Shu Ling will travel in 12 hours.

### Answers

Answer: y = 40*12 = 480

Step-by-step explanation:

I need help asap pleaseeee

### Answers

**Answer: **B

**Step-by-step explanation:**

Based on the function, this is most likely a function of absolute value.

The domain is how far the "x" values go, so as the function shows, it is all real numbers.

As for the range, it's how far the "y" values go, which doesn't go below 0, but only increases, so you can express this as y >= 0

So, the answer is B.

**Answer:**

the answer for this question is b

THIS IS NOT MY WORK!

### Answers

The **solution **to the given **inequality **is x < 3.5. The **graph **that represents the solution is D.

Solving linear inequalities

From the question, we are to determine the **graph **that represents the **solution **to the given **inequality **

The given **inequality **is

5 + 8x < 3(2x + 4)

To determine the graph that represents the solution to the inequality, we will solve the inequality

Solving the inequality

5 + 8x < 3(2x + 4)

5 + 8x < 6x + 12

Collect like terms

8x - 6x < 12 - 5

2x < 7

Divide both sides by 2

x < 7/2

x < 3.5

Hence, the **solution **is x < 3.5.

The **graph **of the **inequality **is a **number line **from -10 to 10 with an open circle at 3.5 pointing towards the negative direction. That is, D.

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The pH scale measures the acidity of a liquid as a function of its hydrogen ion

(H+) concentration.

pH = -log[H+]

How does the H+concentration of a solution with a pH of 4 compare with

that with a pH of 2?

O A. The H+ concentration in a solution with a pH of 4 is 0.1 times that

of a solution with a pH of 2.

O B. The + concentration in a solution with a pH of 4 is 0.01 times

that of a solution with a pH of 2.

C. The H+ concentration in a solution with a pH of 4 is 100 times

that of a solution with a pH of 2.

D. The + concentration in a solution with a pH of 4 is 10 times that

of a solution with a pH of 2.

### Answers

Therefore, the H attention in a result with a pH of 4 is0.01 times that of a result with a **pH** of 2. So, the answer is B.

What's ph.?

pH is a measure of the acidity or stipulation(alkalinity) of a result. It's a logarithmic scale that** measures** the attention of hydrogen ions(H) in a result. The pH scale ranges from 0 to 14, where 0 is the most **acidic**, 7 is neutral, and 14 is the utmost introductory.

Given by the question.

According to the pH **formula**, pH = -log[H+], we can rearrange the equation to solve for [H+]:

[H+] = [tex]10^{(-ph)}[/tex]

Substituting pH = 4 and pH = 2 into this **equation**, we get:

[H+] at pH 4 = [tex]10^{(-4)}[/tex] = 0.0001 M

[H+] at pH 2 = [tex]10^{(-2)}[/tex] = 0.01 M

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**Answer:**

B

**Step-by-step explanation:**

The pH scale is a logarithmic scale that measures the concentration of hydrogen ions (H+) in a solution. A decrease of one pH unit corresponds to a tenfold increase in the concentration of hydrogen ions.Using the pH formula pH = -log[H+], we can solve for the concentration of hydrogen ions in a solution with a pH of 4 and a pH of 2.For a solution with a pH of 4:pH = -log[H+]

4 = -log[H+]

log[H+] = -4

[H+] = 10^-4 = 0.0001 moles/literFor a solution with a pH of 2:pH = -log[H+]

2 = -log[H+]

log[H+] = -2

[H+] = 10^-2 = 0.01 moles/literSo, the [H+] concentration of a solution with a pH of 4 is 100 times less than that of a solution with a pH of 2. In other words, a solution with a pH of 2 is 100 times more acidic than a solution with a pH of 4.

a principal amount P is invested at a rate of r*100% per year. After 2 years the amount of the investment becomes

### Answers

After two years, the **amount** of the investment becomes 1.02P.

**What is simple interest?**

Simple **Interest** (S.I.) is the used of calculating the interest amount for a particular principal amount of money at some rate of interest. Thus a person takes a loan of Rs. 5000, at a rate of 10 p.a. for 2years, the person's **interest** for two years is interest on the borrowed money.

Thus, The Simple Interest = (**Principal** × Rate × Time) / 100

A = P (1 + rt)

A=final amount

P=initial principal balance

r=annual interest rate

t=time (in years)

We are given that;

Principal amount= P

Rate = r*100%

=0.01r

Time = 2 year

Now,

A=P(1+rT)

=P(1+0.01*2)

=1.02P

Therefore, after the **interest** of given time answer will be 1.02P

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How many are there?

H+H+H+H+H+

### Answers

**Answer: There are 5 H's on the board but I do not see how this is a school question.**

**Step-by-step explanation:**

Answer: according to my calculations, there is exactly 2 H’s

(So easy)

Step-by-step explanation:

Can someone help with this? Thank you!

### Answers

The value of **x is 24 **and different **angles** of **hexagon **will be -

A = 160

B = 142

C = 120

D = 156

E = 31

F = 111

Describe angle.

An angle is a **geometric shape **that is defined as the amount of rotation that occurs between two straight lines or planes. Angles are measured in degrees, with 360° representing a full **circle**.

We need to apply the inverse tangent function to **determine **the **angle **at which the sun strikes the flagpole. We are aware that the triangle's adjacent side is 42 feet long and its **opposite side **is 25 feet tall (the height of the flagpole) (the length of the shadow).

Given the figure is **hexagon**,

the sum of **angles **of a **hexagon is 720,**

Upon adding the given **angles**,

mA = (7x-8)°

mB (4x+46)°

mC = (5x)

mD = (6x+12)°

mE = (x+7)°

mF = (5x-9)°

⇒ 7x - 8 + 4x + 46 + 5x + 6x + 12 + x + 7 + 5x - 9 = **720**

⇒ 28x + 48 = 720

**⇒ 28x = 672**

**⇒ x = 24**

Therefore, the **angles **will be -

A = 160

B = 142

C = 120

D = 156

E = 31

F = 111

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Charlie sees a rare bird in a tall tree. He estimates the height of the tree to be 125

feet and the angle of elevation to the bird from where he stands on the ground to

be 61°. The branches on the tree make it difficult for Charlie to watch the bird, so

he walks several steps backwards and further away from the tree to get a better

view. He now estimates his angle of elevation to be 47°. How many feet did Charlie

step back to gain a better view of the bird? Round your answer to the nearest

hundredth.

### Answers

Charlie **step back** 19.56 ft to gain a better **view **of the bird.

What is the angle of elevation?

The angle formed by the line of sight and the horizontal plane for an object above the **horizontal **is called **angle of elevation**.

Given that, Charlie sees a bird seating at a tree of 125 ft, at an **angle of elevation** 61°, he moves backs at now the **angle of elevation** is 47°

We need to find the **distance **he moved back,

Please refer to the image attached,

BC = AB / tan 61°

BC = 125 / 61°

BC = 69.28 ft

Similarly,

BD = AB / tan 47°

BD = 125 / tan 47°

BD = 116.56 ft

The **distance **Charlie moved back is DC,

DC = BD - BC

DC = 116.56 ft - 69.28 ft

DC = 19.56 ft

Hence, Charlie **step back** 19.56 ft to gain a better **view **of the bird.

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A student wrote the matrix below to represent the solution to a system of equations.

1

0

0 0 0 0

0 5 -2

0 3 0

-1

### Answers

The **system** has a **unique** **solution**, which is (-2, 1, 1), the correct answer is** (a) no solution**.

**What is the system of equations?**

The **solution set **to a **system of equations **will be the **coordinates** of the ordered pair(s) that satisfy all **equations** in the **system**. In other words, those **values** of x and y will make the **equations** true.

The given matrix represents the row echelon form of the augmented matrix of a system of equations. We can use this matrix to determine the solution(s) of the system by back-substitution.

Looking at the row echelon form of the augmented matrix, we can write the corresponding system of equations:

x - 4y - 3z = -7

y - 2z = -1

5z = 5

The last equation tells us that z = 1. Substituting z = 1 into the second equation, we get y - 2(1) = -1, which gives y = 1. Substituting z = 1 and y = 1 into the first equation, we get x - 4(1) - 3(1) = -7, which gives x = -2.

So, the system has a unique solution, which is (-2, 1, 1).

Therefore, the correct answer is (a) (-2, 0, -1) is not the solution to the system, and neither is (b) an infinite number of solutions or (d) (5, -2, -1).

Hence, the **system** has a **unique** **solution**, which is (-2, 1, 1), the correct answer is** (a) no solution**.

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Use substitution to solve the system of equations. Show your work (30 Points!!!)

### Answers

Using the **substitution** method, the **values** of x and y are 1 and 10 respectively

What is simultaneous equation?

There are times when you come across two or more **unknown quantities** and two or more equations relating to them. These are called **Simultaneous Equations**.

The given **equations** are

4x + y = 14

y=8+2x

By **substitution**, put y=8+2x in **equation** 1

4x+8+2x=14

6x=14-8

6x=6

x=1

Recall that y=8+2(1)

y=8+2

Therefore the **value** of y=10

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The **values **are given as;

x = 1

y = 10

How to determine the value

From the information given, we have that;

4x + y = 14

y = 8 + 2x

**Substitute** the value of y in** equation** (1), we get;

4x + 8 + 2x = 14

collect the like terms

4x + 2x = 14 - 8

Add or subtract the like terms

6x = 6

Make 'x' the subject

x = 1

Substitute the value

y = 8 + 2x

y = 8 + 2(1)

expand the bracket

y = 8 + 2

Add the values

y = 10

Hence, the **values **are 1 and 10

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A ladder leans against the wall of a building. The ladder measures

48 inches and forms an angle of 70 with the ground. How far from

the ground, in inches, is the top of the ladder? How far from the

wall, in inches, is the base of the ladder? Round to two decimal

places as needed.

### Answers

We can use trigonometry to solve this problem. Let's first define some variables:

Let x be the distance, in inches, from the ground to the base of the ladder.

Let y be the distance, in inches, from the wall to the base of the ladder.

Let z be the distance, in inches, from the ground to the top of the ladder.

We can then use the trigonometric ratios of the angle formed by the ladder with the ground (70 degrees) to write two equations relating these variables:

tan(70) = z / x

cos(70) = y / x

We can solve for x and z in terms of y using these equations:

x = y / cos(70)

z = x * tan(70) = y * tan(70) / cos(70)

Plugging in the values, we get:

x = y / cos(70) = 48 / cos(70) ≈ 15.04 inches (rounded to two decimal places)

z = x * tan(70) = y * tan(70) / cos(70) ≈ 47.93 inches (rounded to two decimal places)

Therefore, the top of the ladder is about 47.93 inches from the ground, and the base of the ladder is about 15.04 inches from the wall.

Suppose f is a linear function and f ( x ) varies at a constant rate of change of 2.5 with respect to x . Suppose we know that f ( 1.6 ) = 2.6.

a. Suppose x varies from x=1.6 to x = 1.8

i. How much did x change by?

Δx=

ii. What is the corresponding change in f(x)?

Δf(x)=

iii. What is the value f(1.8)?

f(1.8)=

b.Write a function formula for f(x). (Hint: use the same thinking as in (a), but now imagine x varying from x=1.6 to any new value of x)

f(x)=

### Answers

If **x **changes from 1.6 to 1.8 is Δx = 0.2, the **corresponding **change in f(x) with respect to x is 2.5 at Δx = 0.2 and the **value** of f(1.8) = 3.66.

As per the question given,

Suppose we have a **linear **function f(x) that **changes **with respect to X at a **constant **rate of 2.5. If you know that f(1.6) **equals** 2.6, you can use this information to **answer **interesting questions.

**First**, if x changes from 1.6 to 1.8, how much has x changed? The answer is Δx = 0.2 because 1.8 **minus **1.6 equals 0.2.

**Second**, what is the corresponding change in f(x)? The rate of change of f(x) with **respect **to x is 2.5, so at Δx = 0.2, f(x) changes by Δf(x) = 2.5 times 0.2, which equals 0.5.

**Finally**, what is the value of f(1.8)? You can use the linear **function **equation to find this value. Using the **fact **that f(1,6) equals 2.6 and the rate of change of f(x) with respect to x is 2.5, we can derive the equation f(x) = 2.5x - 1.14. replacing x = 1.8 to this **equation **you get f(1.8) = 3.66.

Then you have it! By knowing the **rate **of change of a linear function and **learning **basic **algebra**, you can answer these **questions** and better understand how the function behaves for **different **values of x.

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Please help! 75 point math question, Thank you!

### Answers

$2.62 expected value because that is the mean, it is not a fair game because most of the time you lose money

**Answer:**

41.0

**Step-by-step explanation:**

3483

como incubar de mi

Consider the geometric sequence whose nth term is. This sequence can also be defined recursively with and . If you need to compute the first 8 terms, would you prefer to use the explicit definition or the recursive definition? Which definition would you prefer to use to calculate the 100 th term?

### Answers

The first 8 **terms** can be computed by **recursive form**. The 100th term can be computed by **explicit definition**.

What definition is prefered for a case of geometric sequence?

**Geometric sequences** are sequences generated by exponential **functions**, whose forms are described below:

**Recursive form**

aₙ₊₁ = r · aₙ

**Explicit definition**

a(t) = a₁ · rⁿ⁻¹

Where:

r - Common raten - Index of the n-th element of the series.a₁ - First element of the geometric series.

The **recursive form** can be recommended to compute a sucesive **set** of **elements** of a geometric sequence, whose first element is the first element of the **series**, whereas the **explicit definition** can be used to determine any element of the geometric sequence.

The following conclusions are listed below:

The first 8 **terms** can be found by **recursive form**.The 100th term can be found by **explicit definition**.

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What language is generated by the following grammars? In each case justify your answer. a) s 0S1 | 01 b) s + S S | -SS | a c) S S (S)S | d) s a Sb S | b S a S | e e) s a | S+S | SS S*|(S)

### Answers

The **language **is generated by the following grammars are

a) s 0S1 | 01 is **string**

b) s + S S | -SS | a is **arithmetic expressions**

c) S S (S)S | is **string**

d) s a Sb S | b S a S | e is **string**

e) s a | S+S | SS S*|(S) is **expression**

Grammar is a set of rules that define the structure of a **language**. It provides a set of guidelines for constructing sentences that are **grammatically **correct.

a) The grammar s -> 0S1 | 01 generates a **language **consisting of all strings that begin with 0, end with 1, and have any number of 0's and 1's in between.

The **justification **for this is that the grammar provides rules for constructing such strings by either appending a 0 and a 1 to an existing string or by directly generating the string "01".

Thus, all **strings **in the language can be generated by the given grammar.

b) The **grammar **s -> +SS | -SS | a generates a language consisting of all valid arithmetic expressions using the symbols +, -, and a single variable "a".

The justification for this is that the grammar provides rules for constructing such expressions by **recursively **applying the rules to generate sub-expressions.

Thus, all valid arithmetic expressions can be generated by the given grammar.

c) The grammar S -> SS(S)S | ε generates a language consisting of all strings of balanced **parentheses**.

The justification for this is that the grammar provides rules for constructing such strings by recursively generating balanced strings and surrounding them with parentheses.

Thus, all **balanced **strings of parentheses can be generated by the given grammar.

d) The grammar s -> aSbS | bSaS | e generates a language consisting of all strings that contain an **equal **number of "a"s and "b"s.

The justification for this is that the grammar provides rules for constructing such strings by recursively generating pairs of "a" and "b" surrounded by two new **substrings**.

Thus, all strings containing an equal number of "a"s and "b"s can be generated by the given grammar.

e) The grammar s -> a | S+S | SS | S* | (S) generates a language consisting of regular expressions over the **alphabet **{a, +, *, (, )}.

The justification for this is that the grammar provides rules for constructing regular expressions using the provided operators and the variable S.

Thus, all regular **expressions **over the given alphabet can be generated by the given grammar.

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Judy took $30 with her to spend on popcorn and drinks for herself and her friends at the movie theater. The price for each bag of popcorn was $5. The price of each drink was $3. This can be represented by the

equation 5x + 3y = 30

(a) Sketch the graph that represents the situation. Use one axis to represent the number of bags of popcorn and the other axis to represent the number of drinks.

Points

Concept Addressed

(b) Explain your graph and what the intercepts

### Answers

The **graph** of the **linear function** is attached below and the x and y- intercepts are 6 and 10 respectively.

What is the linear function to model this problem

The** graph** of a** linear function** is a straight line. A linear function is a function that has a constant rate of change, which means that its graph will be a straight line.

The general form of a** linear function** is y = mx + b, where m is the slope or the rate of change of the function, and b is the y-intercept, which is the point where the line intersects the y-axis. The slope represents the steepness of the line and can be positive, negative, or zero. If the slope is positive, the line will slant upwards from left to right, and if the slope is negative, the line will slant downwards from left to right. If the slope is zero, the line will be horizontal.

a.

The graph of the** linear function** 5x + 3y = 30 is attached below.

b.

From the graph below, the y - intercept is 10 and x - intercept 6. This shows that graph touches the y - axis at 10 and the x - axis at 6.

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A rectangular garden is 9 ft longer than it is wide. Its area is 400 ft2. What are its dimensions?

### Answers

The **dimensions** of the rectangular garden are 16.5 ft by 7.5 ft.

What is area?

**Area** is a quantity that measures the size of a **two-dimensional surface** or shape. It is expressed in square units such as square centimetres (cm2), square metres (m2) or square kilometres (km2). Area is used to describe the size of a garden, a house, a room, a city and much more. It can also be used to calculate the amount of material required for a project, such as paint, carpet or tiles. Knowing the area of a shape can help to calculate costs, and to make sure that enough materials are ordered.

To solve this problem, we need to use the formula A = lw, where A is the area, l is the length, and w is the width. Since the area is 400 ft2, we can set up the equation 400 = l(l - 9). Solving for l, we get l = 16.5 ft. This means the width of the rectangular garden is 7.5 ft. Thus, the dimensions of the rectangular garden are 16.5 ft by 7.5 ft.

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Which of the following is true about the relationship between the population mean and the mean of the sampling distribution? a. population mean = mean of the sampling distribution b. population mean mean of the sampling distribution c. population mean mean of the sampling distribution

### Answers

a. **population mean **= mean of the sampling distribution is true about the relationship between the population mean and the mean of the sampling distribution

The overall** population** of a category with the characteristics that a researcher is interested in is referred to as the populace in statistics. While the population's average and the sample's means are equivalent, the population's standard deviation, sample size, and population density all affect the sample's **standard error.** The average of the sample distribution serves as a reasonable approximation of the population means.

The **average mean **of the sample distribution is therefore equal to the population mean. Option b is erroneous as it implies that the population's mean is higher than the mean of the **sampling distribution**. The population mean is not less than the mean of the sampling distribution, as implied by Option C, which is also incorrect.

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use logarithmic differentiation

### Answers

**Answer:**

step 1: Multiply both sides by 7.436564x.

2+40.171074= 7.436564

**Step-by-step explanation:**

Kendra has $1,757 in an account that earns 12% interest compounded annually.

To the nearest cent, how much interest will she earn in 1 year?

### Answers

The amount of the interest in 1 year that Kendra will earn from an account of $1,757 that earns 12% **interest compounded** annually is $210.84.

What is compound interest?

**Compound interest** is the interest system that accumulates interest and uses the accumulated interest and the principal to compute the interest for subsequent periods.

In the first period, the amount of the interest is the same for both compound and **simple interests**.

However, for subsequent periods, the amounts for compound interest and simple interest will differ slightly.

The principal = $1,757

The annual interest rate = 12%

The investment period = 1 year

The **compound interest **= $210.84 ($1,757 x 12%)

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Calculate the length of the diagonal for the given rectangular prism. Round to the nearest tenth Length = 10cm Width = 4cm Height = 10cm

### Answers

the length of the **diagonal** is approximately10.8cm.

The find the length of the diagonal of the rectangular prism, we can use the **Pythagorean theorem**, which states that for a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse (the longest side).

In this case, the diagonal is the hypotenuse of a right triangle whose legs are the length and width of the rectangular prism. So we can calculate it as

diagonal = sqrt([tex]lenght^{2}[/tex] +[tex]width^{2}[/tex] + [tex]height^{2}[/tex])

= sqrt([tex]10^{2}[/tex] + [tex]4^{2}[/tex]+ [tex]10^{2}[/tex])

= sqrt(116)

≈ 10.8 cm

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A type of fish for your aquarium costs $4 each. You can spend no more than $24. How many of these fish can you buy? Write an inequality to model the problem. Then solve the inequality to find the number of fish.

### Answers

The **maximum number** of **fish** that can be **bought** is 6, and the answer is option C) 4f ≤ 24.

**What is inequality?**

An **inequality** is a mathematical statement that shows a **relationship** between two values, indicating that one is **less than** or **greater than** the other, or less than or equal to, greater than or **equal** to, or not equal to the other. **Inequalities** are represented using symbols such as <, >, ≤, ≥, or ≠.

In the given problem, the inequality that models the problem is:

4f ≤ 24

This is because f represents the number of fish that can be bought, and each fish costs $4. The total amount that can be spent is no more than $24. Multiplying the cost per fish ($4) by the number of fish bought (f), the cost of the fish can be represented as 4f. Therefore, 4f should be less than or equal to $24.

To solve this inequality for f, we can divide both sides by 4:

4f ≤ 24

f ≤ 6

Therefore, the **maximum number** of **fish** that can be **bought** is 6, and the answer is option C) 4f ≤ 24.

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