2. For each of the following functions, determine the average rate of change over the specified interval.
From x=4 to x=10

Answer:

The quadratic equation is in the standard form: ax^2 + bx + c = 0. To use the quadratic formula to find the solutions, we need to identify the values of a, b, and c in the equation:

a is the coefficient of the x^2 term, which is -3 in this case.

b is the coefficient of the x term, which is -5 in this case.

c is the constant term, which is 9 in this case.

Therefore, the values of a, b, and c in the quadratic equation -3x^2 - 5x + 9 = 0 are:

a = -3

b = -5

c = 9

The linear function that models this problem is y = 25x + 25 and the cost for 6 hours is $175

To solve this problem, we have to find the slope and y-intercept of the line.

m = y2 - y1 / x2 - x1

Taking two points from the line;

A = (2, 75)

B = (5, 150)

Substituting the values into the formula;

m = 150 - 75 / 5 - 2

m = 25

Using the slope;

y = mx + c

150 = 25(5) + c

150 = 125 + c

c = 150 - 125

c = 25

The equation of line is ;

y = 25x + 25

The cost for 6 hours will be

y = 25(6) + 25

y = 175

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Answer:

A

Step-by-step explanation:

To find the transformation, we need to take a point and the point to which it has been translated.

Let's take A and A'.

⇒ A = (2, -2)

⇒ A' = (-2, 1)

Change in x : -4 (shifted 4 units left)

Change in y : 3 (shifted 3 units up)

This means to obtain A'B'C'D' from ABCD, the graph has to be :

shifted 4 units left and 3 units up

Answer:

Step-by-step explanation:

The equation of a circle with a center at (4,−9) and a radius of 5 is

(x−4)^2+(y+9)^2=25. The correct Option C.

A circle is a set of all points which are equally spaced from a fixed point in a plane. The fixed point is called the center of the circle. The distance between the center and any point on the circumference is called the radius of the circle.

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

(x−h)^2+(y−k)^2=r^2 .

where h = 4

k = -9

r = 5

The equation of the circle

(x−4)^2+(y−(-9))^2 = 5^2

The equation of the circle

(x−4)^2+(y+9)^2 = 25

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Answer:

Step-by-step explanation:

Which rational function models the time, y, in hours, that it takes the train to travel between the two cities at an average speed of x miles per hour?  

✔ y = 160/x

What does the number 160 represent in the rational function that models the situation?

✔distance, in miles, between the cities

amount of fuel, in gallons, used by the train

time, in minutes, needed to travel between the cities  

Answer: NOTE Pictures Are not in order! Make sure you check page number.

Step-by-step explanation:

On page 6 the answers are ”C” and ”5lbs”

On page 7 the answers are “C”, “$ 418”, “ The cat”, and then “ $960”.

On page 8 the answers are : < Just copy paste this below.>

“Based on the calculation, they should purchase the second television as the cost is cheaper.

Based on the information given in the question, the cost of the first television will be:

= $330 + ($14 × 8)

= $330 + $102

= $442

The cost of the second television will be:

= $369 + ($9 × 8)

= $369 + $72

= $441

Therefore, the second television is cheaper.”

                   ^    

<Just copy paste this. >           [Then check. “A,B,C,”.]

On page 9 the answer is “ D “.

Step-by-step explanation:

Improper fractions have a numerator that is larger than the denominator

1 = 6/6

 so  1  1/6  = 7/6

Answer:

30 feet^2

Step-by-step explanation:

eeeeet eeeees what eeeeet eeeees

Using geometry, we can find that the length of the line segment AC as per the distance between two points is= 18 ft.

What are line segments?

In geometry, a line segment's borders are determined by two distinct points on the line. Any portion of a line that connects two places is referred to as a line segment.

A segment has an endpoint, whereas a line doesn't.

What are some examples of line segments in the real world?

a scale, a ruler, a stick, and a boundary line.

Thus, the length of the line segment AC is = 18ft.

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Answer:

\(\frac{3}{2}\)

Step-by-step explanation:

Did you mean \(log_{25}\) 125, so that your base is 25?  If so then all you do is ask  yourself "25 to what power is 125?" since \(25^{\frac{3}{2} }\) = 125, then the  \(log_{25}\) 125 = 3/2

Answer:

Step-by-step explanation:

In the Intermediate Phase of mathematics education, one topic that demonstrates a hierarchical and logical progression in patterns, functions, and algebra is the concept of "Linear Equations."

The topic of Linear Equations in the Intermediate Phase builds upon the foundation laid in earlier grades and serves as a stepping stone towards more advanced algebraic concepts. Here is an overview of the topic sequence and content area spread for Linear Equations:

Introduction to Variables and Expressions:

Students are introduced to the concept of variables and expressions, learning to represent unknown quantities using letters or symbols. They understand the difference between constants and variables and learn to evaluate expressions.

Solving One-Step Equations:

Students learn how to solve simple one-step equations involving addition, subtraction, multiplication, and division. They develop the skills to isolate the variable and find its value.

Solving Two-Step Equations:

Building upon the previous knowledge, students progress to solving two-step equations. They learn to perform multiple operations to isolate the variable and find its value.

Writing and Graphing Linear Equations:

Students explore the relationship between variables and learn to write linear equations in slope-intercept form (y = mx + b). They understand the meaning of slope and y-intercept and how they relate to the graph of a line.

Systems of Linear Equations:

Students are introduced to the concept of systems of linear equations, where multiple equations are solved simultaneously. They learn various methods such as substitution, elimination, and graphing to find the solution to the system.

Word Problems and Applications:

Students apply their understanding of linear equations to solve real-life word problems and situations. They learn to translate verbal descriptions into algebraic equations and solve them to find the unknown quantities.

The content area spread for Linear Equations includes concepts such as variables, expressions, equations, operations, graphing, slope, y-intercept, systems, and real-world applications. The progression from simple one-step equations to more complex systems of equations reflects a logical sequence that builds upon prior knowledge and skills.

By following this hierarchical progression, students develop a solid foundation in algebraic thinking and problem-solving skills. They learn to apply mathematical concepts and procedures in a systematic and logical manner, paving the way for further exploration of patterns, functions, and advanced algebraic topics in later phases of mathematics education.

The amount of tea that is left after the student has drank is; ⁴/₅d mL

In word problems on fractions, what we do is that we will solve different types of problems on multiplication of fractional numbers and division of fractional numbers.

For example;

4/7 of a number is 84. Find the number.

Since 4/7 of a number = 84

The, the Number = 84 × 7/4

Number = 147

Now, in this question, we are given;

Quantity of tea in cup = d ml

Amount of tea a student takes out to drink = ¹/₅ of the quantity in the cup.

Thus;

Amount of tea left = d - ¹/₅d

Amount of tea left = ⁴/₅d

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Answer:

(0,34)

Step-by-step explanation:

For each rise of 14 in the x direction, this graph rises by -8 in the y direction. This means that, when x is 0, and the graph intersects the y axis, the y value will be 50-8-8=34. Therefore, the y intercept of this line is (0,34). Hope this helps!

Answer:

The answer is (0,34)

If the data were transformed and points with the coordinates

(x, log(y)) were plotted then the points (1, 1.079), (2, 1.919), (3, 2.763) were plotted.

To plot the points with the coordinates (x, log(y)), we need to take the logarithm of the y-values.

Let's calculate the logarithms for the given data:

x     y            log(y)

1     12          log(12) = 1.079

2     83        log(83) = 1.919

3     580      log(580) = 2.763

Therefore, the points that would be plotted are (1, 1.079), (2, 1.919), (3, 2.763)

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The simplified boolean expression for F5 is F5 = P OR (NOT R). In this expression, P represents one boolean variable, and (NOT R) represents the negation of another boolean variable R. The OR operator combines the two variables, resulting in the final boolean expression F5.

To simplify the boolean expression F5 = (P AND Q) OR (R AND NOT P), we can apply Boolean algebra rules to reduce it to its simplest form.

Step 1: Apply De Morgan's laws

NOT (R AND NOT P) can be simplified as (NOT R) OR P.

Step 2: Distributive property

F5 = (P AND Q) OR ((NOT R) OR P)

Step 3: Apply associative property

F5 = (P AND Q) OR (P OR (NOT R))

Step 4: Apply absorption law

F5 = P OR (P OR (NOT R))

Step 5: Apply idempotent law

F5 = P OR (NOT R)

By simplifying the expression, we have reduced it to its simplest form while preserving its logical meaning. The simplified expression can be used to analyze and evaluate logical circuits or systems where the variables P and R are used.

Remember, boolean algebra allows us to manipulate and simplify complex logical expressions by applying various rules and properties. These simplifications help in understanding and analyzing logical operations in digital systems and circuits.

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The question probable may be:

Consider the boolean variables P, Q, and R. Determine the boolean expression for the variable F5 using the given variables:

F5 = (P AND Q) OR (R AND NOT P)

Using the boolean variables P, Q, and R, can you simplify the expression F5 to its simplest form by applying Boolean algebra rules?

Step-by-step explanation:

a) No

because

1/x + 1/y

= y + x/xy

b) No

because

x³ = x × x × x

c) Yes

because

3x - 12 = 0

3(x - 4) = 0

x - 4 = 0

d) No

because

-2(x - 3)

= -2x + 6

e) Yes

because

x² + 4x² = 5x²

f) Yes

because

x(y + z)

= xy + zx

Answer:

x = 1

Step-by-step explanation:

\(5(10^x+2)=60 \\ \\ 10^x+2= \frac{60}{5} \\ \\ 10^x+2= 12 \\ \\ {10}^{x} = 12 - 2\\ \\ {10}^{x} = {10} \\ \\ {10}^{x} = {10}^{1} \\ \\ x = 1\)

Let w = x^3

Square both sides to find that w^2 = (x^3)^2 = x^(3*2) = x^6

In short: w^2 = x^6

The given expression 2+x^3-3x^6 turns into 2+w-3w^2 and rearranges into -3w^2+w+2

Set this equal to zero and use the quadratic formula. We'll plug in

So,

\(w = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\w = \frac{-1\pm\sqrt{(1)^2-4(-3)(2)}}{2(-3)}\\\\w = \frac{-1\pm\sqrt{25}}{-6}\\\\w = \frac{-1\pm5}{-6}\\\\w = \frac{-1+5}{-6} \ \text{ or } \ w = \frac{-1-5}{-6}\\\\w = \frac{4}{-6} \ \text{ or } \ w = \frac{-6}{-6}\\\\w = -\frac{2}{3} \ \text{ or } \ w = 1\\\\\)

If w = -2/3, then that rearranges to the following

w = -2/3

3w = -2

3w+2 = 0

This makes (3w+2) a factor of -3w^2+w+2

If w = 1, then it rearranges to w-1 = 0.

This makes (w-1) a factor of -3w^2+w+2

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

To summarize the previous section, we found the factors of -3w^2+w+2 were:

It leads to (3w+2)(w-1)

We must stick a negative out front because the leading coefficient is negative.

Therefore, -3w^2+w+2 = -(3w+2)(w-1)

You can use the FOIL rule to confirm.

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

Recall we made w = x^3

Let's replace each w with x^3

-(3w+2)(w-1)

-(3x^3+2)(x^3-1)

This tells us that 2+x^3-3x^6 factors to -(3x^3+2)(x^3-1)

The next task is to factor x^3-1 using the difference of cubes factoring rule.

a^3 - b^3 = (a-b)(a^2 + ab + b^2)

x^3 - 1^3 = (x-1)(x^2 + x*1 + 1^2)

x^3 - 1 = (x-1)(x^2 + x + 1)

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

So,

2+x^3-3x^6

-3x^6 + x^3 + 2

-(3x^3+2)(x^3-1)

-(3x^3+2)(x-1)(x^2 + x + 1)

-(2 + 3x^3)(-(1-x))(1 + x + x^2)

(2 + 3x^3)(1 - x)(1 + x + x^2)

Take careful notice that x-1 turned into -(1-x) in the 3rd step. The negative out front for -(1-x) cancels out with the original negative out front.

Answer:

  False

Step-by-step explanation:

You want to know if it is true that ...

To solve an equation you undo what is being done to the variable. You go about that by applying the order of operations backwards.

If an equation has a single instance of the variable, and if the operations done to the variable are ones that have inverse operations, then it is often true that the equation can be solved by applying inverse operations in the reverse order that operations are performed.

This is obviously true in trivial cases:

  ax + b = c . . . . b is added to the product

This is solved by subtracting b (the inverse of adding b), then multiplying the product by the inverse of the coefficient of the variable.

Strictly speaking, we are not executing the order of operations backwards. We are applying inverse operations in the reverse order that operations were originally done.

Throughout the solution process, the order of operations remains applicable. "Backwards" refers only to using the original sequence of operations to identify an appropriate order in which to apply inverse operations.

Often, the solution process begins by simplifying the equation and/or eliminating fractions. These processes are executed in accordance with the order of operations and the properties of equality. There is nothing backwards about it.

The ratio 81:108 can be simplified to 9:13. To find the simplest form of the ratio, you can divide both numbers by the greatest common factor. The greatest common factor of 81 and 108 is 9, so you can divide both numbers by 9 to get the simplified ratio of 9:13.

The selling price of the antique through an online auction website is $680.

Given that,

Kyle sold an antique through an online auction website.

The website host charges Kyle $15, plus 2.5% of the final selling price of the antique.

After selling the antique, Kyle had to pay the website host $32.

Let x be the selling price of the antique.

We get an equation,

15 + (2.5% × x) = 32

15 + 0.025x = 32

0.025x = 17

x = $680

Hence the selling price is $680.

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Answer:

47

Step-by-step explanation:

x is one of the three angles of a triangle. The other two are given as 95 and 38.

Together all three have to make 180 degrees.

x + 95 + 38 = 180

x + 133 = 180

x = 180 - 133

x = 47

Answer:

47

Step-by-step explanation:

as we know that sum of a triangle is always=180°

so x+38°+95°=180

so x+133=180

and x=180-133

so x= 47

The correct statement that F: "I agree. The ratios of consecutive x-values are constant, and the y-values are increasing at a constant rate."

To determine whether the function shown in the table is exponential, we need to analyze the relationship between the x-values and y-values.

The table shows x-values that are increasing by a power of 2 (1, 2, 4, 8, 16, 32, 64) and y-values that are increasing by a constant rate of 1 (0, 1, 2, 3, 4, 5, 6).

The constant ratio between consecutive x-values indicates exponential growth or decay.

In this case, the function is growing exponentially because the y-values are increasing at a constant rate.

Based on this information, we can conclude that the function is exponential.

Therefore, we agree with statement F: "I agree. The ratios of consecutive x-values are constant, and the y-values are increasing at a constant rate."

The other statements are incorrect because they do not accurately describe the relationship between the x-values and y-values in an exponential function.

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Answer:

10%

Step-by-step explanation:

It can be inferred that H0: ? = 15, Ha: ? = 15 doesn't comply because both hypothesis use an equal sign.

From the information, H0: p = 0.5, Ha: p > 0.6 doesn't comply because the two hypothesis use different numbers.

Furthermore, H0: ? = 121, Ha: ? < 121 shows that the pair complies. Also, H0: ? = 123, Ha: ? = 125 doesn't comply as they use the equal sign and different numbers.

Lastly, H0: p? = 0.2, Ha: p? ? 0.2 doesn't comply because p is not a population characteristic.

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The required angle is 24.5°.

Given is a right triangle with perpendicular side 16 and the base = 35 we need to find an acute angle in it,

To find the acute angle in a right triangle given the lengths of the perpendicular side and the base, you can use the tangent function.

The tangent of an angle is defined as the ratio of the length of the perpendicular side to the length of the base side.

In this case, the perpendicular side is 16 and the base is 35.

Let's denote the acute angle as θ.

Using the tangent function, we can set up the equation:

tan(θ) = perpendicular side / base

tan(θ) = 16 / 35

To find the value of θ, we can take the inverse tangent of both sides:

θ = tan⁻¹(16 / 35)

θ = 24.5°

Hence the required angle is 24.5°.

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Answer:

Step-by-step explanation:

D

a value of x cannot give two or more values of y.

Answer:C

Step-by-step explanation:because 19 has two y

Angle L's in the triangle LMN is,

⇒ L = 72.54°

We have to given that;

A triangle LMN is shown in figure.

And, The sides are,

Since, We know that;

A triangle is a three-sided polygon with three vertices, three angles that add up to 180 degrees, and three sides.

Since, We know that;

⇒ sin L = Opposite / Hypotenuse

And, cos L = Base / Hypotenuse

Two rays after combined into an angle have a single terminal. And, latter is known as the vertex of the angle, and the rays are known as its sides, occasionally as its legs, and occasionally as its arms.

Now, We can use trigonometry formula to find the value of angle l we get;

⇒ sin L = Opposite / Hypotenuse

⇒ sin L = LM / LN

⇒ sin L = 18/60

Taking arc sin both side, we get;

⇒ L = 72.54°

Therefore, The value of measure of angle L is triangle LMN is,

⇒ L = 72.54°

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