Find the average rate of change of f(x) = x² - 4x + 1 from x=3 to x = 5.
Simplify your answer as much as possible.

The identity is cos²∅+sin²∅= 1.

This is the boundary or contour length of a 2D geometric shape.

Depending on their size, multiple shapes may have the same circumference. For example, imagine a triangle made up of wires of length L.

The same wire can be used to create a square if all sides are the same length.

The length covered by the perimeter of the shape is called the perimeter. Therefore, the units of circumference are the same as the units of length.

As we can say, the surroundings are one-dimensinal. As a result, you can measure in meters, kilometers, millimeters, etc.

Inches, feet, yards, and miles are other globally recognized units of circumference measurement.

According to our question-

cos(90∘−θ)=sinθ

sin(90∘−θ)=cosθ

cos²∅+sin²∅= 1.

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Triangle ABC can be mapped onto triangle XYZ by a composition of transformations represented by (x,y) → (x+3,y-6) and followed by a dilation with center (0,0) and scale factor 1/3 represented by (x,y) → (x/3,y/3).

Dilation is a geometric transformation that involves changing the size of an object while maintaining its shape and proportions. In dilation, every point of the original figure is expanded or contracted away from a fixed point called the center of dilation.

The ratio of the distance between the center of dilation and each point of the original figure is the same for all corresponding points in the dilated figure. This ratio is called the scale factor, and it determines the degree of dilation.

If the scale factor is greater than 1, the figure is enlarged or dilated, while if the scale factor is less than 1, the figure is reduced or contracted. If the scale factor is negative, the figure is reflected across the center of dilation before being dilated.

Triangle ABC can be mapped onto triangle XYZ by a composition of transformations represented by (x,y) → (x+3,y-6) and followed by a dilation with center (0,0) and scale factor 1/3 represented by (x,y) → (x/3,y/3).

Triangle ABC is similar to Triangle XYZ.

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Margo will pay approximately $45.50 in interest on the loan.

What is interest?

Borrowers must pay lenders simple interest as a fee in exchange for a loan.

First, we need to calculate the interest that Margo will pay on the loan.

Interest = Principal x Rate x Time

where:

Principal = $1300

Rate = 3% per year (or 0.03)

Time = 14/12 years (since there are 12 months in a year)

Plugging in the values, we get:

Interest = $1300 x 0.03 x (14/12)

Interest ≈ $45.50

Therefore, Margo will pay approximately $45.50 in interest on the loan.

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The lower bound of c - d is 42.66 and the upper bound of c - d is 42.7.

Given that

To find the upper and lower bounds of c - d, we first need to find the maximum and minimum possible values of c and d.

For c rounded to 1 decimal place, the actual value of c could be between 47.25 and 47.34.

This is because when we round to 1 decimal place, we take the first decimal place and round up or down depending on the second decimal place.

Similarly, we have

d = 4.59 to 4.64

So, we have

c - d = 47.25  - 4.59 to 47.34 - 4.64

Evaluate

c - d = 42.66 to 42.7

Therefore, the lower bound of c - d is 42.66 and the upper bound of c - d is 42.7.

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Answer:
Morning shifts -

Evening shifts -

Step-by-step explanation:

Morning

mean: (8+15+16+18+18+20+21+21+21+56) / 10 = 21.4

median: 10 numbers + 1 / 2 =5.5th number is median. therefore that is 19 (numbers have to be in chronological order to start counting)

mode: (most common number) 21

Evening

mean: (3+11+12+14+22+22+30+33+34+47) / 10 = 22.8
median: 10 numbers + 1 / 2 =5.5th number is median. therefore that is 22
mode: 22

The statement "we number the vertices counterclockwise 0, 1, ..., n − 1, and number the axes of symmetry counterclockwise, 0, 1, ..., n − 1" is likely referring to a specific mathematical context, such as a polygon or a regular solid.

In this context, "numbering the vertices counterclockwise" means that the vertices (or corners) of the shape are given labels or numbers in a particular order, moving counterclockwise around the shape. For example, in a triangle, the vertices might be labeled as 0, 1, and 2, in counterclockwise order around the triangle.

Similarly, "numbering the axes of symmetry counterclockwise" means that the axes of symmetry (or lines of reflection) in the shape are also given labels or numbers in a particular order, moving counterclockwise around the shape. For example, in a regular pentagon, there are 5 axes of symmetry that can be labeled as 0, 1, 2, 3, and 4 in counterclockwise order around the pentagon.

By establishing this numbering convention, mathematicians and scientists can more easily communicate about the properties and characteristics of the shape or object, and can refer to specific vertices or axes of symmetry in a standardized way.

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Answer:

y = 1/2 x

Step-by-step explanation:

y = mx + b

We need to find the m (slope) and the b (y-intercept)

The slope is the change in y over the change in x.  Use to points on the line to find the slope.

(4,2) and (6,3)

[tex]\frac{3-2}{6-4}[/tex] = [tex]\frac{1}{2}[/tex]

The slope (m) is 1/2.

Use the slope and one of the points to calculate b

I will use the point (4,2)

y = mx + b

2 = 1/2(4) +b

2 = 2 +b  Subtract 2 from both sides

0 = b

The y-intercept (b) is 0

y = _m + _

y = 1/2x + 0  

y = 1/2 x

Helping in the name of Jesus.

Answer:

What are the roots of 2x2-6x+18=0​

Step-by-step explanation:

The value of the principal investment (rounded to the nearest hundredth) is $9,000.39.

Interest rate refers to the percentage amount charged or paid on a loan or investment, usually expressed as an annual percentage rate (APR). It represents the cost of borrowing or the return earned on an investment.

The equation you provided represents the compound interest formula:

A = P[tex](1 + r/n)^(n*t)[/tex]

Given that the annual interest rate is 6% and is compounded quarterly (i.e., n = 4), and the amount in the account after 15 years is $16,497.06, These values can be plugged into the equation to find P.

A = $16,497.06

r = 0.06 (6% expressed as a decimal)

n = 4

t = 15

$16,497.06 = P[tex](1 + 0.06/4)^(4*15)[/tex]

Dividing both sides by [tex](1 + 0.06/4)^(4*15)[/tex] to isolate P, we get:

P = $16,497.06 / [tex](1 + 0.06/4)^(4*15)[/tex]

Plugging in the values and solving for P using a calculator, we get:

P ≈ $9,000.39

So, the value of the principal investment (rounded to the nearest hundredth) is $9,000.39.

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The correct statements for inequality are:

0.087 m³ < 8700 cm³

8 m³ = 8,000,000 cm³

An inequality is a mathematical statement that compares two expressions or values and indicates that one is greater than, less than, or not equal to the other. Inequalities use symbols such as "<" (less than), ">" (greater than), "≤" (less than or equal to), "≥" (greater than or equal to), and "≠" (not equal to) to show the relationship between the two expressions.

Here,

The statement "300 mm³ = 0.03 cm³" is incorrect because 300 mm³ is equal to 0.3 cm³ (since 1 cm = 10 mm).

The statement "7.2 cm³ > 720 mm³" is incorrect because 1 cm³ is equal to 1000 mm³, so 7.2 cm³ is equal to 7200 mm³. Therefore, 7.2 cm³ is less than 720 mm³.

Note: When comparing volumes, it is important to make sure that the units are consistent (e.g. converting all volumes to the same unit of measurement) before making any comparisons.

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Answer:

D

Step-by-step explanation:

A=LW

A=L(36)

Divide by 36

L=A/36

(L=X)

:)

So the price that would maximize revenue is $2.75.

An equation is a mathematical statement that asserts the equality of two expressions. It typically contains one or more variables, which represent unknown values that can be solved for using algebraic methods. An equation can be represented symbolically using mathematical notation, such as using letters, numbers, and mathematical symbols like +, -, x, /, and =. Equations can be used to model and solve a wide variety of problems in mathematics, science, engineering, and other fields.

Here,

We can start by finding the linear equation that relates price and monthly sales. Let's use the point-slope form of a linear equation:

y - y1 = m(x - x1)

where y is the monthly sales, x is the price, m is the slope, and (x1, y1) is a point on the line.

Using the two points we have:

(10, 19000) and (7, 25000)

We can find the slope:

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

m = (25000 - 19000) / (7 - 10)

m = 2000

Now we have:

y - 19000 = 2000(x - 10)

Simplifying:

y = 2000x - 11000

This equation represents the relationship between price and monthly sales.

To maximize revenue, we need to find the price that gives the highest value for the product of price and monthly sales. In other words, we need to find the maximum value of the function:

R(x) = x * y = x(2000x - 11000)

Expanding and simplifying:

R(x) = 2000x² - 11000x

To find the maximum, we need to take the derivative and set it equal to zero:

R'(x) = 4000x - 11000 = 0

Solving for x:

4000x = 11000

x = 2.75

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Sο, if οur estimatiοns are reasοnably accurate, the likelihοοd that a chοsen student is a wοman οr a business majοr is rοughly 0.6.

It describes the likelihοοd οf a specific οccurrence οr grοup οf οutcοmes in a given envirοnment and is a key idea in mathematics and statistics. Prοbability is typically stated as a number between 0 and 1, with 1 denοting a certain event and 0 denοting an impοssibility. By dividing the number οf faοurable οutcοmesv by the tοtal number οf pοssible οutcοmes, οne can determine the prοbability οf an οccurring.

We must sum the prοbabilities οf twο events—the prοbability that the student is a wοman and the prοbability that the student is a business majοr—in οrder tο get the likelihοοd that a certain student is a wοman οr a business majοr.

P(A and B) equals P(A)*P (B)

Using this presumptiοn, we may calculate the likelihοοd that a chοsen student will majοr in business and be a wοman as fοllοws:

P (wοman) = 0.2 * P (business majοr) = 0.1 (10%)

After that, we can enter this value in the previοus equatiοn:

P (wοmen majοring in business) = 0.5 + 0.2 - 0.1 = 0.6 (οr 60%)

Sο, if οur estimatiοns are reasοnably accurate, the likelihοοd that a chοsen student is a wοman οr a business majοr is rοughly 0.6.

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Complete Question:

A college has an undergraduate enrollment of 3500. Of these, 860 are business majors, 1800 are women and 425 are women business majors.

a. Are the events "selecting a woman student" and "selecting a business major" mutually exclusive? Explain.

b. If a college newspaper conducts a poll and selects students at random to answer a survey, find the probability that a selected student is a woman or a business major.

Answer: $0.85 per tomato

Step-by-step explanation:

To find the unit rate, we need to determine the cost of one tomato. We can do this by dividing the total cost of the four tomatoes by the number of tomatoes:

$3.40 ÷ 4 = $0.85

Therefore, the cost of one tomato is $0.85. The unit rate can also be expressed as a fraction, which is:

$0.85 per tomato, or 85 cents per tomato

Answer:

The Unit rate is 0.85

Step-by-step explanation:

You divide it by 4

the statement that "as x decreases, f(x) increases" is not true for an exponential decay function.

An exponent, also known as a power or index, is a mathematical operation that indicates how many times a number (the base) is multiplied by itself.

When graphed, an exponential decay function shows that

as x increases

f(x) approaches 0

but as x decreases

f(x) also approaches 0.

The function decreases from left to right, meaning that as x increases, f(x) decreases towards 0. However, it does not increase as x decreases.

In general, an exponential decay function can be represented as f(x) = a ×e⁻ᵇˣ, where a and b are constants. As x gets larger and larger, the exponential term e⁻ᵇˣ approaches 0, which causes the overall function value f(x) to approach 0 as well. As x gets smaller and smaller, the exponential term e⁻ᵇˣ becomes larger, but the function value f(x) still approaches 0.

Therefore, the statement that "as x decreases, f(x) increases" is not true for an exponential decay function.

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The length of the circle is approximately 9.237 cm.

What are the diagonals of a rhombus?
A rhombus is a quadrilateral with all sides of equal length. It also has the following properties regarding its diagonals:

The diagonals bisect each other at 90 degrees.

The diagonals are perpendicular to each other.

The diagonals have the same length.

The diagonals divide the rhombus into four congruent triangles.

In a rhombus with a side of 8 cm and an angle of 60 degrees, the diagonals are equal in length.

The length of the diagonals can be found using the formula:

[tex]$D_1 = \frac{2a}{\sqrt{3}}$[/tex]

where a is the length of the side. Substituting the value of a as 8 cm, we get:

[tex]$D_1 = \frac{2(8)}{\sqrt{3}} \approx 9.237 \text{ cm}$[/tex]

The length of the circle inscribed in the rhombus is equal to the length of the diagonals.

Therefore, the length of the circle is approximately 9.237 cm.

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The calculated resolved part of vector p (6i-3j+9k) in the direction of a is 2i + 2j - k.

Given that

p = (6i-3j+9k)

a =  (2i+2j-k)​

To find the resolved part of the vector p in the direction of a, we can use the following formula:

r = (p · a) / |a|

where · denotes the dot product and |a| denotes the magnitude of vector a.

First, we need to calculate the magnitude of vector a:

|a| = √(2² + 2² + (-1)²)

|a| = √(4 + 4 + 1)

|a| = √9

|a| = 3

Next, we need to calculate the dot product p · a:

p · a = (6i - 3j + 9k) · (2i + 2j - k)

= 12i + 12j - 6k - 6i - 6j + 3k

= 6i + 6j - 3k

Finally, we can calculate the resolved part of p in the direction of a:

r = (p · a) / |a|

= (6i + 6j - 3k) / 3

= 2i + 2j - k

Therefore, the resolved part of vector p in the direction of a is 2i + 2j - k.

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The diameter of the circle is equal to 2.00 mm rounded to 3 significant figures.

The diameter of a circle is a straight line that passes through the center of the circle and whose endpoints are on the circumference of the circle. We can use the formula for the circumference of a circle to calculate for the diameter.

circumference of is derived using πD or 2πr, where D is the diameter, π is 22/7, and r the radius.

The given circle have 6.3 mm circumference, so;

6.3 mm = 22/7 × D

D = (6.3 mm × 7)22 {cross multiplication}

D = 44.1 mm/22

D = 2.0045

Therefore, the diameter of the circle is equal to 2.00 mm rounded to 3 significant figures.

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Answer:

A) Range = 2

B) 30 Students

C) 8 Marks

Step-by-step explanation:

A) Range = Biggest - Smallest

Frequency = How many students achieved that many marks

Range = 9-7 = 2

C) Mean = Times the Marks by the Frequencies, add them together, then divide by total frequency (No. of students)

6x5=30

7x4=28

8x7=56

9x10=90

10x4=40

30+28+56+90+40=244

Total Frequency = 30

244/30=8.13

Round to nearest mark = 8

Hope this helped

The vertex of the parabola is at (1, -9). The x-intercepts are (-2, 0) and (4, 0), which are the points where the parabola crosses the x-axis. The y-intercept is (0, -8), which is the point where the parabola crosses the y-axis.

What is vertex ?

To find the vertex and the x- and y-intercepts of the equation y = x² - 2x - 8, we can start by completing the square to put the equation in vertex form:

y = x² - 2x - 8

y + 8 = x² - 2x

y + 8 + 1 = x² - 2x + 1

y + 9 = (x - 1)²

So, the vertex of the parabola is at (1, -9). The x-coordinate of the vertex is found by taking the opposite of the coefficient of x in the equation (which is -2) and dividing by two times the coefficient of x (which is 1), giving x = -(-2) / (2*1) = 1. The y-coordinate is found by substituting x = 1 into the equation.

What are the intercepts?

To find the x-intercepts, we set y = 0 and solve for x:

0 = x² - 2x - 8

0 = (x - 4)(x + 2)

So, the x-intercepts are at (-2, 0) and (4, 0).

To find the y-intercept, we set x = 0 and solve for y:

y = 0² - 2(0) - 8

y = -8

So, the y-intercept is at (0, -8).

To graph the parabola, we can use the vertex and intercepts we found.

The vertex is (1, -9), which is the lowest point on the parabola. The x-intercepts are (-2, 0) and (4, 0), which are the points where the parabola crosses the x-axis. The y-intercept is (0, -8), which is the point where the parabola crosses the y-axis.

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Bus 14 is therefore the more reliable bus as a result of its lower IQR.

The dot plot, which depicts the frequency with which each observation appears in the data set, is the first thing we take into account.

The difference between the data set's third and first quartiles is represented by the interquartile range, which is the next factor we take into account.

Because it ignores outliers, the interquartile range is a better indicator of a spread than a data set's range.

We have the following for classes of 15 students:

Since the first seven students make up the first half, the first quartile is represented by the fourth dot, which represents the first half's median.

The final seven students make up the second half, thus the eleventh dot—the first half's median—is in the first quartile.

So, the following are the Bus 14 quartiles:

Q1 = 12.

Q3 = 18.

The IQR is therefore of:

IQR = Q3 - Q1 = 18 - 12 = 6.

Hence, the IQR is:

Q1 = 9.

Q3 = 16.

The IQR is therefore of:

IQR = Q3 - Q1 = 16 - 9 = 7.

Therefore, bus 14 is therefore the more reliable bus as a result of its lower IQR.

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Correct question:

The line plots represent data collected on the travel times to school from two groups of 15 students. A horizontal line starting at 0, with tick, marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 10,16,20, and 28. There are two dots above 8 and 14. There are three dots above 18. There are four dots above 12. The graph is titled Bus 14 Travel Times. A horizontal line starting at 0, with tick, marks every two units up to 28. The line is labeled Minutes Traveled. There is one dot above 8, 9,18, 20, and 22. There are two dots above 6, 10, 12,14, and 16. The graph is titled Bus 18 Travel Times. Compare the data and use the correct measure of variability to determine which bus is the most consistent. Explain your answer. Bus 14, with an IQR of 6 Bus 18, with an IQR of 7 Bus 14, with a range of 6 Bus 18, with a range of 7

The result is, [tex]\dfrac{dy}{dx} = x \times ln\dfrac{x}{(x+3)} + 2x\times ln\dfrac{x}{(x+3)} - \dfrac{(3x^2)}{(x+3)^2}[/tex].

The derivative of a function is a measure of how much the function changes as its input (often time or space) changes. More specifically, the derivative is the instantaneous rate of change of the function at a particular point.

To find the derivative of y = x^2ln(x/(x+3)), we use the product rule and the chain rule.

y = x^2 * ln(x/(x+3))

[tex]y' = 2x \times ln\dfrac{x}{(x+3)} + x^2 \times \dfrac{d}{dx}ln\dfrac{x}{(x+3)}[/tex]

To find the derivative of ln(x/(x+3)), we use the quotient rule:

[tex]\dfrac{d}{dx} ln\dfrac{x}{(x+3)} = \dfrac{1}{\dfrac{x}{(x+3)}} \times \dfrac{d}{dx}\dfrac{x}{(x+3)} - \dfrac{1}{\dfrac{x+3}{x}} \times \dfrac{d}{dx}\dfrac{(x+3)}{x}[/tex]

= [(x+3)/(x^2)] - [x/(x+3)^2]

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

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

Substituting this back into the original equation, we get:

y' = 2x * ln(x/(x+3)) + x^2 * [3(x+1)/(x^2(x+3)^2)]

= 2x * ln(x/(x+3)) + 3(x+1)/(x+3)^2

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Answer: 2%

Step-by-step explanation: i looked it up

a) The probability that the sample mean breaking strength for a random sample of 40 rivets is between 9,850 and 10,150 is equal to 0.8903.

b) No, because n should be greater than 30 in order to apply the central limit theorem. So, right choice for answer is option (b).

The normal distribution forms a bell shaped curve. The parameter of normal distribution is [tex]\mu[/tex] and [tex]\sigma ^2 [/tex]. It is a continuous distribution. Now, Mean value of breaking strength of a rivet = 9,950

Standard deviations = 502

Let's X be variable denotes breaking strength of a rivet.

a) Now, a random sample is choosen from X such that, Sample size, n = 40. By using central limit theorem, sample mean follows approximately normal distribution with mean 9950 and standard deviations = 502/√40 = 79.3732 that is [tex]\bar X \: \tilde \: \: N( 9950, ( 79.3732)²)[/tex]

[tex]P( 9,850 < \bar X < 10,150 ) = P( \bar X < 10,150) -

P( \bar X < 9,850) \\ [/tex]

= [tex] P(\frac{\bar X - \mu }{\sigma} < \frac{10,150 - 9950}{79.3732} )- P(\frac{\bar X - \mu }{\sigma} < \frac{10,150 - 9850}{79.3732}) \\ [/tex]

= P( Z < 2.52 ) - P( Z < -1.26)

= 0.9941 - 0.1038

= 0.8903

b) If sample size,n is 15 instead of 40, then the probability requested in part (a) will not be calculated from the provide information. Because, for applying central limit theorem we have sample size must be grater than 30. So, option(b) is correct.

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Answer:

y = 1/2 x + 3

Step-by-step explanation:

y = mx + b

You need to find the slope (m) and the y-intercept (b) to write the equation

The slope is the change in y over the change in x.

(0,3) (4,5)  The y values are 5 and 3.  The x values are 4 and 0.  You find the change by subtracting.

[tex]\frac{5-3}{4-0}[/tex] = [tex]\frac{2}{4}[/tex] = [tex]\frac{1}{2}[/tex]

The slope (m) is 1/2.

The y-intercept is the point (0,b).  They give us this with the point (0,3)  The y-intercept (b) is 3

y = 1/2 x + 3

Helping in the name of Jesus.

Answer:

Step-by-step explanation:

y=x23 so the dt is moved to perticlulate the dy

Answer:

SE π ≈ 3.14 and round your answer to the nearest hundredth

Thus, for 3.14, the number in the hundredth place is 4. To round it, we will look at the third digit after the decimal, which is 0. We know any number lower than 5 is rounded down to zero, so we will round down and the rounded number will be 3.14. Conclusion: 3.14 rounded to the nearest hundredth is 3.14 itself.

Answer: 3.14

Step-by-step explanation: It’s not possible to round the answer so it would just be 3.14

Answer:

B. Parallel

Step-by-step explanation:

Let S1 be the slope of the line passing through the points

(9, - 5) and (5, - 2).

S1 = (- 2 -(- 5))/(5 - 9)

   = (- 2 + 5)/(- 4)

   = 3/(- 4)

S1 = - 3/4

Let S2 be the slope of the line passing through the points

(- 4, - 2) and (- 8, 1).

S2 = (1 -(- 2))/(-8 -(- 4))

     = (1 + 2)/(- 8 + 4)

     = 3/(-4)

S2 = - 3/4

Since,

the slope of two lines are same.

They are parallel.

Answer:120 degrees

Step-by-step explanation:

2pi = 360 degrees

pi = 180 degrees

2pi/3 = 120 degrees