The 10% condition when constructing a confidence interval for a population proportion states that the sample size should be less than 10% of the population size.
This is expressed mathematically as follows:
n/N < 0.10
Where n is the sample size and N is the population size. For example, if the population size is 500, then we must ensure that the sample size is less than 50 (500 x 0.10 = 50). This is because if the sample size is larger than 10% of the population size, then the confidence interval will not be reliable. In other words, the results of the confidence interval may be inaccurate and biased.
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How do you graph a quadratic function in vertex form with a fraction ??
A quadratic function in vertex form with a fraction can be graphed by first converting the fraction to decimal form.
How do you graph a quadratic function in vertex form?A quadratic function is a function of the form f(x) = ax² + bx + c, where a, b, and c are numbers with a not equal to zero. It should be noted that the graph of a quadratic function is a curve called a parabola.
Then, using the vertex and the slope of the parabola, plot the vertex point and at least two more points on either side of the vertex. Finally, use these points to graph the parabola. The equation of a quadratic in vertex form is y = a(x-h)^2 + k where (h,k) is the vertex.
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Which inequality is represented by this graph?
Answer:
0 > x
Step-by-step explanation:
The open circle indicates that x cannot be equal to zero, which rules out the last two options. Since the line is going to the right (towards numbers greater than 0), x must be greater than zero, which means that the first option is correct.
Answer:
x > 0
Step-by-step explanation:
we see the blue line. it covers everything that is larger than 0.
the dot on 0 is empty, which indicates that the point (here 0) itself is not included.
therefore, the second answer option (x > 0) is correct.
A scale drawn on the map shows that 1 inch represents 20 miles. If two cities
are 2.5 inches apart on the map, what is the distance between them in real
life?
OA. 40 miles
B. 30 miles
O C. 50 miles
OD. 20 miles
SUBMIT
Using the arithmetic operation of multiplication, two cities having a distance of 2.5 inches on map have an actual distance of 50 miles.
What is arithmetic operation?
A subject of mathematics known as arithmetic operations deals with the study and use of numbers in all other branches of mathematics. Basic operations including addition, subtraction, multiplication, and division are included.
The value of 1 inch on scale = 20 miles.
The distance between two cities on map = 2.5 inches
To find the distance between two cities in actual use the arithmetic operation of multiplication.
Let the original distance value in miles be x.
The equation for distance will be -
x = 2.5 × 20
Solve the equation -
x = 2.5 × 20
x = 50
Therefore, the actual distance in miles is 50 miles.
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a real estate office handles an apartment complex with 80 units. when the rent is $360 per month, all 80 units are occupied. when the rent is $405, however, the average number of occupied units drops to 77. assume that the relationship between the monthly rent p and the demand x is linear. (the term demand refers to the number of occupied units.) (a) write a linear equation expressing x in terms of p. x
the relationship between the monthly rent (p) and the demand (x) is p(x) = -15x + 1560.
A real estate office handles an apartment complex with 80 units. When the rent per unit is $360 per month, all 80 units are occupied. However, when the rent is $405per month, the average number of occupied units drops to 77. Assume that the relationship between the monthly rent (p) and the demand (x) is linear.
Since the equation will be linear, we only need two coordinates to write the equation.
They are given, we can assign the values x = no.of units and y = rent:
x1 = 80, y1 = 360
and
x2 = 77, y2 = 405
Find the slope: m = (y2-y1)/(x2-x1)
m = (405-360)÷(77-80) = 45÷(-3)= -15
The slope m= -15
A) Write the equation of the line giving the demand (x) in terms of the rent (p)
Use the point/slope equation.
Using the point/slope formula: y - y1 = m(x - x1)
y - 360= -15(x - 80)
y - 360 = -15x + 1200
y = -15x + 1200 + 360
y = -15x + 1560
or
p(x) = -15x + 1560
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Please help
Fill in each answer box so that the resulting statement is true.
Let y = πx + π. For what value of x is y = π^3?
The value of the variable is; x = π³ - π/π
What are algebraic expressions?Algebraic expressions are described as mathematical expressions that are comprised of variables, factors, constants, terms and coefficients.
These algebraic expressions are also comprised of arithmetic operations, such are; Division
Multiplication, Subtraction, Floor division, Addition, Bracket, Parentheses
From the given information, we have that;
y = π³
y = πx + π
Substitute the values, we have;
π³ = πx + π
Collect like terms
π³ - π = πx
Divide both sides by the coefficient of x
x = π³ - π/π
Thus, the value is x = π³ - π/π
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Alexandria walks 171 meters due west, then 480 meters due south and finally 427 meters due east.
How far is Alexandria from her starting point?
meters
Answer: To find the distance Alexandria is from her starting point, you can use the Pythagorean theorem.
The theorem states that in a right triangle, the square of the hypotenuse (the distance between the starting point and the final point) is equal to the sum of the squares of the other two sides (the distance traveled north or south and the distance traveled east or west).
In this case, the distance traveled north or south is 480 meters (the distance traveled due south) and the distance traveled east or west is 427 meters (the distance traveled due east) - 171 meters (the distance traveled due west) = 256 meters.
So, using the Pythagorean theorem:
Distance = √(480^2 + 256^2)
Distance = √(230400 + 65536)
Distance = √(295936)
Distance = 544.98 meters
So Alexandria is 544.98 meters away from her starting point.
Step-by-step explanation:
Given sin � = 21 29 sinA= 29 21 and that angle � A is in Quadrant I, find the exact value of cot � cotA in simplest radical form using a rational denominator.
The exact value of cot A is .20/21
What is rational denominator?The denominator means moving the radical term square root or cube root to the numerator such that a denominator is a whole number.
We have been given that sin( A ) = 21/29 and angle A is in quadrant 1. We are asked to find the exact value of cot (A) in simplest radical form.
We know that sine relates opposite side of right triangle with hypotenuse.
Since = opposite
hypotenuse
This means that opposite side is 21 units and hypotenuse is 29 units.
We know that cotangent relates adjacent side of right triangle with adjacent side.
Cot = Adjacent
opposite
Now we will find adjacent side using Pythagoras theorem as:
Adjacet² = hypotenuse² - opposite²
Adjacent² = 29² - 21²
Adjacent² = 841 - 441
Adjacent² =400
Let us take positive square root on both sides:
√Adjacent² = √400
Adjacent = 20
Therefore, adjacent side of angle A is 20 units.
Cot (A) = 20/21
The exact value of cot A is .20/21
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The movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer.
Three polynomials are factored below, but some coefficients and constants are missing. All of the missing values of a, b, c, and
d are integers.
1. x² - 8x + 15 = (ax + b)(cx + d)
2. 2x³ - 8x² - 24x = 2x(ax + b)(cx + d)
3.6x² + 14x + 4 = (ax + b)(cx + d)
Fill in the table with the missing values of a, b, c, and d.
b
a
1. 1
2. 1
3.
Submit
Pass
-5
1
C
1
2
d
Question ID: 504360
-6
Save an
150
d = 4
b = -3. c = 1
a = 3 and d = 1
What are three polynomials factorized below?
To get the missing values in the table, we will factorize the given expression and compare the factored expression with the expression containing the missing constants.
1) For the expression x²+2x-8, on factorizing we have;
x²+2x-8
= (x²+4x)-(2x-8)
Factoring out the common terms from both parenthesis;
= x(x+4)-2(x+4)
= (1x+4)(1x-2)
= (1x-2)(1x+4)
Comparing the resulting expression with (ax+b)(cx+d)
a = 1, b = -2, c = 1 and d = 4
2) For the expression 2x³+2x²-24x
Factoring out the common term we will have;
= 2x(x²+x-12)
= 2x(x²-3x+4x-12)
= 2x{x(x-3)+4(x-3)}
= 2x{(x+4)(x-3)}
= 2x(1x-3)(1x+4)
Comparing the resulting expression with 2x(ax+b)(cx+d)
a = 1, b = -3. c = 1 and d = 4
3) For the expression 6x²-15x-9 we will have;
On simplifying,
= 6x²+3x-18x-9
= 3x(2x+1)-9(2x+1)
= (3x-9)(2x+1)
Comparing the resulting expression with (ax+b)(cx+d)
a = 3, b = -9, c = 2 and d = 1
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8 times the sum of 2 and a number
The word problem "8 times the sum of 2 and a number" is equal to 8(2 + n).
What is an algebraic expression?In Mathematics, an algebraic expression simply refers to a mathematical equation that can be used for showing the relationship which exist between two (2) or more variables, numerical quantities, as well as in conjunction with different mathematical operations.
Assuming the variable n represents the unknown numerical value (number), we would translate the word problem into an algebraic expression as follows;
Algebraic expression = 8 × (2 + n)
Algebraic expression = 8(2 + n).
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Why is 5/100 not equal to 0.5
Answer: Because instead it equals 0.005
Step-by-step explanation:
The reason 5/100 is not equal to 0.5 is that they are written in different forms.
5/100 is written in the form of a fraction, where the numerator (5) represents the number of parts out of a whole, and the denominator (100) represents the total number of parts in the whole. So, 5/100 represents 5 parts out of 100 parts, which is equal to 0.05.
On the other hand, 0.5 is written in decimal form. It represents the number of parts out of 1, where the digits to the right of the decimal point represent the number of tenths, hundredths, thousandths and so on. So, 0.5 represents 5 tenths, or 5 parts out of 10 parts.
In conclusion, while they represent the same value, they are written in different forms and it's important to know how to read and convert them correctly.
which pair of numbers does not have a least common multiple less than 100
What is the next fraction in this sequence? Simplify your answer. 3/4, 2/3 , 7/12 , 1/2 ,
Answer:
1/3
Step-by-step explanation:
Using the fractions given, make common denominators.
After you do the first hint, look for a pattern in the top separate from a pattern in the bottom.
Bay City Balloons studied the demand in Bay City before opening a hot air balloon company. The owners expected to book 375 hot air balloon rides in their first ninety days of business. Instead, they booked 392 rides in that period. What is the percent error for the company's estimate?
+4.533 is the percent error for the company's estimate
What is percentage?A value or ratio that may be stated as a fraction of 100 is referred to as a percentage in mathematics. If we need to compute a percentage of a number, we should divide it by its whole and then multiply it by 100. The proportion, therefore, refers to a component per hundred. Per 100 is what the term percent signifies. The letter "%" stands for it.
Given, Bay City Balloons studied the demand in Bay City before opening a hot air balloon company. The owners expected to book 375 hot air balloon rides in their first ninety days of business. Instead, they booked 392 rides in that period.
Since percentage error:
Percent error is the difference between the estimated value and the actual value in comparison to the actual value and is expressed as a percentage. In other words, the percent error is the relative error multiplied by 100.
Thus in our case
Percentage error = ((392 - 375)/ 375) * 100
Percentage error = 4.533
therefore, the percent error for the company's estimate is +4.533.
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A _____ is a statistical technique for combining results from multiple studies in order to identify overall patterns in the studies as a whole.
A meta-analysis is a statistical method for combining the findings average of several research to find overarching trends in the studies as a whole.
A statistical method known as meta-analysis is used to aggregate data from various research in order to find overall trends across all of the investigations. Finding patterns in research that have been carried out using various methods and in various contexts is a specialty of this strategy. Researchers can determine the overall impact of a specific intervention or treatment by performing a meta-analysis, which combines the findings of various studies into a single effect size or summary statistic. This method is employed in a variety of fields, such as psychology, education, and medicine. Meta-analysis is an effective method that may be used to pinpoint larger study trends and to bolster the body of evidence supporting any given hypothesis or course of action.
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the following data represent glucose blood levels (mg/100 ml) after a 12-hour fast for a random sample of 70 women (reference: american journal of clinical nutrition, vol. 19, pp. 345-351). 45 66 83 71 76 64 59 59 76 82
80 81 85 77 82 90 87 72 79 69
83 71 87 69 81 76 96 83 67 94
101 94 89 94 73 99 93 85 83 80
78 80 85 83 84 74 81 70 65 89
70 80 84 77 65 46 80 70 75 45
101 71 109 73 73 80 72 81 63 74
For this problem, use six classes.
(a) Find the class width
(b) Make a frequency table showing class limits, class boundaries, midpoints, frequencies, relative frequencies, and cumulative frequencies. (Give relative frequencies to 4 decimal places.
(c) Draw a histogram.
(d) Draw a relative-frequency histogram.
(e) Categorize the basic distribution shape.
(f) Draw an ogive. (Graph each point and the closed line segments connecting the points to create your graph.)
The class width is 10.
What is Histogram ?A histogram is a representation of statistical data that makes use of rectangles to illustrate the frequency of data items in a series of equal-sized numerical intervals. The independent variable is represented along the horizontal axis and the dependent variable is plotted along the vertical axis in the most popular type of histogram.
Given:
Order the data increasingly
45,45,46,59,59,63,64,65,65,66,67,69,69,70,70,70,71,71,71,72,72, 73,73,73,74,74,75,76,76,76,77,77,78,79,80,80,80,80,80,80,81,81
81,81,82,82,83,83,83,83,83,84,84,85,85,85,87,87,89,89,90,93,94
94,94,96,99,101,101,109
Now, the range: maximum-minimum
= 109-45 = 64
Now, Divide the range by the number of classes you wish to distribute the data,
= 64/6
= 10.66 (round to 11)
Create intervals of type [a,b] starting with the first datum, and then add the class width to create the interval classes.
[45,56), [56,67), [67,78), [78,89), [89,100), [100,111)
Then, Create a frequency table that includes class boundaries (the smallest and largest data in the class), midpoints (adding the endpoints and dividing by 2), frequencies (the amount of data that fall into the class), relative frequencies (the frequency of the class divided by the total number of data, in this case 70), and cumulative frequencies (the sum of the frequencies of the class plus the previous ones)
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Two similar rectangular prisms. The smaller prism has a base length of 12 meters. The larger prism has a base length of 15 meters. The equations over the larger prism are the following.
Surface Area = 1575 square meters
Volume = 4050 cubic meters
Find the surface area and volume of the smaller similar solid
Answer:
Step-by-step explanation:
First person to answer the questions on the picture get brainiest!!!!
1. Yes. A dilation is performed on the figure. The ratio of the two figures is same. 2. No. The ratios of the sides of the figures are not equal and no transformation is applied.
What are transformations?The transformation, or f: X X, is the name given to the function, f, that maps to itself. After the transformation, the pre-image X becomes the picture X. Any operation, or a combination of operations, such as translation, rotation, reflection, and dilation, can be used in this transformation. A function can be moved in one way or another using translation, rotation, reflection, and dilation. A function can also be scaled using rotation around a point. Two-dimensional mathematical figures move about a coordinate plane according to transformations.
1. Yes. A dilation is performed on the figure. The ratio of the two figures is same.
2. No. The ratios of the sides of the figures are not equal and no transformation is applied.
3. Yes. A translation is performed on the figure to shift the transformation with 2 units.
4. No. The ratios of the sides of the figure are not same.
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Here is work Daniela did to solve y=3x+b
For x. Explain why this is incorrect
The value that Daniela got is wrong because she should have divided through by 3 to get x instead of subtracting
How to solve for x in the equation
We have y = 3x + b
We are to find the value of x from the equation that we have above
But first we are to take the other terms to one part of the equation so that x can stand on its own
3x = y - b
divide through the equation by 3
3x / 3 = y - b / 3
x = y - b / 3
This is the solution Daniela should have gotten
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Find the distance of the line segment joining the two points: (2,0) (0,-2)
The distance of the line segment joining the two points is [tex]2\sqrt{2}[/tex].
What is the distance between two points ( p,q) and (x,y)?The shortest distance (length of the straight line segment's length connecting both given points) between points ( p,q) and (x,y) is:
[tex]D = √[(x-p)² + (y-q)²] D = \sqrt{(x-p)^2 + (y-q)^2} \: \rm units.[/tex]
Given;
The two points;
(2,0) (0,-2)
Now,
D=[tex]\sqrt{(0-2)^2+(-2-0)^2}[/tex]
=2[tex]\sqrt{2}[/tex]
Therefore, the distance between two points will be 2[tex]\sqrt{2}[/tex]
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this one is easy
Evaluating expressions
make six expressions and show how you got the answer
The Expressions with solution are shown below.
What is Expression ?A mathematical operation such as subtraction, addition, multiplication, or division is used to combine terms into an expression. In a mathematical expression, the following terms are used:
An absolute numerical number is referred to as a constant.Variable: A symbol without a set value is referred to as a variable.Term: A term can be made up of a single constant, a single variable, or a mix of variables and constants multiplied or divided.Coefficient: In an expression, a coefficient is a number that is multiplied by a variable.Given:
As, the expression is contains the variables, numbers and operation.
So, the expression can be
1. 3x + 4 + 6x
= 9x + 4 (adding the like terms)
2. x+5y
3. 4(2y+3)
= 8y + 12
4. 3y×2
= 6y
5. 20x / 5x
= 4
6. 42 / 12x
= 7/2x
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2.1.3 Quiz: What Is a Function?
Question 2 of 10
If f(x) = 2x² +5√(x-2), complete the following statement:
f(6) =_______
A statement, rule, or legislation that establishes the link between the independent variable and the dependent variable (the dependent variable). f ( 6 ) = 82 .
What is the definition of a function?Domain of [tex]$2 x^2+5 \sqrt{x-2}:\left[\begin{array}{cc}\text { Solution: } & x \geq 2 \\ \text { Interval Notation: } & {[2, \infty)}\end{array}\right]$[/tex]
Range of [tex]$2 x^2+5 \sqrt{x-2}:\left[\begin{array}{cc}\text { Solution: } & f(x) \geq 8 \\ \text { Interval Notation: } & {[8, \infty)}\end{array}\right]$[/tex]
Interception zones on the axis [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : None .
Asymptotes of [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : None .
Points of Extremity [tex]$2 x^2+5 \sqrt{x-2}$[/tex] : Minimum (2,8) .
[tex]f(x) = 2x^{2} +5\sqrt{(x-2} \\f (6) = 2 *(6^{2} ) + \sqrt{6 - 2} \\ = 82[/tex]
the type of conduct or activity appropriate to a person, thing, or institution; the reason why something is made or existing; role. any formal event or occasion that is held in public or among friends. a component that affects or is influenced by other components: Availability and demand determine price.
As a set of inputs with one output for each, a function is defined as a relationship between them. A function, expressed simply, is an association between inputs where each input is connected to one and only one output. Each function has a range or codomain. f(x), where x is the input, is a common way to refer to a function.
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In a statistics class containing 250 students, each student is instructed to toss a coin 20 times and record the value of , the sample proportion of heads. The instructor then makes a histogram of the 250 values of obtained. In a second statistics classcontaining 200 students, each student is told to toss a coin 40 times and record the value of , the sample proportion of heads. The instructor then makes a histogram of the 200 values of obtained. Which of the following statements regarding the two histograms of -values is true? A.The first class’s histogram is more biased since it is derived from a smaller number of tosses per student.B.The first class’s histogram has greater spread (variability) since it is derived from a smaller number of tosses per student.C.The first class’s histogram has less spread (variability) since it is derived from a larger number of students
B.The first class’s histogram has greater spread (variability) since it is derived from a smaller number of tosses per student.
The question is asking which statement is true regarding the two histograms of -values. The first class has 250 students and they are each instructed to toss a coin 20 times. The second class has 200 students and they are told to toss a coin 40 times. The first class is tossing the coin less times than the second class, so the spread (variability) is greater for the first histogram since it is derived from a smaller number of tosses per student.
The first class has 250 students and they are each instructed to toss a coin 20 times. The second class has 200 students and they are told to toss a coin 40 times.
The first class is tossing the coin less times than the second class, so the spread (variability) is greater for the first histogram since it is derived from a smaller number of tosses per student.
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Please help
Determine whether the following statement makes sense or does not make sense and explain your reasoning.
The word imaginary in imaginary numbers tells that these numbers are undefined
View image
The correct answer is OB. The statement does not make sense as a and b are real numbers in the imaginary number of the form a + bi. Hence, the imaginary numbers are defined and interpreted in the set of real numbers also.
What is imaginary number and determine it?Imaginary numbers are defined as the set of numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit.
It is a mathematical concept that extends the real numbers system, by introducing a new number i which is defined as the square root of -1. It allows us to solve equations which are not solvable in real numbers.
The word "imaginary" is used to denote the presence of the imaginary unit i, which is not a real number but is used to extend the real numbers. It does not mean that these numbers are undefined.
Hence, the statement "The word imaginary in imaginary numbers tells that these numbers are undefined" is not true, as the imaginary numbers are defined and interpreted in the set of real numbers also.
In summary, the statement is not making sense as the imaginary numbers are defined in the set of complex numbers.
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Can some help me solve this geometry question?
The length of NQ for this problem is given as follows:
NQ = 5.
How to obtain the length NQ?In this problem, we have two similar triangles, meaning that their side lengths are proportional.
The lengths for this problem are given as follows:
NO = 8.NQ = x.QR = x - 1.QP = 5.The similar triangles are given as follows:
PQR and PNO.
The equivalent side lengths are given as follows:
5 and 5 + x.x - 1 and 8.Hence the proportional relationship is given as follows:
5/(5 + x) = (x - 1)/8
Applying cross multiplication, we have that:
(x + 5)(x - 1) = 40
x² + 4x - 5 = 40
x² + 4x - 45 = 0.
Hence:
(x + 9)(x - 5) = 0.
The positive solution is:
x - 5 = 0
x = 5.
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Answer the following
The type of slope for each function is given as follows:
Zero, Positive, Positive, Undefined.
How to classify the slope of a line?There are four types of lines, and their slopes are classified as follows:
Constant horizontal line: slope of zero.Constant vertical line: undefined slope.Increasing line: positive slope.Decreasing line: negative slope.There is a mistake in the options or in the graph, as both graphs b and c are increasing lines.
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A person in a car is driving at 60 km/h toward a ferry whose whistle is blowing at 400 Hz. (a) What frequency does she hear? (b) The ferry leaves the dock and heads directly away from the driver at 15 km/h, still blowing its whistle. What frequency does she hear now?
a. The frequency which she hears is 440Hz.
b. The person in the car would hear a frequency of 391.98 Hz when the ferry leaves the dock and heads directly away from the driver.
(a) The frequency that the person in the car hears is affected by the motion of the car and the sound source (the ferry). The effect is called the Doppler effect, which causes the perceived frequency of sound to change depending on the observer and the relative motion of the source.
If the car is moving toward the ferry, the frequency will be higher than the actual frequency of the sound. The perceived frequency is given by the formula:
f' = f (v + v_s) / (v + v_s)
where
f = the actual frequency of the sound (400 Hz)
v = the velocity of sound (approximately 340 m/s)
[tex]v_s[/tex] = the velocity of the car (60 km/h = 16.67 m/s)
So,
f' = 400(340 + 16.67) / (340 - 16.67) = 400(356.67) / 323.33 = 440 Hz
Hence, the frequency which she hears is 440Hz.
(b) The perceived frequency when the ferry leaves the dock and heads directly away from the driver is also affected by the Doppler effect.
f' = f (v - v_s) / (v - v_s)
where
f = the actual frequency of the sound (400 Hz)
v = the velocity of sound (approximately 340 m/s)
v_s = the velocity of the ferry (15 km/h = 4.17 m/s)
So,
f' = 400(340 - 4.17) / (340 + 4.17) = 400(335.83) / 344.17 = 391.98 Hz
The person in the car would hear a frequency of 391.98 Hz when the ferry leaves the dock and heads directly away from the driver.
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TRUE OR FALSE if you have a logical statement in five variables how many rows do you need in the truth table that you would use to evaluate it? answer with an integer.
We need 32 rows in the truth table that would use to evaluate a logical statement in five variables.
To find out the number of rows in a truth table with 5 variables, we should know what is a truth table.
A truth table is a study of a logic function by listing all possible values the function can attain. The top row illustrates logical variables and combinations in increasing complexity leading up to the final function.
The number of rows that a truth table needs is decided by the number of basic statement letters involved in the set formulas that will be involved in the set of formulas.
The formula to calculate the number of rows is equal to 2ⁿ, where n is the number of basic statement letters involved.
Number of rows = 2ⁿ
If the number of variables in a truth table is 5, then the number of rows = 2⁵
2⁵ = 2×2× 2× 2× 2
= 32
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XYZ is located on the coordinate plane below. If the triangle is transformed following the rule (x, y) to (x, -y), will congruence be maintained? Why or why not?
Answer: yes
Step-by-step explanation: because it will stay the same
Assume f(x) has a continuous second derivative. Show that f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx
Assume f(x) has a continuous second derivative. The expression "f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx" is TRUE.
How can we show that the expression true?To show that f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx, we will use Taylor's theorem with a remainder in the integral form.
Taylor's theorem states that for any function f(x) that has a continuous kth derivative, we can express the function at any point x = b as:
f(b) = f(a) + f'(a)(b-a) + (1/2!)f''(a)(b-a)^2 + ... + (1/k!)f^(k)(c)(b-a)^k
where c is some value between a and b.
If we take the limit as k approaches infinity, we get:
f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx
This is known as the Taylor's theorem with a remainder in the integral form.
This equation shows that f(b) is equal to f(a) plus the linear approximation of f'(a)(b-a) plus the remaining error term ∫ₐᵇf"(x)(b-x)dx which is the integral of the second derivative of f(x) times (b-x) from a to b.
This is true because f"(x) is continuous on the interval [a, b], so the integral of f"(x) over that interval exists. And because the second derivative of f(x) is continuous, the integral can be evaluated.
Therefore, assuming that f(x) has a continuous second derivative, we can conclude that f(b) = f(a) + f'(a)(b-a) + ∫ₐᵇf"(x)(b-x)dx
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