Answer:
120 - 15 = 105 pages for the chapters 105 ÷ 7 = 15 chapters Therefore, there are 15 chapters in Roger's novel. The answer is not given in the options provided, so I cannot select one.
What is the frequency of the sinusoidal graph?
the frequency of the sinusoidal graph is 1/π.
Given sinx=3/5 and is in quadrant 2, what is the value of tan x/2 ?
Answer:
[tex]\tan \left(\dfrac{x}{2}\right)=3[/tex]
Step-by-step explanation:
Trigonometric ratios are the ratios of the sides of a right triangle.
[tex]\boxed{\begin{minipage}{9.4 cm}\underline{Trigonometric ratios} \\\\$\sf \sin(\theta)=\dfrac{O}{H}\quad\cos(\theta)=\dfrac{A}{H}\quad\tan(\theta)=\dfrac{O}{A}$\\\\where:\\ \phantom{ww}$\bullet$ $\theta$ is the angle. \\ \phantom{ww}$\bullet$ $\sf O$ is the side opposite the angle. \\\phantom{ww}$\bullet$ $\sf A$ is the side adjacent the angle. \\\phantom{ww}$\bullet$ $\sf H$ is the hypotenuse (the side opposite the right angle). \\\end{minipage}}[/tex]
The sine trigonometric ratio is the ratio of the side opposite the angle to the hypotenuse.
Given sin(x) = 3/5, the side opposite angle x is 3, and the hypotenuse is 5.
As we have two sides of the right triangle, we can calculate the third side (the side adjacent the angle) using Pythagoras Theorem.
[tex]\boxed{\begin{minipage}{9 cm}\underline{Pythagoras Theorem} \\\\$a^2+b^2=c^2$\\\\where:\\ \phantom{ww}$\bullet$ $a$ and $b$ are the legs of the right triangle. \\ \phantom{ww}$\bullet$ $c$ is the hypotenuse (longest side) of the right triangle.\\\end{minipage}}[/tex]
Therefore:
[tex]\implies A^2+3^2=5^2[/tex]
[tex]\implies A^2+9=25[/tex]
[tex]\implies A^2+9-9=25-9[/tex]
[tex]\implies A^2=16[/tex]
[tex]\implies \sqrt{A^2}=\sqrt{16}[/tex]
[tex]\implies A=4[/tex]
Use the cosine trigonometric ratio to find the value of cos(x), remembering that cosine is negative in Quadrant II.
[tex]\implies \cos x=-\dfrac{4}{5}[/tex]
Now we have the values of sin(x) and cos(x) in Quadrant II, we can use the tangent half angle formula to find the value of tan(x/2).
[tex]\begin{aligned}\implies \tan \left(\dfrac{x}{2}\right)&=\dfrac{\sin x}{1+\cos x}\\\\&=\dfrac{\frac{3}{5}}{1-\frac{4}{5}}\\\\&=\dfrac{\frac{3}{5}}{\frac{1}{5}}\\\\&=\dfrac{3}{5} \cdot \frac{5}{1}\\\\&=3\end{aligned}[/tex]
Therefore, the value of tan(x/2) is 3.
If agent D was able to reduce their average Handling Time by 10% what would thier average handling time be
The new average handling time for Agent D would be 90% of their original handling time.
What are Percentages?A percentage is a way of expressing a number as a fraction of 100. It is typically represented using the percent sign (%) and is often used to describe the amount or proportion of something in relation to a whole. Percentages are commonly used in a variety of fields, such as finance, mathematics, and statistics.
Let's say the original average handling time for Agent D was "x" units (e.g. seconds, minutes, etc.). If Agent D was able to reduce their average handling time by 10%, their new average handling time would be:
New average handling time = x - 0.1x
Simplifying the expression on the right-hand side:
New average handling time = 0.9x
Therefore, the new average handling time for Agent D would be 90% of their original handling time.
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Consider a triangle ABC like the one below suppose that C equals 32 vehicles 44 and C equals 27° the figure is not drawn to scale solve the triangle
On solving the provided question we can say that As a result, the triangle is resolved.
what is trigonometry?The study of the relationship between triangle side lengths and angles is known as trigonometry. The concept first originated in the Hellenistic era, during the third century BC, due to the application of geometry in astronomical investigations. The subject of mathematics known as exact techniques deals with certain trigonometric functions and their potential applications in computations. There are six commonly used trigonometric functions in trigonometry. Sine, cosine, tangent, cotangent, secant, and cosecant are their separate names and acronyms (csc). The study of triangle properties, particularly those of right triangles, is known as trigonometry. As a result, geometry is the study of the properties of all geometric shapes.
We now have all of the information we require to solve the triangle. We now have:
a = 55.815 sin(A) (A
b = 55.815 sin(B) (B)
c = 32
A + B + C = 180°
B = 153° - A
To find the values of A and B, we can use a calculator. We get:
A ≈ 83.814°
B ≈ 42.186°
32 / sin(27°) = a / sin(A)
a ≈ 54.482
AB ≈ 54.482
BC ≈ 39.343
AC ≈ 22.414
A ≈ 83.814°
B ≈ 42.186°
C ≈ 27°
As a result, the triangle is resolved.
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The Venn diagram shows below the number of customers in a restaurant who ordered a starter or a dessert. A customer is picked at random. If they ordered a dessert, what is the probability that they did *not* order starter? Give your answer as fraction in its simplest form.
Answer:
Step-by-step explanation:
23 people ordered dessert.
8 of these ordered only a dessert.
P(not starter | ordered dessert) [tex]=\frac{8}{23}[/tex]
Solve the systems by graphing.
Y=1/4 x-5
y=-X+4
Answer: (7.2, -3.2)
Step-by-step explanation:
First, we will graph these equations. See attached. One has a y-intercept of -5 and then moves four units right for every unit up (we get this from the slope of 1/4). The other has a y-intercept of 4, and moves right one unit for every unit down (we get this from the slope of -1).
The point of intersection is the solution, this is the point at which both graphed lines cross each other. Our solution is:
(7.2, -3.2) x = 7.2, y = -3.2
A company makes two kinds of engineering pencils , Type I and Type II ( deluxe ) . Type I needs 2 min of sanding and 6 min of olishing . Type needs 5 min of sanding and 3 min of polishing . The sander can run no more than 66 hours per week and the olisher can run no more than 73 hours a week . A $ 3 profit is made on Type I and $ 5 profit on Type II . How many of each type be made to maximize profits ?
After solving by linear programming, the business needs create 100 Type I pencils and 80 Type II pencils to increase revenue.
LINEAR PROGRAMMING: WHAT IS IT?
A mathematical method called linear programming is used to maximise a linear objective function under the restrictions of linear equality and inequality. In a mathematical model whose requirements are expressed by linear connections, it is used to identify the best result.
With linear programming, this issue can be resolved.
Please define x as the quantity of Type I pencils produced and y as the quantity of Type II pencils produced.
Profit = 3x + 5y is the formula for the goal function.
2x + 5y 660 are the restrictions (sanding constraint)
(Polishing constraint): x ≥0 y≥ 0; 6x + 3y 730
Under these limitations, we wish to maximize the profit function.
The largest profit comes when x = 100 and y = 80,
with a profit of $740, according to software that can solve linear programming issues or a graphing calculator.
Thus, the business needs create 100 Type I pencils.80pencil for type II
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Suppose that A and B are independent events such that P (A) - 0.10 and P (B) - 0.60.
Find P(A n B) and P (A U B).
Answer:
Step-by-step explanation:
Since A and B are independent events, we can use the formula:
P(A ∩ B) = P(A) x P(B)
P(A ∩ B) = 0.10 x 0.60
P(A ∩ B) = 0.06
So the probability of both events A and B occurring is 0.06.
To find P(A U B), we can use the formula:
P(A U B) = P(A) + P(B) - P(A ∩ B)
P(A U B) = 0.10 + 0.60 - 0.06
P(A U B) = 0.64
Therefore, the probability of either A or B occurring (or both) is 0.64.
72x5/12=blankx5x1/12=blank x 1=360/12=blank
Answer:
Step-by-step explanation:
Starting with 72x5/12:
72x5/12 = (72/12) x 5 (simplifying the fraction)
= 6 x 5
= 30
Now, we have:
30 = ?x5x1/12
Multiplying both sides by 12, we get:
30 x 12 = ? x 5 x 1
360 = ? x 5
Dividing both sides by 5, we get:
72 = ?
Therefore, the missing value is 72.
On thurday, lisa had 5$ in her bank account. she went to target to purchase stickers for her class. each pack of stickers cost 2$
write an inequality that represents s, the number of stickers purchased that resulted in her account ending in -10.
Four transformations of the function f (x ) = 2x are given below. For each transformation, drag the expression that shows the result of that transformation into the box under it.
For the four transformations based on f(x)= [tex]2^{x}[/tex],
1)6f(x)= 6 [tex]2^{x}[/tex]
2)f(6x)= [tex]2^{6x}[/tex]
3)f(x+6)=[tex]2^{x+6}[/tex]
4)f(x)+6 = [tex]2^{x}[/tex]+6
What are transformations?
Transformations in any given function is changing its original form to nre form by flipping, rotating, shifting, enlarging and compressing the function. We can move the given function up or down as per the given conditions by adding up or subtracting the constant in y axis. We can move the given function left or right as per the given conditions by adding up or subtracting the constant in x axis. We can stretch or compress the function about x or y axes. Also we can also flip,reverse the function, reflect about axes or enlarge the functions.
Here given that function f(x)= [tex]2^{x}[/tex]
From the given graph we can identify few points for the given function:
(-1,0.5); (0,1); (1,2); (2,4); (3,8); (4,16); (5,32); (6,64) and so on.
Now to identify the transformations, we can substitute the tranformed value of x in the function:
1)6 f(x) = 6 . [tex]2^{x}[/tex] {as we know that f(x)= [tex]2^{x}[/tex]}
∴6 f(x) will be equal to 6 . [tex]2^{x}[/tex]
2)f(6x) : for this we can replace 'x' by '6x'
f(6x) = [tex]2^{6x}[/tex]
3)f(x+6): for this function replace 'x' by 'x+6'
f(x+6)=[tex]2^{x+6}[/tex]
4)f(x)+6 : substitute f(x)= [tex]2^{x}[/tex], we get
f(x)+6 = [tex]2^{x}[/tex]+6
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The superintendent of a school district wants to predict next year‘s middle school lunch count the graph shows the results of a survey randomly selected middle school students she conducted a survey and which randomly select the middle school students rest do you typically buy a school lunch each week approximately 63% of students responded they do not typically buy a school lunch if the district has 5000 middle school students next year about how many students plan to buy lunch one and two days a week 
1850 students are expected to buy lunch one or two days a week next year.
what is a percentage?A ratio or figure stated as a fraction of 100 is called a percentage. The sign "%" is frequently used to indicate it as a percentage or a component of a total.
If 63% of the students do not typically buy a school lunch, then 37% of the students do typically buy a school lunch.
Let's assume that x students plan to buy lunch one or two days a week.
Then, the number of students who do typically buy a school lunch can be estimated as:
0.37(5000) = 1850
Let's assume that p% of the students plan to buy lunch one or two days a week. Then, we can set up the following equation:
p% of (5000) = x
To solve for x, we need to convert the percentage to a decimal by dividing by 100:
p/100 × 5000 = x
Simplifying the equation, we get:
50p = x
We can substitute this equation into the original equation to get:
0.37(5000) = 50p
Simplifying and solving for p, we get:
1850 = 50p
p = 37
Therefore, approximately 37% of the 5000 middle school students, or 1850 students, are expected to buy lunch one or two days a week next year.
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What is the perimeter of a rectangle with a base of 9 ft and a height of 10 ft?
Answer: The perimeter of a rectangle is found by adding up all four sides. For this rectangle with a base of 9 ft and a height of 10 ft, the two base sides have a length of 9 ft each, and the two height sides have a length of 10 ft each. Therefore, the perimeter is:
P = 2(9 ft) + 2(10 ft) = 18 ft + 20 ft = 38 ft
So the perimeter of the rectangle is 38 feet.
Step-by-step explanation:
The time between arrivals of vehicles at a particular intersection follows an exponential probability distribution with a mean of 12 seconds. a. Sketch this exponential probability distribution. b. What is the probability that the arrival time between vehicles is 12 seconds or less? c. What is the probability that the arrival time between vehicles is 6 seconds or less? d. What is the probability of 30 or more seconds between vehicle arrivals?
After answering the presented question, we can conclude that the probability of 30 or more seconds between vehicle arrivals is approximately 0.082.
What is probability?Probability is a measure of how likely an event is to occur. It is represented by a number between 0 and 1, with 0 representing a rare event and 1 representing an inescapable event. Switching a fair coin and coin flips has a chance of 0.5 or 50% because there are two equally likely outcomes. (Heads or tails). Probabilistic theory is an area of mathematics that studies random events rather than their attributes. It is applied in many fields, including statistics, economics, science, and engineering.
exponential probability distribution
[tex]P(X ≤ 12) = ∫[0,12] f(x) dx = ∫[0,12] (1/12) * e^(-x/12) d\\P(X ≤ 12) = [-e^(-x/12)] [0,12] = -e^(-1) + 1 ≈ 0.632\\P(X ≤ 6) = ∫[0,6] f(x) dx = ∫[0,6] (1/12) * e^(-x/12) dx\\P(X ≤ 6) = [-e^(-x/12)] [0,6] = -e^(-1/2) + 1 ≈ 0.393\\[/tex]
Therefore, the probability that the arrival time between vehicles is 6 seconds or less is approximately 0.393.
P(X ≥ 30) = 1 - P(X < 30) = 1 - P(X ≤ 30) = 1 - ∫[0,30] f(x) dx
[tex]= 1 - ∫[0,30] (1/12) * e^(-x/12) dx[/tex]
[tex]P(X ≥ 30) = 1 - [-e^(-x/12)] [0,30] = e^(-2.5) ≈ 0.082[/tex]
Therefore, the probability of 30 or more seconds between vehicle arrivals is approximately 0.082.
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Analyze the diagram below and complete the instructions that follow.
Find sin 45°.
A.
112
B. √√√2
L
45°
sin 45° = opposite/hypotenuse = x/√2x = √2/2. The answer is B. √2/2.
What is trigonometry?
Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It focuses on the study of trigonometric functions, which are functions that relate the angles of a triangle to the ratios of the lengths of its sides.
The triangle shown in the diagram is a right triangle with one angle of 45 degrees, which means that the other two angles must measure 45 degrees each as well.
The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In this case, we can label the sides of the triangle as follows:
The side opposite the 45-degree angle is x.
The side adjacent to the 45-degree angle (and opposite the other 45-degree angle) is also x.
The hypotenuse is the longest side of the triangle and is labeled as √2x.
Using the definition of sine, we have:
sin 45° = opposite/hypotenuse = x/√2x = √2/2
Therefore, the answer is B. √2/2.
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The Griffins bought a $292,000 house. They made a down payment of $48,000 and took out a mortgage for the rest. Over the course of 30 years they
monthly payments of $1462.91 on their mortgage until it was paid off.
Question 13
(a) What was the total amount they ended up paying for the house (including
the down payment and monthly payments)?
$
(b) How much interest did they pay on the mortgage?
$
The Griffins ended up paying a total of $575,647.60 for the house. The Griffins paid a total of $331,647.60 in interest over the 30-year period
What is interest rate?Interest rate refers to the percentage of the principal amount charged by a lender to a borrower for the use of money over a certain period of time.
According to question:(a) The total amount they ended up paying for the house can be calculated by adding the down payment to the total amount paid in monthly mortgage payments over the 30-year period:
Total amount paid = down payment + (monthly payment x number of payments)
Number of payments = 30 years x 12 months/year
= 360
Total amount paid = $48,000 + ($1462.91 x 360)
= $48,000 + $527,647.60
= $575,647.60
Therefore, the Griffins ended up paying a total of $575,647.60 for the house.
(b) The total amount of interest paid on the mortgage can be calculated by subtracting the amount borrowed (i.e., the purchase price minus the down payment) from the total amount paid over the 30-year period:
Total interest paid = total amount paid - amount borrowed
Amount borrowed = $292,000 - $48,000 = $244,000
Total interest paid = $575,647.60 - $244,000 = $331,647.60
Therefore, the Griffins paid a total of $331,647.60 in interest over the 30-year period.
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Which system of equations represents the graph?
y = 3x - 5 and 2x + 4y = 8
y = 3x - 5 and 4x + 2y = 8
y = 2x - 5 and 4x + 2y = 8
y = 2x - 5 and 2x + 4y = 8
Part B
What is the apparent solution to the system of equations in the graph?
(1, 2)
(2, 1)
(4, 0)
(0, 2)
Part A: The system of equations represented by the graph: y = 3x - 5 and 2x + 4y = 8.
Part B: The solution of the system of equations : (2, 1).
Explain about the system of equations:Determining the significance of the variables employed in a system of equations entails solving the set of equations.
A specific system of equations may have a variety of solutions,
unique responseNo remedythere are several optionsLet's examine three approaches to solving a set of equations, presuming that they are linear equations with two variables.
Method of Substitution Method of EliminationGraphical ApproachFrom the graph shown.
The blue line shows the equation: 2x + 4y = 8
At x =0, y= 2
At y =0, x = 4
Red line shows the equation: y = 3x - 5
At x = 0, y = -5.
Part B: solution to the system of equations.
From the graph, where two lines intersect is the solution of the system of equations.
That is point (2,1).
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The combined city/highway fuel economy of a 2016 Toyota 4runner 2wd 6-cylinder 4-L automatic 5-speed using regular gas is a normally distributed random variable with a range of 21mpg to 26mpg answer A and B URGENT
a)The range of 95% of the data is from 21 mpg to 26 mpg, which is a range of 5 mpg.
b)We need a sample size of 543 to estimate the mean with 98% confidence and an error of 0.25 mpg
What is Empirical Rule for a normal distribution?
If a dataset is normally distributed, we can expect that about 68% of the data points will fall within one standard deviation of the mean, about 95% of the data points will fall within two standard deviations of the mean, and about 99.7% of the data points will fall within three standard deviations of the mean. This rule is a useful guideline for understanding the spread of data in a normal distribution.
(a) Using Method 3 (the Empirical Rule for a normal distribution), we know that for a normally distributed random variable, approximately 68% of the data falls within one standard deviation of the mean, 95% of the data falls within two standard deviations of the mean, and 99.7% of the data falls within three standard deviations of the mean.
Since the range of the combined city/highway fuel economy of a 2016 Toyota 4Runner 2WD 6-cylinder 4-L automatic 5-speed using regular gas is from 21 mpg to 26 mpg, the midpoint of the range is (21 + 26) / 2 = 23.5 mpg.
Using the Empirical rule, we know that approximately 95% of the data falls within two standard deviations of the mean. Therefore, the range of 95% of the data is from 21 mpg to 26 mpg, which is a range of 5 mpg.
We can set up the following equation to solve for the standard deviation, σ:
2σ = 5
σ = 5 / 2
σ = 2.5
Therefore, the estimated standard deviation is 2.5 mpg. Rounded to 4 decimal places, the estimated standard deviation is 2.5000 mpg.
(b) The formula for the margin of error is:
Margin of error = z-value×(standard deviation / √(sample size))
We want the margin of error to be 0.25 mpg and the confidence level to be 98%. Since we are using a z-value, we can look up the z-value for a 98% confidence level in a standard normal distribution table.
The z-value for a 98% confidence level is approximately 2.33 when rounded to 3 decimal places.
Plugging in the given values, we have:
0.25 = 2.33×(2.5 / √(sample size))
Solving for the sample size, we get:
√(sample size) = 2.33 × (2.5 / 0.25)
√(sample size) = 23.3
sample size = (23.3)²
sample size = 542.89
Rounded to the nearest whole number, we need a sample size of 543 to estimate the mean with 98% confidence and an error of 0.25 mpg.
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Jake and Steve are calculating the volume of a triangular prism. Who calculated the volume incorrectly? What is the student's error?
Answer: Jake solved it incorrectly.
Step-by-step explanation: His equation was supposed to be (7/2*35/2) *18
What lis the length of bc?
Answer: C (23)
Step-by-step explanation:
Since the triangle is isosceles, BA = BC
x + 17 = 2x -6
x = 23
Answer:
C(23)
Step-by-step explanation:
Since line AB = line BC
x+17=2x-6, by collecting like terms x=23
Are these ratios equivalent?
6 teal sweatshirts to 8 purple sweatshirts
15 teal sweatshirts to 13 purple sweatshirts
True
False
Answer:
False
Step-by-step explanation:
Purple: 13÷8= 1.625
Teal: 15÷6= 2.5
The ratios don't have the same scale factor. Therefore they are not equivalent fractions.
Light travels 1.8x10^7 kilometers in one minute. How far does it travel in 6 minutes?
Divide the following 11/15by 7/18
Answer:
4567
578९=8877
5790=9766
Answer:
11/15 : 7/18 = 15 / 11 : 18 / 7= 270 / 77
Step-by-step explanation:
You met Jonathan while waiting for your plane in the airport at Brownsville. He is
working in the marketing research department for a company that manufactures
and sells memory chips for microcomputers. He has established the following price-
demand and revenue functions:
P(x) = 75 - 3x
R(x) = x P(x)
Where P(x) is the wholesale price in dollars at which x million chips can be
sold, and R(x) is in millions of dollars. Both functions have a domain 1 ≤ x ≤
20.
a) Jonathan wants to sketch a graph of the revenue function in a rectangular
coordinate system.
b) Jonathan wants to find the value of x that will produce the maximum
revenue. He wants also to find the maximum revenue.
c) Finally, he wants to know the wholesale price per chip that produces the
maximum revenue?
a. The revenue function R(x) = 75x - 3x² graph plotted below.
b. Value of x is 12.5 and maximum revenue produces $468.75
c. Wholesale price/chip that produces the maximum revenue is $37.50
Define the term revenue?A company's or business's revenue is the total amount of money it receives from sales of its products or services over a given time period.
a) To sketch a graph of the revenue function R(x), we first need to calculate R(x) using the given formula:
⇒ R(x) = x P(x)
⇒ R(x) = x(75 - 3x)
⇒ R(x) = 75x - 3x²
Now we can plot this function on a rectangular coordinate system. Below is an sketch of the graph.
b) Taking the derivative of R(x) and setting it equal to 0:
⇒ R'(x) = 75 - 6x = 0
⇒ 6x = 75
⇒ x = 12.5
So, x = 12.5 is the value of x that will produce the maximum revenue. To find the maximum revenue, we can substitute x = 12.5 into the revenue function R(x) = x (75 - 3x)
⇒ R(12.5) = 12.5 (75 - 3 (12.5)) = 468.75
⇒ R(12.5) = $ 468.75
Therefore, the maximum revenue is $468.75
c) To find the wholesale price per chip that produces the maximum revenue, we can substitute x = 12.5 into the price function, P(x) = 75 - 3x
⇒ P(12.5) = 75 - 3(12.5)
⇒ P(12.5) = $37.50 per chip
Therefore, the wholesale price per chip that produces the maximum revenue is $37.50.
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This answer in a fraction
The experimental probability that the next student will register for German is 9/79.
What is probability?
To find the experimental probability that the next student will register for German, we need to divide the number of students who have registered for German by the total number of students who have registered so far:
P(German) = number of students who have registered for German / total number of students who have registered
P(German) = 108 / (108 + 360 + 21 + 459) [Adding all the students who registered for each language]
P(German) = 108 / 948
P(German) = 9/79
Therefore, the experimental probability that the next student will register for German is 9/79.
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Suppose a random sample of 80 measurements is selected from a population with a mean of 25 and a variance of 200. Select the pair that is the mean and standard error of x. Rstudio
a) [25, 2.081]
b) [25, 1.981]
c) [25, 1.681]
d) [25, 1.581]
e) [80, 1.681]
[ 25 , 1.581 ] is the pair that is the mean and standard error of x.
What does standard error mean?
A statistical concept known as the standard error uses standard deviation to assess how well a sample distribution represents a population.
The standard error of the mean describes the statistical variation between a sample mean and the population's actual mean. Measures of variability include standard error and standard deviation: The standard deviation describes variation within a single sample.
n = 80
μ = 25
σ² = 200
mean of x = μ = 25
standard error = √ σ²/n
= √200/80
= 1.581
= [ 25 , 1.581 ]
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Find g(x), where g(x) is the translation 5 units right of f(x)= – 7(x–5)2+3.
g(x) is the translation 5 units right of f(x)= – 7(x–5)²+3.
A function called f(x) accepts an input of "x" and outputs "y". You can write it out as y = f. (x).‘x’ is a variable that represents an input to a function.
To translate a function, we need to replace x with (x-a) in the function f(x) where ‘a’ is the amount of translation.
To translate a function 5 units right, we need to replace x with (x-5) in the function f(x).
So, g(x) = f(x-5) = -7(x-5-5)²+ 3 = -7(x-10)²+ 3.
Therefore, g(x) is the translation 5 units right of f(x)= – 7(x–5)²+3.
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Calculate five-number summary and construct box and whisker plot from the following data: Ans: 30, 40, 50, 60 & 70; No skewed Daily wages (Rs.) 10-30 30-50 50-70 70-90 90-110 110-130 130-150 No. of workers 53 85 56 4 3 21 16 Aus: 10 150-170 2
Five-number summary: Minimum = 30, Q1 = 35, Median = 50, Q3 = 65, Maximum = 70. Bοx and whisker plοt: Bοx spans frοm 35 tο 65 with median at 50, whiskers extend frοm 30 tο 70, nο οutliers.
What are the steps tο calculate five-number summary and cοnstruct a bοx and whisker plοt?Tο find the five-number summary and cοnstruct a bοx and whisker plοt, we need tο first οrganize the given data in ascending οrder:
30, 40, 50, 60, 70
The five-number summary cοnsists οf the minimum value, the first quartile (Q1), the median (Q2), the third quartile (Q3), and the maximum value.
Minimum value: 30
Q1 (first quartile): the median οf the lοwer half οf the data set, which is (30 + 40)/2 = 35
Median (Q2): the middle value οf the data set, which is 50
Q3 (third quartile): the median οf the upper half οf the data set, which is (60 + 70)/2 = 65
Maximum value: 70
Sο, the five-number summary is:
Minimum = 30
Q1 = 35
Median = 50
Q3 = 65
Maximum = 70
To construct a box and whisker plot, we draw a number line that includes the range of the data (from the minimum value to the maximum value), and mark the five-number summary on the number line. Then we draw a box that spans from Q1 to Q3, with a vertical line inside the box at the median (Q2). In addition, we draw "whiskers" from the box to the minimum and maximum values.
The box and whisker plot for the given data is as follows:
20 40 60 80 100
|----------|----------|----------|----------|
+-----+
| |
| |
| |
| |
+-----+
The box spans from 35 to 65, with a vertical line inside the box at 50. The whiskers extend from 30 to 70. There are no outliers in the data, so there are no points beyond the whiskers.
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Write a paragraph proof of the Triangle Proportionality Theorem.
(Theorem 8.6)
__ __
Given: BD || AE
Prove: BA/CB = DE/CD
The Triangle Proportionality Theorem, also known as the Side Splitter Theorem, states that if a line is parallel to one side of a triangle, then it divides the other two sides proportionally.
Triangle Proportionality Theorem:
To prove this theorem, we begin by drawing a ΔABC with a line DE parallel to side AB. We then draw lines BD and CE, which intersect the parallel line DE at points F and G, respectively. By the properties of parallel lines, we know that ∠ADE and ∠ABD are congruent, and ∠AED and ∠ADB are congruent. Similarly, ∠CDE and ∠BDC are congruent, and ∠CED and ∠DCB are congruent.
We can then use the properties of similar triangles to show that ΔADE and ΔABC are similar, as are ΔCDE and ΔACB. This means that the ratios of corresponding side lengths are equal:
BA/DE = CA/CE and CB/DE = AB/BD
We can then substitute CA - BA for CB in the first equation, and BD for AB in the second equation:
BA/DE = (CA - BA)/CE and CB/DE = BD/(CA - BA)
Cross-multiplying both equations, we obtain:
BA * CE = DE * (CA - BA) and CB * DE = BD * (CA - BA)
Adding the two equations, we get:
BA * CE + CB * DE = (DE + CE) * CA
Dividing both sides by CB * DE, we obtain:
BA/CB = (DE + CE)/CE * CA/DE = DE/CD
Thus, we have proven the Triangle Proportionality Theorem.
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How will the product change if one number is increased by a factor of 12 and the other is decreased by a factor of 4
If one number is increased by a factor of 12 and the other is decreased by a factor of 4, the product of the two numbers will be multiplied by a factor of 3.
Let's suppose we have two numbers, A and B, and we want to know how their product will change if one number is increased by a factor of 12 and the other is decreased by a factor of 4.
The initial product of the two numbers is:
A x B
If we increase A by a factor of 12, the new value of A will be 12A. If we decrease B by a factor of 4, the new value of B will be B/4. Therefore, the new product of the two numbers will be:
(12A) x (B/4) = (12/4) x A x B = 3AB
So the new product of the two numbers will be three times the initial product. In other words, if one number is increased by a factor of 12 and the other is decreased by a factor of 4, the product of the two numbers will increase by a factor of 3.
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