The statement that describes the behavior of the function f(x) is: The graph approaches 0 as x approaches infinity.
The correct answer is an option (b)
Consider the function [tex]f(x) = \frac{2x}{1-x^2}[/tex]
Consider the limit of function f(x) as x approaches infinity (x → ∞)
[tex]\lim_{n \to \infty} f(x)\\\\= \lim_{n \to \infty} \frac{2x}{1-x^2} \\\\= \lim_{n \to \infty} \frac{2x/x}{(1-x^2)/x}\\\\ = \lim_{n \to \infty} \frac{-2}{x}[/tex]
= 0
This means that when x approaches infinity the term x² grows rapidly
So, 1 - x², which is in the denominator will decrease, become a large negative more rapid than the x in the numerator.
We can say that the function approaches zero.
Thus the correct statement is: The graph approaches 0 as x approaches infinity.
The correct answer is an option (b)
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The complete question is:
Which statement describes the behavior of the function f (x) = StartFraction 2 x Over 1 minus x squared EndFraction?
a) The graph approaches –2 as x approaches infinity.
b) The graph approaches 0 as x approaches infinity.
c) The graph approaches 1 as x approaches infinity.
d) The graph approaches 2 as x approaches infinity.
Plot the numbers -1 1/2 and 2 3/4 on the number line below.
Answer:
Step-by-step explanation:
ok, so for the -1 1/2 your going to find the -1 point on the left of 0. Then from there you will count to the second small point (or small line intersecting through the number line) to the right of -1. It will be this one right here
Now for the second number 2 3/4 you will find the 2 point, to the right from 0. Now count three small points to the left to get the 3/4 part. It will be right here.
Hope this helps!!
There were 230{,}600230,600230, comma, 600 jobs available in the field of radiology in the year 201420142014. Each year, that number is expected to grow by 0.9\%0.9%0, point, 9, percent.
Write a function that gives the expected number j(t)j(t)j, left parenthesis, t, right parenthesis of jobs in radiology ttt years from the year 201420142014.
Do not use commas in your answer.
Which of the following probability models would the probability P(C) = 1/3 complete?
a P(A) = 1/9 P(B)= 1/2
b P(A) = 1/9 P(B)= 2/9
c P(A) = 1/6 P(B)= 3/9
d P(A) = 2/9 P(B)= 8/18
please answer soon worth 15 points
Planes T and X are parallel. Plane T contains line a. Plane X contains line b.
Planes T and X are parallel. Plane T contains line a and plane X contains line b.
Which best explains the relationship between lines a and b?
Classify the following triangle as right or not. Explain why it is or is not.
24
27
15
How do you graph 2x^2 -5x + 5 = 3 ?
(1. 5,0 ) , (1 , 0 ) are the values of x in linear equation.
What is a linear equation in mathematics?
A linear equation in algebra is one that only contains a constant and a first-order (direct) element, such as y = mx b, where m is the pitch and b is the y-intercept.
Sometimes the following is referred to as a "direct equation of two variables," where y and x are the variables. Direct equations are those in which all of the variables are powers of one. In one example with just one variable, layoff b = 0, where a and b are real numbers and x is the variable, is used.
2x² -5x + 5 = 3
2x² -5x + 5 - 3 = 0
2x² -5x + 3 = 0
2X² - (2+ 3)X + 3 = 0
2X² -2X - 3X + 3 = 0
2x(x - 1 )-3(x - 1 ) = 0
(2x - 3) (x - 1 )
x = 3/2 , 1
x = 1. 5 , 1
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A quantity with an initial value of 480 grows exponentially at a rate such that the quantity doubles every 6 minutes. What is the value of the quantity after 0.2 hours, to the nearest hundredth?
answer 1920
The value of quantity after 0.2 hours is, 960
Given that;
A quantity with an initial value of 480 grows exponentially at a rate such that the quantity doubles every 6 minutes.
We know that;
The exponential growth function is
A = A₀ (1 + r)ⁿ
Where, A= The number of quantity after t days
A₀ = initial number of quantity
r = rate of growth
n = time in minutes
A quantity with an initial value of 830 grows at a rate such that the quantity doubles in 0.2 hours = 12 minutes
And, A = 2 x 480 = 960
A₀ = 480
n = 12 minutes
r = ?
⇒ A = A₀ (1 + r)ⁿ
⇒ 960 = 480 (1 + r)¹²
⇒ 2 = (1 + r)¹²
⇒ [tex]r = \sqrt[12]{2} - 1[/tex]
Now, to find the quantity after 12 minutes, we plug A = 830, t = 12 minutes in exponential function;
⇒ A = A₀ (1 + r)ⁿ
⇒ [tex]A = 480 (1 + \sqrt[12]{2} - 1)^{12}[/tex]
⇒ A = 480 (2)
⇒ A = 960
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Juan catches 80% of the passes thrown to him in football. If the quarterback throws to him 15 times during a game, what is the probability he will catch atleast 10 of them?
the probability that Juan will catch at least 10 passes out of 15 is approximately 0.9951, or about 99.51%.
We may utilise the binomial distribution formula to resolve this issue:
P(X ≥ 10) = 1 - P(X < 10)
where P(X 10) is the likelihood that Juan catches fewer than 10 passes and X is the total number of passes he catches.
By applying the binomial distribution formula, we may determine P(X 10):
P(X < 10) = Σ(k=0 to 9) [(15 select k)×(0.8)k×(0.2)×(15-k)]
where (15 select k) is the binomial coefficient and (0.8)k×(0.2)(15-k) is the probability of catching k passes and missing (15-k) passes, and (15 choose k) is the number of possibilities to choose k items out of 15.
P(X 10) can be calculated using a calculator or a spreadsheet and is roughly equal to 0.0049.
P(X 10) is therefore equal to 1 - P(X 10) 1 - 0.0049 0.9951.
So the probability that Juan will catch at least 10 passes out of 15 is approximately 0.9951, or about 99.51%.
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The average rate of change of the given function between the two given values is: 0.82
How to find the average rate of change of a function?The Average Rate of Change of a function is one that is defined as the average rate at which one quantity is changing with respect to another thing changing. Thus, an average rate of change function is basically a process that helps to calculate the amount of change in one item divided by the corresponding amount of change in another.
It is given by the formula:
f'(x) = [f(b) - f(a)]/(b - a)
We are given the function:
f(x) = √(3x - 11)
Thus from x1 = 4 to x2 = 6, we will have:
f(4) = √(3*4 - 11) = 1
f(6) = √(3*6 - 11) = √7
Thus:
f'(x) = ((√7) - 1)/(6 - 4)
f'(x) = 0.82
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42 SING A chef makes batches of dumpling wrappers for shrimp dumplings. She uses 2 cups of flour for each batch and a total of an additional § cup of flour to keep the dough from sticking. She uses 21 cups of flour to make n batches of dumpling wrappers. How many batches of dumpling wrappers does the chef make?
The chef makes 11 batches of dumpling wrappers.
What is dumpling wrapper?
Dumpling wrappers are thin pieces of dough that are used to wrap around fillings to make dumplings. They are a staple in many Asian cuisines, including Chinese, Japanese, Korean, and Thai. The dough is typically made with flour, water, and salt, and is rolled out into thin circles or squares. The size and thickness of the wrappers can vary depending on the type of dumpling being made and the regional cuisine. Dumpling wrappers can be used to make a variety of dumplings, such as pork and cabbage dumplings, shrimp dumplings, potstickers, and gyoza.
Let's assume that the chef makes "n" batches of dumpling wrappers.
For each batch of dumpling wrappers, the chef uses 2 cups of flour.
So, the total amount of flour used for making "n" batches of dumpling wrappers is 2n cups of flour.
In addition to the 2 cups of flour for each batch, the chef also uses an additional § cup of flour to keep the dough from sticking. So, the total amount of flour used for "n" batches is 2n + § cups of flour.
According to the problem statement, the chef uses a total of 21 cups of flour to make "n" batches of dumpling wrappers. So, we can set up the following equation 2n + § = 21
To solve for "n," we need to first convert the § cup of flour to a decimal. § is equivalent to 0.25 cups of flour.
So, the equation becomes 2n + 0.25 = 21
Subtracting 0.25 from both sides,
2n = 20.75
Dividing both sides by 2,
n = 10.375
Since we can't have a fraction of a batch, we can round up to the nearest whole number. Therefore, the chef makes 11 batches of dumpling wrappers.
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The chef makes 11 batches of dumpling wrappers.
What is dumpling wrapper?
Dumpling wrappers are thin pieces of dough that are used to wrap around fillings to make dumplings. They are a staple in many Asian cuisines, including Chinese, Japanese, Korean, and Thai. The dough is typically made with flour, water, and salt, and is rolled out into thin circles or squares. The size and thickness of the wrappers can vary depending on the type of dumpling being made and the regional cuisine. Dumpling wrappers can be used to make a variety of dumplings, such as pork and cabbage dumplings, shrimp dumplings, potstickers, and gyoza.
Let's assume that the chef makes "n" batches of dumpling wrappers.
For each batch of dumpling wrappers, the chef uses 2 cups of flour.
So, the total amount of flour used for making "n" batches of dumpling wrappers is 2n cups of flour.
In addition to the 2 cups of flour for each batch, the chef also uses an additional § cup of flour to keep the dough from sticking. So, the total amount of flour used for "n" batches is 2n + § cups of flour.
According to the problem statement, the chef uses a total of 21 cups of flour to make "n" batches of dumpling wrappers. So, we can set up the following equation 2n + § = 21
To solve for "n," we need to first convert the § cup of flour to a decimal. § is equivalent to 0.25 cups of flour.
So, the equation becomes 2n + 0.25 = 21
Subtracting 0.25 from both sides,
2n = 20.75
Dividing both sides by 2,
n = 10.375
Since we can't have a fraction of a batch, we can round up to the nearest whole number. Therefore, the chef makes 11 batches of dumpling wrappers.
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Calculate the height of the pole
Step-by-step explanation:
See image
Given f(x)=x2−4‾‾‾‾‾‾√ and g(x)=3x2, what expression represents (f∘g)(x)?
Enter your answer, in simplest form, in the box.
(f∘g)(x)=
The value of (f∘g)(x) is 3x² - 2.
What is a function?
Each element of X receives exactly one element of Y when a function from one set to the other is used. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. Initially, functions represented the idealized relationship between two changing quantities.
Here, we have
Given: f(x) = √x²-4 and g(x) = 3x²
We have to find the value of (f∘g)(x).
(f∘g)(x) = f(g(x))
(f∘g)(x) = f(3x²)
f(3x²) = √9x⁴- 4
f(3x²) = 3x² - 2
Hence, the value of (f∘g)(x) is 3x² - 2.
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A farmer sells four of his farm products, Maize, potatoes, carrots and tomatoes in each of the two
towns in Rwanda ( Huye and Muhanga) to three classes of customers: Consumers, wholesalers
and retailers in bags as given below:
HUYE
Product
Maize Potatoes Carrots Tomatoes
Consumers 4 6 7 4
Retailers 3 2 1 6
Wholesalers 4 3 5 3
MUHANGA
Product
Maize Potatoes Carrots Tomatoes
Consumers 4 5 3 6
Retailers 7 8 4 4
Wholesalers 2 4 0 1
In order to sell his products in the towns, a farmer pays a commission to salesman, town managers
and Division Manager as shown below.
Salesman
Town
manager
Division
manager
Huye 6% 5% 2%
Muhanga 4% 3% 3%
The selling price per bag is given as follows
Products
Selling price per bag in
Rwf
Maize 200
Potatoes 1,000
Carrots 500
Tomatoes 700
Required: ---------------- 2 Marks per Each
a) Find the total sales in units by product and customer type.
b) Determine the difference between the two towns in sales (in Units) by product and
customer type.
c) Calculate the total sales in Rwf by each town.
d) Find the sales in Rwf by customer type for each town.
e) Compute the amount of commission to be paid by type of commission and type of
customer.
a) To find the total sales in units by product and customer type, we need to multiply the number of bags sold by the selling price per bag. The table below shows the calculations:
Town Product Consumers Retailers Wholesalers
Huye Maize 800 600 800
Potatoes 6,000 2,000 3,000
Carrots 3,500 500 2,500
Tomatoes 2,800 4,200 2,100
Muhanga Maize 800 1,000 0
Potatoes 5,000 8,000 4,000
Carrots 1,500 2,000 0
Tomatoes 4,200 2,800 700
b) To determine the difference between the two towns in sales (in Units) by product and customer type, we subtract the sales in Muhanga from the sales in Huye. The table below shows the calculations:
Town Product Consumers Retailers Wholesalers
Maize 0 -400 800
Huye Potatoes 0 -1,000 0
Carrots 2,000 0 2,500
Tomatoes -1,400 1,000 1,400
Total 600 -400 4,700
Product Consumers Retailers Wholesalers
Maize 0 400 0
Muhanga Potatoes 0 6,000 1,000
Carrots -2,000 -500 0
Tomatoes 1,400 -1,400 -700
Total -600 4,500 300
c) To calculate the total sales in Rwf by each town, we need to add up the sales in units for each town and product and then multiply by the selling price per bag. The table below shows the calculations:
Town Total sales in Rwf
Huye 24,400
Muhanga 23,700
d) To find the sales in Rwf by customer type for each town, we need to multiply the number of bags sold by the selling price per bag and then add up the sales for each customer type. The table below shows the calculations:
Town Customer Maize Potatoes Carrots Tomatoes
Huye Consumers 800 6,000 3,500 2,800
Retailers 600 2
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Good luck to you friend!
CAN SOMEONE HELP WITH THIS QUESTION?
Answer:
39 1/3 = 118/3 square units
Step-by-step explanation:
You want the area under the curve y = -x² +9x on the interval [3, 5].
IntegralThe fundamental theorem of calculus tells us the integral of the function will be its anti-derivative. That is, for f(x) = -x² +9x, we want the function F(x) whose derivative is f(x). The reverse of the power rule for derivatives can be used:
[tex]F(x)=-\dfrac{x^3}{3}+9\dfrac{x^2}{2}[/tex]
AreaThe area of interest is ...
F(5) -F(3) = -1/3(5³ -3³) +9/2(5² -3²) = -98/3 +72
F(5) -F(3) = 39 1/3
The area under the curve between x=3 and x=5 is 39 1/3 square units.
Write about plane of contact , pole and polar plane ?
Explain all briefly!
The concepts of plane of contact, pole, and polar plane are related to the study of conic sections in geometry.
Plane of Contact:
The plane of contact is a plane that passes through the points of intersection of a cone and a plane. In other words, when a plane intersects a cone, the points where the plane and cone touch determine a plane of contact. This plane is important in determining the properties of the conic section that is formed by the intersection.
Pole:
The pole is a point that is used to define a polar plane. It is a point that is not on the plane and is located on the axis of the cone. When a plane passes through the pole, it intersects the cone in a conic section. The pole is a useful concept in determining the properties of the conic section.
Polar Plane:
The polar plane is a plane that passes through the pole and is perpendicular to the plane of contact. It is the plane that contains the polar axis, which is the axis of the cone passing through the pole. The polar plane is important in determining the properties of the conic section that is formed by the intersection of the cone and the plane of contact.
In summary, the plane of contact, pole, and polar plane are all concepts that are used in the study of conic sections. The plane of contact is a plane that passes through the points of intersection of a cone and a plane. The pole is a point on the axis of the cone that is used to define a polar plane. The polar plane is a plane that passes through the pole and is perpendicular to the plane of contact. These concepts are important in determining the properties of conic sections and are used in various applications, including optics, astronomy, and engineering.
Answer:
Step-by-step explanation:
Sure! Let's start with the plane of contact.
Plane of Contact:
The plane of contact is a term commonly used in geometry and physics. When two objects come into contact with each other, their surfaces touch at a particular area. This area of contact is known as the plane of contact. It is a two-dimensional surface that represents the point of contact between the two objects.
Imagine two books placed on a table. The area where the bottom surface of each book touches the table is the plane of contact. It is like a flat sheet that separates the objects and defines the region where they interact with each other.
Pole and Polar Plane:
The concepts of pole and polar plane are closely related to the plane of contact. Let's explore them:
Pole:
In geometry, the pole is a point that represents the point of contact between a line and a plane. When a line is in contact with a plane, the point of contact is called the pole. It is the specific point where the line intersects the plane.
For example, if you have a line placed on a tabletop, the point where the line touches the surface is its pole.
Polar Plane:
The polar plane is the plane that contains all the lines passing through a specific point called the pole. In other words, the polar plane is a plane that is perpendicular to the line at its pole.
Continuing with the previous example, if you have a line touching the tabletop at a certain point, the polar plane would be a plane that is perpendicular to the line at that point. It includes all the lines passing through that point.
The pole and polar plane are connected in such a way that every line passing through the pole lies on the polar plane, and the polar plane contains the pole.
In summary, the plane of contact represents the area where two objects touch each other, while the pole represents the point of contact between a line and a plane. The polar plane is a plane that contains all the lines passing through the pole.
Hope it helps!!
Find the value of x, y, and z in the parallelogram below.
The values of x , y and z in the parallelogram are 12, -20 and -33 respectively
How to find the angles of a parallelogram?A parallelogram is a quadrilateral with opposite side equal to each other and opposite sides parallel to each other.
The opposite angles of parallelogram are equal . Consecutive angles in a parallelogram are supplementary .
Therefore,
-4y + 5 = 85
-4y = 85 - 5
-4y = 80
divide both sides by -4
y = 80 / -4
y = - 20
Therefore, let's find x and z as follows:
8x - 1 + 85 = 180
8x = 180 - 84
8x = 96
divide both sides by 8
x = 96 / 8
x = 12
Let's find z
- 3z - 4 + 85 = 180
-3z = 180 - 81
-3z = 99
divide both sides by -3
z = 99 / -3
z = -33
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Please find the answer for both questions nicely
Answer: d) the amount of clay for each student. 8
Step-by-step explanation: For the sketch, draw 8 pieces of paper being divided evenly. For the expression write 2 divided by 1/4 = 8.
Determine the value(s) for which the rational expression −6a−10−5a−8 is undefined. If there's more than one value, list them separated by a comma, e.g. a=2,3.
Since the rational expression is incorrectly constructed, I am unable to determine what it actually means.
I'll attempt to respond to this in a general approach so that you may try to apply it to the specific issue.
Let's say that this is our expression of reason:
[tex]\frac{-6a-10}{-5a-8}[/tex]
There is no undefined value because the numerator is -12d and this component is unaffected by any value of d.
-5a-8 is the denominator.
Any rational expression that has a denominator equal to zero is now undefined. (we can not divide by zero)
Thus, a value that is undefined will be when
-5a-8 = 0
d = 8/5
This means that the number d = 8/5 is invalid because it would cause the denominator to be equal to 0.
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2/9 ÷2/8_____ reduce the answer to the lowest terms
A survey found that women's heights are normally distributed with mean 63.3 in. and standard deviation 3.8 in. The survey also found that men's heights are normally distributed with mean 67.3 in. and standard deviation 3.8 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 56 in. and a maximum of 64 in. Complete parts (a) and (b) below.
a. Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement park?
The percentage of men who meet the height requirement is ?
The percentage of men who meet the height requirement is 19.09%. This result suggests that a relatively small proportion of men meet the height requirements for the characters at the amusement park.
To find the percentage of men who meet the height requirement, we need to calculate the probability that a randomly selected man has a height between 56 and 64 inches.
Using the z-score formula, we can standardize the height range as follows:
z = (64 - 67.3) / 3.8 = -0.87
z = (56 - 67.3) / 3.8 = -3.00
Using a standard normal distribution table, we can find that the area to the left of z = -0.87 is 0.1922 and the area to the left of z = -3.00 is 0.0013.
Therefore, the percentage of men meeting the height requirement is:
P(-0.87 < Z < -3.00) = P(Z < -0.87) - P(Z < -3.00) = 0.1922 - 0.0013 = 0.1909 or 19.09%.
It suggests that the park may be employing more women than men as characters, as the mean height for women is below the maximum height requirement.
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The frequency table that summarizes the data by the biologist and the additional plants is c. third table.
How to find the frequency table?From the data by the biologist and the additional plants, we see that there are 11 plants with flowers and 7 with caterpillars.
The number of flowering plants with caterpillars is 6 plants. Those without flowers would then be:
= 11 - 6
= 5 plants
This is shown by the second frequency table.
In conclusion, the correct frequency table showing the data by the biologist is the second frequency table.
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Randy and Brenda organize one of their family reunions and have developed the budget shown in the circle graph.
If the total budget for the family reunion is $1,200, then how much will be spent on food and drinks?
A $360
B $660
C $540
D $420
Answer:
C $540
Step-by-step explanation:
Food: .30 × $1,200 = $360
Drinks: .15 × $1,200 = $180
-------
$540
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(Enlarge photo)
Answer:B
Step-by-step explanation:the triangle has 3 sides
A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 7 freshmen, 8 sophomores, 7 juniors, and 7 seniors are eligible to be on the committee, in how many ways can the committee be chosen?
6. Social Security numbers are of the form XYZ - MN - ABCD.
a. The first three digits XYZ are referred to as the area number. The number is associated
with where you registered for your social security card. X is a number 1 - 7. Y and Z can
be any number 0 - 9. Also the area code 666 has never been used. How many area code
numbers are possible?
b. The middle two digits MN are referred to as the group number. The group number can
be any number from 01 to 99 (it cannot be 00). The last 4 digits ABCD are the serial
number. The serial number can be any from 0001 to 9999 (it cannot be 0000).
Additionally, the social security number 123-45-6789 is only used in advertisements.
How many social security numbers are possible?
a) The number of possible area code numbers is: 69,299. b) the number of possible social security numbers is: 68,665,031,401
How to determine the How many area code numbers are possiblea. The first three digits can be any number from 001 to 999, except for 666. Since X can be any number from 1 to 7, there are 7 choices for X. For Y and Z, there are 10 choices each (0-9). Therefore, the number of possible area code numbers is:
(7 choices for X) x (10 choices for Y) x (10 choices for Z) - 1 (for 666) = 69,299
b. The middle two digits can be any number from 01 to 99, which gives 99 choices. The last 4 digits can be any number from 0001 to 9999, which gives 9999 - 1 (for 0000) = 9998 choices. Therefore, the number of possible social security numbers is:
(69,299 choices for the area number) x (99 choices for the group number) x (9998 choices for the serial number) - 1 (for 123-45-6789) = 68,665,031,401
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Change 14,277 s to h, min, and s.
Answer:
3hours 237 minutes 5700 seconds
The diameter of the lead in a pencil p must be within 0.01 millimeters of 1 millimeter
The diameter of the lead in a pencil p must be within 0.01 millimetre of 1 millimeter. In the form of inequality, this can be expressed as 0.01<p<1.
A mathematical expression with values that on the left and right are not equal is referred to as having an inequality. In essence, an inequality examines any two values as well as demonstrates whether one is bigger than, less than, or neither greater nor less than the comparable quantity. Numbers are compared using inequalities to determine the range of quantities that fall inside the parameters of a variable. The diameter of the lead in a pencil p must be within 0.01 millimetre of 1 millimeter. This can be expressed as 0.01<p<1 inequality.
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Your question is incomplete but most probably your full question was,
The diameter of the lead in a pencil p must be within 0.01 millimetre of 1 millimeter. Express this statement of an inequality.
george and jenna each completed a 12 miles run. this graph shows the total distance in miles that genna ran over time
In the first hour of their run, Jenna ran 1 mile more than George.
Given graph shows the data on Jenna's running status.
It shows that Jenna ran 12 miles in 2 hours of time.
Jenna's running status in the first one hour = 12/2 = 6 miles.
Given that George ran at a constant of 5 miles per hour.
To find out how many miles will Jenna ran more than George in the first one hour = 6 -5 = 1 mile.
From the above analysis, we can conclude that the Jenna will run 6 miles in the first one hour and George will run 5 miles in the first one hour. So, Jenna will run 1 mile more than George.
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Given question is not having enough information, the complete question is attached below,
Esports the Titian determined that the probability of a certain rugby team winning its next match is 11/23. Find the odds of the next match
The odds of the next match from the probability of 11/23 is 11 : 12
Finding the odds of the next matchFrom the question, we have the following parameters that can be used in our computation:
Probability = 11/23
For a probability represented as a/b, the odds is
Odds = a : b - a
In this case
a = 11 and b = 23
Using the above as a guide, we have the following:
Odds = 11 : 23 - 11
Evaluate the difference
So, we have
Odds = 11 : 12
Hence, the value of the odds is 11 : 12
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5. a. A box contains 20 pens of which m are red. If 5 more pens of which 3 are red are added, the probability of selecting two red pens at random without replacement is 7/20 Find the value of m. b. The first four consecutive terms of a linear sequence are x,y, (2x + 1), (2y-3). i. Show that S4= 3(x+y) - 2. ii. Find the fifth term, U5
a. There are 4 red pens in the box.
b i. Simplifying, we get:
S₄ = 3(x + y) - 2
ii. The fifth term is 4y - 2x + 1.
What is probability?The chance of an event occurring is called the probability of the event happening. It tells us how likely it is for an event to happen; it does not tell us what is going to happen. There is an even chance of an event happening (happen/not happen).
a. Let the number of non-red pens be n, then we have m + n = 20. Also, after adding 5 more pens, we have m + 3 red pens and n + 2 non-red pens. The probability of selecting two red pens at random without replacement from these 25 pens is given by:
(m + 3)/(20 + 5) * (m + 2)/(20 + 4) = 7/20
Simplifying this equation, we get:
(m + 3)(m + 2) = 14 * 5
Expanding the left side and simplifying, we get:
m² + 5m - 36 = 0
Factoring this equation, we get:
(m + 9)(m - 4) = 0
Since m cannot be negative, we have:
m = 4
Therefore, there are 4 red pens in the box.
b. i. The sum of the first n terms of an arithmetic sequence can be given by:
Sₙ = n/2[2a + (n - 1)d]
where a is the first term, d is the common difference, and n is the number of terms. Using this formula, we can find S₄ as follows:
S₄ = 4/2[x + y + (2x + 1) + (2y - 3)]
= 2(3x + 3y - 1)
= 6(x + y) - 6 - 2
Simplifying, we get:
S₄ = 3(x + y) - 2
ii. The common difference of the sequence is given by:
d = y - x
Therefore, the fifth term can be expressed as:
U₅ = (2x + 1) + 4d
= (2x + 1) + 4(y - x)
= 4y - 2x + 1
Therefore, the fifth term is 4y - 2x + 1.
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a. There are 4 red pens in the box.
b i. Simplifying, we get:
S₄ = 3(x + y) - 2
ii. The fifth term is 4y - 2x + 1.
What is probability?The chance of an event occurring is called the probability of the event happening. It tells us how likely it is for an event to happen; it does not tell us what is going to happen. There is an even chance of an event happening (happen/not happen).
a. Let the number of non-red pens be n, then we have m + n = 20. Also, after adding 5 more pens, we have m + 3 red pens and n + 2 non-red pens. The probability of selecting two red pens at random without replacement from these 25 pens is given by:
(m + 3)/(20 + 5) * (m + 2)/(20 + 4) = 7/20
Simplifying this equation, we get:
(m + 3)(m + 2) = 14 * 5
Expanding the left side and simplifying, we get:
m² + 5m - 36 = 0
Factoring this equation, we get:
(m + 9)(m - 4) = 0
Since m cannot be negative, we have:
m = 4
Therefore, there are 4 red pens in the box.
b. i. The sum of the first n terms of an arithmetic sequence can be given by:
Sₙ = n/2[2a + (n - 1)d]
where a is the first term, d is the common difference, and n is the number of terms. Using this formula, we can find S₄ as follows:
S₄ = 4/2[x + y + (2x + 1) + (2y - 3)]
= 2(3x + 3y - 1)
= 6(x + y) - 6 - 2
Simplifying, we get:
S₄ = 3(x + y) - 2
ii. The common difference of the sequence is given by:
d = y - x
Therefore, the fifth term can be expressed as:
U₅ = (2x + 1) + 4d
= (2x + 1) + 4(y - x)
= 4y - 2x + 1
Therefore, the fifth term is 4y - 2x + 1.
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