In mathematical induction, prove that n lines separate the plane into (n^2 + n + 2)/2 regions if no two of these lines are parallel and no three pass through a common point.
Mathematical induction is a mathematical proof method that is used to demonstrate a statement or formula for all values of n, where n is a positive integer. If we use induction, we can show that the formula is true for n = 1, and we can also show that if the formula is true for n = k, then it is also true for n = k + 1.
Proof for n lines separating the plane into (n^2 + n + 2)/2 regions is given below:
Base Case: The theorem is true for n = 1. When we draw one line in the plane, we see that it splits the plane into two regions. Hence the formula is true for n = 1.
Induction Hypothesis: We believe that the formula is true for k lines. That is, k lines split the plane into (k^2 + k + 2)/2 regions.
Induction Step: We want to demonstrate that the formula is also true for k + 1 lines. We first take an arbitrary line from these k + 1 lines, which we call l. We notice that this line splits the plane into two regions.
Now, for the remaining k lines, we make an induction argument. We are sure that the formula is true for k lines. Thus, the k lines split the plane into (k^2 + k + 2)/2 regions. We know that these k lines intersect the line l at k points. Thus, by adding line l, we create k + 1 regions on the plane between these lines.
We now consider the line l itself. It can't cross any of the other k lines, or it would not meet our requirements. Therefore, it crosses each of the k existing lines, generating k + 1 areas. Thus, with the inclusion of line l, the number of regions on the plane is (k^2 + k + 2)/2 + k + 1 = (k^2 + 3k + 4)/2.
The formula for k + 1 is (k + 1)^2 + (k + 1) + 2 = k^2 + 3k + 4, and it is thus identical to the formula for the number of regions when k lines are drawn on the plane.
Therefore, the statement is true for all positive integers n.
Therefore, we have proved that n lines separate the plane into (n^2 + n + 2)/2 regions if no two of these lines are parallel and no three pass through a common point, by using mathematical induction.
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I know I'm using this app a ton today.
...A store pays $261 for a diving board and marks the price up by 45%. What is the amount of the mark-up?
Answer:
If the store marks up the price of the diving board by 45%, the selling price will be 100% + 45% = 145% of the cost price.
Let's calculate the selling price:
Selling price = Cost price + Mark-up
Mark-up = Selling price - Cost price
Mark-up = (145% of cost price) - cost price
Mark-up = 0.45 * cost price
We know that the store paid $261 for the diving board, so:
Mark-up = 0.45 * $261 = $117.45
Therefore, the amount of the mark-up is $117.45.
well, what's 45% of 261?
[tex]\begin{array}{|c|ll} \cline{1-1} \textit{\textit{\LARGE a}\% of \textit{\LARGE b}}\\ \cline{1-1} \\ \left( \cfrac{\textit{\LARGE a}}{100} \right)\cdot \textit{\LARGE b} \\\\ \cline{1-1} \end{array}~\hspace{5em}\stackrel{\textit{45\% of 261}}{\left( \cfrac{45}{100} \right)261}\implies \text{\LARGE 117.45}[/tex]
A 45-Seater bus is fully booked. The cost of an adult ticiket is R375 and the cost of a student ticket is R200. How many students were on the bus, If total ticket sales amounted to R9875? (The bus can seat 45 passenger
The number of students on the bus is 40.
How many students are in the bus?The first step is to form two equations that represent the information in the question:
a + b = 45 equation 1
375a + 200b = 9875 equation 2
Where:
a = number of adults in the bus
b = number of students in the bus
The elimination method would be used to determine the number of students on the bus.
Multiply equation 1 by 375
375a + 375b = 16,875 equation 3
Subtract equation 2 from equation 3
175b = 7000
Divide both sides of the equation by 175
b = 7000 / 175
b = 40
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Each morning, Sleepwell Hotel offers its guests a free continental breakfast with pastries and orange juice. The hotel served 180 gallons of orange juice last year. This year, the hotel served 70% more orange juice than it did the previous year. How much was served this year?
The hotel served 306 gallons of orange juice this year.
To find the amount of orange juice served this year, we need to add 70% more of the amount served last year to the amount served last year. Let's denote the amount served last year as "x". Then we can set up the equation:
Amount served this year = x + 0.7xSimplifying this equation gives us:
Amount served this year = 1.7xWe know from the problem that the amount served last year was 180 gallons. Plugging this into our equation, we get:
Amount served this year = 1.7(180)Simplifying this equation gives us:
Amount served this year = 306Therefore, the hotel served 306 gallons of orange juice this year.
In summary, we used the information given in the problem to set up an equation and solve for the amount of orange juice served this year. We first found the amount served last year, and then added 70% more of that amount to get the total amount served this year.
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Can someone assist me?The cheerleading squad at Morristown High School had 20 members with the old coach. Now, with the new coach, there are 45% more members on the squad. How many members are on the squad now?
Answer:
29
Step-by-step explanation:
First, let's find out how many members are added to the squad with the new coach:
45% of 20 = 0.45 x 20 = 9
So, with the new coach, there are 9 more members added to the squad.
Now, to find the total number of members on the squad with the new coach, we just need to add the old number of members (20) to the number of new members added (9):
20 + 9 = 29
Therefore, there are now 29 members on the cheerleading squad at Morristown High School with the new coach.
A car travels at an average speed of 100 km/h and covers a certain distance in 3 hours 40 minutes. At what average speed must it travel to cover the same distance in 3 hours and 30 minutes
show steps
The car must travel at an average speed of 105 km/h to cover the same distance in 3 hours and 30 minutes.
How to deal with the Speed and time?Let's start by calculating the distance covered by the car in 3 hours 40 minutes.
3 hours 40 minutes is equal to 3.67 hours.
Distance = Speed × Time
Distance = 100 km/h × 3.67 h
Distance = 367 km
Now, let's find the average speed the car must travel to cover the same distance in 3 hours and 30 minutes.
3 hours and 30 minutes is equal to 3.5 hours.
Distance = Speed × Time
367 km = Speed × 3.5 h
Speed = Distance / Time
Speed = 367 km / 3.5 h
Speed = 105 km/h
Therefore, the car must travel at an average speed of 105 km/h to cover the same distance in 3 hours and 30 minutes.
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Complete the recursive formula of the arithmetic sequence -16, -33, -50, -67,. −16,−33,−50,−67,. Minus, 16, comma, minus, 33, comma, minus, 50, comma, minus, 67, comma, point, point, point. C(1)=c(1)=c, left parenthesis, 1, right parenthesis, equals
c(n)=c(n-1)+c(n)=c(n−1)+c, left parenthesis, n, right parenthesis, equals, c, left parenthesis, n, minus, 1, right parenthesis, plus
The following is the recursive formula for the arithmetic sequence in this issue:
c(1) = -16.
c(n) = c(n - 1) - 17.
An arithmetic sequence is a series of numbers where each term is obtained by adding a fixed constant, known as the common difference, to the previous term. For example, in the sequence 2, 5, 8, 11, 14, 17, each term is obtained by adding 3 to the previous term.
The formula for finding the nth term of an arithmetic sequence is: a(n) = a(1) + (n-1)d, where a(1) is the first term, d is the common difference, and n is the term number. For example, to find the 10th term of the sequence 2, 5, 8, 11, 14, 17, we would use the formula a(10) = 2 + (10-1)3 = 29. Arithmetic sequences have many practical applications, such as in finance, where they can be used to calculate the interest earned on an investment over time.
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Mrs. Simpkins is buying candy for her daughters class. She wants to give each student 5 pieces of candy and there are 23 students in her daughters class. How much candy will she need to get?
Answer:
115
Step-by-step explanation:
5 times 23= answer
X: -1, 0, 1, 2
g(x): 3, 10, 17, 24
What is the rate of change over the interval [-1,2]? Explain how you know
Answer:
あたしの最後はあなたがいい (いい)
あなたとこのままおサラバするより
死ぬのがいいわ
死ぬのがいいわ
三度の飯よりあんたがええのよ
あんたとこのままおサラバするよか
死ぬのがいいわ
死ぬのがいいわ
それでも時々 浮つく
Step-by-step explanation:
What is the Smallest Positive Integer with at least 8 odd Factors and at least 16 even Factors?
Answer:
Step-by-step explanation:
120
The mail carrier has to deliver 3 boxes. The first box has a mass of 80 kilograms, the second box has a mass of 40 kilograms, and the third box has a mass of 60 kilograms. What is the total mass of all 3 boxes in grams?
180 grams
1,800 grams
18,000 grams
180,000 grams
Answer: 180,000 g
Step-by-step explanation:
1st add all masses together:
80kg + 40kg + 60 kg = 180 kg
then we need to convert kilograms to grams:
1 kg = 1000g
180kg * (1000g / 1kg) = 180,000 g
To calculate the total mass (in grams) of the three boxes, we first need to convert their masses from kilograms to grams and then add them together.
The mass of the first box is 80 kg, which is equivalent to 80,000 grams (because 1 kg = 1000 grams).
The mass of the second box is 40 kilograms, or 40,000 grams.
The mass of the third box is 60 kilograms, or 60,000 grams.
To find the total mass of the three boxes, we add these values:
80,000 grams + 40,000 grams + 60,000 grams = 180,000 grams
Therefore, The total mass of the three boxes in gram is 180,000
The correct answer is D) 180,000
imagine you took an assessment on your math ability at one time point and then the same assessment a month later. if your math ability was the same between time 1 and time 2, and nothing substantial happened during that time, such as getting a tutor, which type of reliability for the math ability assessment was achieved? group of answer choices
The test has demonstrated a good level of test-retest reliability.
If a student took an assessment on their math ability at one time point and then the same assessment a month later, with no substantial changes such as getting a tutor, and their math ability was the same between time 1 and time 2, then the assessment has achieved Test-Retest Reliability.Test-Retest Reliability: Test-Retest reliability is the measure of consistency of a test over time. A test has test-retest reliability if a person performs similarly on the same test taken at two different times.A reliable test must always provide consistent results. Therefore, if the math ability was the same between time 1 and time 2, and no substantial changes occurred during that time, then the test has demonstrated a good level of test-retest reliability.
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A man sells an article at rs 600and makes a profit of 20%. Calculate his profit percentage
Answer:
120
Step-by-step explanation:
20 percent of 600 is 120 so he will get 120
Select all the correct answers.
Which three equations are equivalent to |2x + 1| − 10 = -3?
x = -4 or x = 3
2x − 1 = -7 or 2x − 1 = 7
|2x + 1| = 7
x = 4 or x = -3
|2x − 1| = 7
2x + 1 = -7 or 2x + 1 = 7
Answer:
|2x + 1| = 7
|2x − 1| = 7
2x + 1 = -7 or 2x + 1 = 7
Step-by-step explanation:
or if u have no time
C, E, F.
These might not be in order but hey they are correct amiright :/
If r(x) is a rational function in simplest form where the degree of the numerator is 3 and the degree of the denominator is 1, then
r(x) has no horizontal asymptote
r(x) has a nonzero horizontal asymptote
r(x) has a horizontal asymptote at y=0
If r(x) is a rational function in simplest form where the degree of the numerator is 3 and the degree of the denominator is 1, a)then r(x) has a horizontal asymptote at y=0.
This is because the degree of the denominator is greater than the degree of the numerator, which means that as x gets very large or very small, the denominator will dominate the behavior of the function.
As a result, the function will approach zero, and thus, there is a horizontal asymptote at y=0. If the degree of the numerator were greater than or equal to the degree of the denominator, then the function could have a horizontal asymptote at a nonzero value or no horizontal asymptote at all.
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Write the expression in complete factored form.
3b(4 - 8) - n(u - 8) =please help
Answer:
12b (1-2) -n (u-8) this is the answer
A triangle can have sides with the following measures: 1, 1, 2
True
False
Answer: false
Step-by-step explanation:
Four pipes can fill a tank in 16 hours. How long will it take to fill the tank if twelve
pipes of the same dimensions are used ?
Answer:
5.333 hours
Step-by-step explanation:
We know
4 Pipes fill a tank in 16 hours.
How long will it take to fill the tank if 12 pipes of the same dimensions are used?
We Take
16 x 1/3 = 5.333 hours
So, it takes about 5.333 hours to fill the tank.
Find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 3.
The volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 3 is 12 cubic units.
To calculate the volume, we can use the formula V = l × w × h, where l = length, w = width and h = height.
The three faces lie on the coordinate planes, so we can use the equation x + 2y + 3z = 3 to find the coordinates of the vertex. We can solve for the values of x, y, and z and plug them into the formula.
We will use the substitution method:
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What is the value of x in the triangle to the right? (7x+3) 85 50
Answer: x = 6
Step-by-step explanation:
(7x+3)+85+50 = 180
(7x+3)+135 = 180
7x+3 = 180 - 135 = 45
7x = 45-3 = 42
x = 42 / 7 = 6
x = 6
please help me !!!!!!
The equation representing total fraction strip is ³/₃ + ¹/₃ = ⁴/₃.
option B.
What is a fraction?
A fraction is a mathematical representation of a part of a whole or a ratio between two numbers. It consists of a numerator, which represents the number of parts being considered, and a denominator, which represents the total number of parts in the whole.
For this case, 1 is divided into, and 1 divide into 3.
To obtain the total fractions, we will add the individual fractions as shown below;
For this first fraction = ¹/₃ + ¹/₃ + ¹/₃
For the second fraction = ¹/₃
Total fraction = 3(¹/₃ + ¹/₃ + ¹/₃) + ¹/₃
Total fraction = ³/₃ + ¹/₃ = ⁴/₃
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Find the value of x.
22
39
X
The value of x in the right triangle when calculated is approximately 13.8 units
Calculating the value of x in the triangleGiven the right-angled triangle
The side length x can be calculated using the following sine ratio
So, we have
sin(39) = x/22
To find x, we can use the fact that sin(39 degrees) = x/22 and solve for x.
First, we can use a calculator to find the value of sin(39 degrees), which is approximately 0.6293.
Then, we can set up the equation:
0.6293 = x/22
To solve for x, we can multiply both sides by 22:
0.6293 * 22 = x
13.8446 = x
Rewrite as
x = 13.8446
Approximate the value of x
x = 13.8
Therefore, x is approximately 13.8 in the triangle
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11x + 9y=-20 x= -5y-6
Use substitution method pls
The solution to the system of equations is (x, y) = (1, -1) where the given equations are 11x+9y=-20 and x=-5y-6.
What is substitution method?The substitution method is a technique used in algebra to solve systems of equations by replacing one variable with an expression containing another variable. The goal is to eliminate one of the variables so that we can solve for the other one.
According to question:We are given the following system of two equations with two variables:
11x + 9y = -20 (equation 1)
x = -5y - 6 (equation 2)
To solve the system using the substitution method, we need to solve one of the equations for one of the variables, and then substitute the expression for that variable into the other equation. Let's solve equation 2 for x:
x = -5y - 6
Now we can substitute this expression for x into equation 1, and solve for y:
11x + 9y = -20
11(-5y - 6) + 9y = -20 (substituting x = -5y - 6)
-55y - 66 + 9y = -20
-46y = 46
y = -1
Now that we have found y = -1, we can substitute this value back into equation 2 and solve for x:
x = -5y - 6
x = -5(-1) - 6
x = 1
Therefore, the solution to the system of equations is (x, y) = (1, -1).
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What is the decimal of 2 75/100
2.75 is the decimal of fraction .
In math, what is a fraction?
The amount is represented mathematically as a quotient, where the numerator and denominator are split. In a simple fraction, both are integers. A complicated fraction includes a fraction, either in the denominator or the numerator.
The numerator and denominator must be smaller in a proper fraction. A fraction is a number that is a component of a whole. A whole is appraised by dissecting it into many sections. Half of a whole number or item, for instance, is represented by the number 12.
= [tex]2\frac{75}{100}[/tex]
= [tex]2\frac{3}{4}[/tex]
= 11/4
= 2.75
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Some number minus 24 is equal to the product of the number and -18
Answer:
x = 24/19
Step-by-step explanation:
Let's use algebra to solve this problem.
Let's say the number we're looking for is x.
According to the problem, "Some number minus 24" can be written as "x - 24".
And "the product of the number and -18" can be written as "x*(-18)" or simply "-18x".
So the equation we need to solve is:
x - 24 = -18x
We can solve for x by adding 18x to both sides of the equation:
x + 18x - 24 = -18x + 18x
Simplifying the left side of the equation gives:
19x - 24 = 0
Then we can add 24 to both sides of the equation to isolate the x term:
19x - 24 + 24 = 0 + 24
Simplifying the left side of the equation gives:
19x = 24
Finally, we can solve for x by dividing both sides of the equation by 19:
x = 24/19
Lucy initially invested $1,000 in a stock. The value of the stock increased exponentially over time by a rate of 3%. After 5 years, what is the value of the stock
Answer:
Were sorry! Answer is not available right now check in later.
Step-by-step explanation:
Suppose the entire pink arc measures 200 degrees and the entire blue arc measures 50 degrees, what would be measure of the manilla angle be?
Move the pink point so it becomes tangent to the circle
Answer #1 within this context
The answer of the given question based on the circle on finding the measure of the manilla angle the answer will be the measure of the manila angle is 110° degrees.
What is Circle?A circle is closed, two-dimensional shape that consists of all points in plane that are equidistant from given point, called center of the circle. The distance from center to any point on circle is called radius of the circle.
The circumference of circle is distance around the circle, and it is equal to 2π times radius. Circles are fundamental concept in geometry and are used in many fields, like physics, engineering, and architecture. They are also used to represent cycles, cycles of time, and the cyclical nature of life.
If the entire pink arc measures 200° degrees and the entire blue arc measures 50° degrees, then the sum of the measures of the three arcs (pink, blue, and manila) is 360° degrees, which is the total measure of a circle.
To find the measure of the manila angle, we need to subtract the measures of the pink and blue arcs from 360° degrees:
360° degrees - 200° degrees - 50° degrees = 110° degrees
Therefore, the measure of the manila angle is 110° degrees.
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find a homogeneous linear differential equation with constant coefficients whose general solution is given.
A homogeneous linear differential equation with constant coefficients has the form a_
n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = 0,
where a_n, a_{n-1}, etc. are all constants. The general solution of this equation is given by y = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t} + ... + c_n e^{\lambda_n t}, where c_1, c_2, etc. are constants and \lambda_1, \lambda_2, etc. are the roots of the characteristic equation a_n \lambda^n + a_{n-1} \lambda^{n-1} + ... + a_1 \lambda + a_0 = 0.
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What value of z should we use when making a 98% confidence interval for p? О 2.33 1.75 o It's impossible to make a 98% CI O 2.88
The z value when making 98% confidence interval for p will be 2.33.
When making a confidence interval for a proportion, we use the standard normal distribution, and the value of z depends on the level of confidence we want to achieve.
In this case, we want to make a 98% confidence interval, which means that we want to be 98% confident that the true proportion falls within our interval.
To determine the value of z, we can use a z-table or a calculator. The z-value corresponding to a 98% confidence level is 2.33. Therefore, we use 2.33 as our value of z when making a 98% confidence interval for p.
It is not impossible to make a 98% confidence interval, and the value of z is not 2.88. The z-value of 2.88 corresponds to a much higher confidence level of approximately 99.5%. Using a higher confidence level means we can be more confident that our interval contains the true proportion, but it also means that our interval will be wider, which reduces its precision.
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Ryan buys some jumpers to sell on a stall. He spends £190 buying 80 jumpers. He sells 50% of the jumpers for £12 each. He then puts the rest of the jumpers on a Buy one get one half price offer. He manages to sell half the remaining jumpers using this offer. How much profit does Ryan make?
Ryan makes a profit of £240. the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.
Ryan spends £190 to buy 80 jumpers. He sells 50% of the jumpers, i.e. 40 jumpers, at £12 each. This brings the total sales to £480. Then, he puts the remaining 40 jumpers on a Buy one get one half price offer. He sells 20 of the remaining jumpers using this offer. Therefore, the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.
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I need help on this asap!
The solutions for the systems of inequalities are:
a) (0, -50), (0, -100), (0, -125)
b) (0, 20), (0, 23) , (0, 24).
How to identify 3 solutions of each system?
When we have a system of inequalities, a solution is a point (x, y) that solves both ienqualities at the same time.
The first one is:
y ≤ x - 8
y < -3x - 9
Here y must be smaller than x, then we can define x like x = 0, and really small values for y, like y = -50, replacing that we will get:
-50 ≤ 0 - 8 = -8
-50 < - 3*0 - 9 = -9
Both of these are true, so (0, -50) is a solution, and trivially, other solutions of the system of inequalities can be things like (0, -100) and (0, -125) are other two solutions.
For the second system:
y > 5x + 1
y > 3
Let's do the same thing, x = 0 and y gets really large values, like y = 20
20 > 5* + 1 = 1 this is true.
20 > 3 this is true.
so (0, 20) is a solution, and also are (0, 23) and (0, 24).
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