Answer:
Step-by-step explanation:
Part A: you can just restate parts of the text to make a vaild argument or you can make you own
Part B: he should measure N, O, P and Qas
Triangles MNO and QPO are congruent. To find the width of the creek, Andre should find the measurement of NP
What is congruent triangles?When two triangles are congruent, their corresponding three sides and their three angles match precisely.
Given that
O is the midpoint of NP,
NP is perpendicular to MN, and
NP is perpendicular to PQ.
When two lines intersect at a specific point, the opposite angles formed in this situation are equal.
a) Then ∠MNP =∠QPO=90°
∠MON=∠QOP
Also, NO = OP (by the definition of midpoints)
Therefore, Triangles MNO and QPO are congruent by ASA rule
And,
b) To find the width of the creek, Andre should find the measurement of NP, because according to the figure the width is represented by NP. He should find NO and OP and then add it.
To learn more about congruent triangle, refer to:
brainly.com/question/1675117
#SPJ2
What are the roots of the quadratic equation 2x² 7x +6 0 *?
The roots of the quadratic equation are 3/2 and 2.
What is a polynomial?Polynomial is an equation written as the sum of terms of the form kx^n.
where k and n are positive integers.
We have,
2x² - 7x + 6
We can find the roots by middle-term factorization.
2x² - (4 + 3)x + 6 = 0
2x² - 4x - 3x + 6 = 0
2x(x- 2) - 3(x - 2) = 0
We have,
2x - 3 = 0 and x - 2 = 0
Now,
2x = 3
x = 3/2
x - 2 = 0
x = 2
Thus,
The roots are 3/2 and 2.
Learn more about polynomials here:
https://brainly.com/question/2284746
#SPJ4
Which expression is equivalent to b-2/ab-3? assume a=0, b=0.
O a/b5
O 1/ab5
O a3b/1
O b/a
The expression b/a is equivalent to the given expression b⁻²/ab⁻³ when a≠ 0 and b ≠ 0. So, option D is correct.
What are the equivalent expressions?The two or more expressions are said to be equivalent expressions if their value is the same even if they look different.
This is the same concept as equivalent fractions.
Calculation:The given expression is b⁻²/ab⁻³.
This expression is a fraction.
Simplifying the given expression:
b⁻²/ab⁻³
we know a⁻ⁿ = 1/aⁿ
So, we can write b⁻² = 1/b² and b⁻³ = 1/b³
Then, b⁻²/ab⁻³ = (1/b²)/a(1/b³)
⇒ b⁻²/ab⁻³ = 1/b² × 1/a × b³ = b × 1/a
∴ b⁻²/ab⁻³ = b/a
Thus, the equivalent expression for the given expression is b⁻²/ab⁻³ = b/a.
Learn more about the equivalent expression here:
https://brainly.com/question/15775046
#SPJ1
What is the inverse of 16?
Additive inverse of 16 is -16.
Multiplicative Inverse of 16 is 1/16.
What is Inverse ?To produce a result equal to 0, add the original number and the additive inverse by simply reversing the sign of the integer. On the basis of negation of the original number, the characteristics of additive inverse are listed below. For instance, if x is the starting number, then -x is x's additive inverse.
A multiplicative inverse or reciprocal for a number x is a number that, when multiplied by x, produces the multiplicative identity, or 1. It is represented by 1/x or x/1. A fraction a/b has a multiplicative inverse of b/a. Divide 1 by the real number to obtain the multiplicative inverse of the number.
A function that "undoes" another function is called an inverse. In other words, if f(x) generates y, then y entered into the inverse of f creates x. An invertible function is one that has an inverse, and the inverse is indicated by the symbol [tex]f^{-1}[/tex].
Additive inverse of 16 is -16.
Multiplicative Inverse of 16 is 1/16.
To learn more about Inverse refer to :
https://brainly.com/question/12693983
#SPJ4
State whether the product of √3 and √9 is rational or irrational. Explain your answer. I’ll mark brainliest
Answer: √3 is irrational whereas √9 is rational
Step-by-step explanation:
3x3=9. 3^2=9. but, ?x?=3 and ?^2=3. there is no number for that
The product of [tex]\sqrt{3}[/tex] and [tex]\sqrt{9}[/tex] is irrational
•
An irrational number is a type of real number which cannot be represented as a simple fraction and [tex]\sqrt{3}[/tex] falls under this category. As fractions are just ratios, being multiplied by another number will not affect the ratio, thus making the product irrational.
How do you graph xy =- 4?
The formula for a horizontal line at y=4 is as follows: There is no slope. 4.0 is the y-intercept.
What are the Slope and intercepts ?
In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction.
The slope expresses how quickly the dependent variable changes when the independent variable changes. The Greek capital letter D or D is frequently used by mathematicians and economics to represent change. Slope contrasts the change in the vertical axis, y, with the change in the horizontal axis, x.
On the x-axis, negative numbers are below the origin, and on the y-axis, they are to the left of the origin. If the x-value is negative, begin at the origin and go to the left; if the y-coordinate is negative, move downward.
Line graphs, also known as XY graphs, are used to visually represent data (the outcomes of an experiment) in order to demonstrate the link between various factors, such as the severity of the disease and time.
To know more about Slope please click here
https://brainly.com/question/29265771
#SPJ4
What is the solution of 2x² 3x 2 0?
The solution of the function 2x² + 3x + 2 = 0 is the zeros obtained after factorizing them which is found to be as -2 and -1 / 2.
The given equation is , 2x² + 3x + 2 = 0
On factorizing this we will get ,
2x² +4x - x +2 = 0
=> 2x(x +2) - ( x + 2) = 0
=> (2x-1) (x + 2) = 0
Therefore , the zeroes are -2 and -1 / 2.
Factorization is a process in which we write a given equation in terms of their prime factors. The result of factorization is known as zeros off a polynomial and are the solution for the given equation.
To learn more about factorization
brainly.com/question/20293447
#SPJ4
What is another way to describe an equilateral triangle?
Answer:
The answer is: Acute.
Step-by-step explanation:
All sides of a equilateral triangle are 60 degrees, meaning it is acute.
find the market and the selling price of the following item. Roundup answers to the nearest sent blank chemistry say costs and 27.5 marked up 32% on cards what is the markup
if x is binomial with n = 5 and p = 0.05, the chance that x is at least 1 is 0.6240
In a case of binomial distribution of of x where n = 5 and p = 0.05, the probability that x is at least 1 is not 0.6240, it is 0.2262.
Therefore, the answer is False.
The formula for binomial distribution is
P = nCx × pˣ × (1 - p)⁽ⁿ⁻ˣ⁾
where P is the probability
n is the total number of trials
x is the number of success trial
p is the probability of success in a single trial
From given details
n = 5
p = 0.05
P( x is at least 1) = 1 - P( x is less than 1)
P( x is at least 1) = 1 - P( x is 0)
P( x is at least 1) = 1 - 5C0 × 0.05⁰ × (1 - 0.05)⁽⁵⁻⁰⁾
P( x is at least 1) = 1 - 1 × 1 × 0.95⁵
P( x is at least 1) = 1 - 0.7738
P( x is at least 1) = 0.2262
--The question is incomplete, answering to the question below--
"If x is binomial with n = 5 and p = 0.05, the chance that x is at least 1 is 0.6240. True or False."
To know more on binomial distribution
https://brainly.com/question/14214595
#SPJ4
How do you compute the mean and standard deviation of the random variable with the given discrete probability distribution?
For a given discrete probability distribution, The random variable also contains discrete values, The mean and standard deviation are computed in terms of Expected values E(X).
What do you mean by random variable?The mathematical formalization of a number or object that is subject to chance events is known as a random variable. It is a function or mapping between a space of potential outcomes, commonly real numbers, to a space of measurement.
What is mean and standard deviation?The arithmetic mean, also known as the arithmetic average or simply the mean or average, is the sum of a set of numbers divided by the total number of numbers in the set. The collection frequently consists of a series of findings from a survey, experiment, or observational study.
The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed throughout a wider range, a low standard deviation suggests that the values tend to be close to the established mean.
For discrete probability distribution with P(x) as the probability of the event 'x'.
Mean = E(X) = x*P(x).
Variance = [tex]E(X^{2})-(E(X))^2[/tex]
standard deviation = [tex]\sqrt {Variance}[/tex]
To learn more about probability distribution visit:
https://brainly.com/question/14210034
#SPJ4
How do you find the height of a rectangle when given the area and the base?
Answer:
Heigh of rectangle: Area divided by base
Step-by-step explanation:
Area of rectangle: Base x Height
So, to find the height, we need to take the Area divided by the Base, and we will get the height of a rectangle!
How do you find the horizontal and vertical asymptote of a rational function?
The vertical asymptote of a function can be determined by setting its denominator equal to zero (0) and solving for x.
The horizontal asymptote of a function can be found by determining the degrees of the numerator and denominator.
What is an asymptote?In Mathematics, an asymptote can be defined as a line which the graph of a given function approaches but would never touch. Additionally, the vertical asymptote of a function simply refers to the value of x which makes its denominator equal to zero (0).
In order to graph any rational function, you should determine the values for which it is undefined. This ultimately implies that, a function is considered as undefined when the value of the denominator is equal to zero, which represents vertical asymptote lines.
Read more on asymptote here: brainly.com/question/28003042
#SPJ1
Julie wanted to match Lisa's obstacle course record of 74.6 seconds. She has already spent forty and one-fourth seconds on wall climbing and 10.56 seconds on the ropes. How much time did she have left to match the record?
23.79 seconds
29.69 seconds
34.35 seconds
64.04 seconds
Julie has only 23.79 seconds left to match the record.
How to calculate time left?
We know that times deals with a continued sequence of existence and events that occurs in an apparently irreversible succession from the past
From the question, the given times are:
Time to match Lisa's record = 74.6 seconds
Time to clin the wall = 10.56 seconds
Time already spent on climbing wall = 4.025 seconds
Time left is calculated thus
74.600 seconds =(40.025+10.56) seconds
=74.600-50.585
=23.79 seconds
Therefore time left is 23.79 seconds
Learn more about time on https://brainly.com/question/28050940
#SPJ1
How many circles are drawn?
we can draw in finite number circle from one point. so there are n number of possibility to draw circle from on given single point.
what is circle?
Every point in the plane that is a certain distance away from a certain point forms a circle (center). It is, thus, a curve formed by points moving in the plane at a fixed distance from a point. A circle is a closed two-dimensional object where every pair of points in the plane are equally spaced out from the "center." A line that goes through the circle creates a specular symmetry line. At every angle, it is also rotationally symmetric about the center.
Area of circle =
[tex]\pi r^2\\r = 10\\A = 3.14 X 20 X 20\\A = 628unit sq\\[/tex]
To know more about circle visit:
https://brainly.com/question/29142813
#SPJ4
solve:
y=x-4
y=6x-10
Answer:
x=6/5
Step-by-step explanation:
x-4=6x-10
-5x=-6
x=6/5
y=6/5-4=-14/5
Answer:
x = 1.2
y = -2.8
Step-by-step explanation:
y = x - 4 and y = 6x - 10
so x - 4 = 6x - 10
x - 6x = 4 - 10
-5x = -6
x = -6/-5
x = 1.2
y = x - 4 = 1.2 - 4 = -2.8
When the coordinates 1 1 4 4 7 1 and 4 − 2 are joined which shape is formed 5 points group of answer choices parallelogram rectangle trapezoid Square?
Given the provided coordinates of the points, The shape that is formed is a parallelogram because the slopes of the opposite sides are same.
What do you mean by slope?In mathematics, a line's slope, also known as its gradient, is a numerical representation of the line's steepness and direction. Divide the difference in elevation between two points by the distance between them to find the percent slope, then multiply the result by 100. The term "rise" refers to the difference in elevation between two sites. The run is the separation between the points. Percent slope is therefore equal to (rise / run) x 100.
What do you mean by coordinates?A coordinate system in geometry is a method for determining the precise location of points or other geometrical objects on a manifold, such as Euclidean space, using one or more numbers, or coordinates.
slope between (1,1) and (4,4),
m1 = 4-1/4-1 =3/3 =1
slope between (4,-2) and (7,1),
m2= 1+2/7-4 = 3/3 =1
m1=m2
similarly, slope between (1,1) and (4,-2),
m3= -2-1/4-1 = -3/3 =-1
slope between (7,1) and (4,4)
m4= 1-4/7-4 = -3/3 =-1
m3=m4
So, slopes of opposite sides are same that means they are parallel, The shape is parallelogram.
To learn more about slope visit:
brainly.com/question/3605446
#SPJ4
Makayla is decorating the outside of a box in the shape of a triangular prism. The figure below shows a net for the box.
11.4 cm
11.4 cm
7 cm
9 cm
12 cm
7 cm
What is the surface area of the box, in square centimeters, that Makayla decorates?
The surface area of the triangular prism box Makayla decorates is 454.8 square centimeters
What is the surface area of a solid?The surface area of a solid is obtained by adding together the surface areas of the faces or boundaries that encloses the solid.
The dimensions of the triangular prism are;
Rear rectangle;
Length = 12 cm
Width = 7 cm
Area of rear rectangle = 12 cm × 7 cm = 84 cm²Dimensions of side triangles;
Base length = 9 cm
Height = 7 cm
Area of each side triangle = 9 cm × 7 cm = 63 cm²
Sum of the area of both triangles = 2 × 63 cm² = 126 cm²The dimensions of the rectangle at the bottom of the rectangular prism
Length = 12 cm
Width = 9 cm
Area of the rectangle at the bottom of the prism = 12 cm × 9 cm = 108 cm²Dimensions of the slant surface rectangle at the top of the triangular prism are;
Length = 12 cm
Width = 11.4 cm
Area of the top slant rectangle = 12 cm × 11.4 cm = 136.8 cm²The surface area of the box, A, is the sum of the individual surface areas, which can be found as follows;
A = 84 + 126 + 108 + 136.8 = 454.8
The area of the surface area of the box is 454.8 cm²Learn more about the surface area of solids here:
https://brainly.com/question/10167689
#SPJ1
Find the sum of the first 7 terms of the following series, to the nearest integer.
Answer:
32, 40, 50, ...
The sum of the first 7 terms of the following series is 52
What is a geometric sequence?'A geometric sequence is a sequence where every term has a constant ratio to its preceding term. A geometric sequence with the first term a and the common ratio r and has a finite number of terms.'
According to the given problem,
Sequence = 32,40,50
Let the common ratio be r,
Let the first term of the sequence be a,
⇒ a × r = second term
⇒ 32 × r = 40
⇒ r = 40/32
⇒ r = 5/4
We know,
Sum of n number of terms in a geometric sequence can be represented by,
sn=a(1-[tex]r^{7}[/tex])/1-r
⇒ [tex]s_{7}[/tex] =32(1 - (5/4)[tex]^{7}[/tex]/1 - 5/4
⇒ [tex]s_{7}[/tex] = 32 (1 - (5/4)[tex]^{7}[/tex]/3/5
⇒ [tex]s_{7}[/tex] = 5*32(1 - (5/4)[tex]^{7}[/tex])/3
⇒ [tex]s_{7}[/tex] = 160(0.972)/3
⇒ [tex]s_{7}[/tex] = 156/3
=52
Hence, we can conclude, the sum of the first 7 terms of the geometric sequence is 52
To learn more about geometric sequence refer to :
https://brainly.com/question/24643676
#SPJ1
if the variance between treatments increases and the variance within treatments decreases, what will happen to the f-ratios and the likelihood of rejecting the null hypothesis in an anova test?
When the null hypothesis is rejected, the F-ratio and probability of rejection both rise.
The F-ratio is calculated using the formula F = variance between treatments/variety within treatments. An increase in the numerator will raise the F-ratio, while a reduction in the denominator will raise it even further. The probability of rejecting the null hypothesis rises with increasing F.When the null hypothesis is rejected, the F-ratio and probability of rejection both rise.
The null hypothesis for an ANOVA test is that the means of the different groups are equal. If the F-ratio is high enough (i.e., if it is significantly different from 1), then it is likely that the means of the different groups are indeed different. Therefore, if the variance between treatments increases and the variance within treatments decreases, then the F-ratio will be higher, and the likelihood of rejecting the null hypothesis will be greater.
Know more about anova test :https://brainly.com/question/29988563
#SPJ4
What is an example of continuity vs discontinuity?
Continuity refers to the property of a function or a curve in which a small change in the input results in a small change in the output, while discontinuity is a break or interruption in a function or curve.
An example of continuity is the function f(x) = x^2, which is a continuous function because, for any value of x, there is a corresponding y-value that is obtained by squaring x. This means that if x changes slightly, the value of y will change in a corresponding way.
An example of discontinuity is the function g(x) = 1/x, which is a discontinuous function because it has a break or singularity at x=0, where the function is undefined. This means that there is a gap in the curve at x=0, and the function is not defined for any value of x that is equal to zero
To know more about functions visit : brainly.com/question/29769447
#SPJ4
The function f(x) = x2 serves as an illustration of continuity because there is always a corresponding y-value that can be calculated by squareing x. This implies that if x marginally changes, the value of y will also change.
The function g(x) = 1/x, which has a break or singularity at x=0, where the function is undefined, is an example of a discontinuous function. This indicates that the curve has a gap at x=0 and that the function is not defined for any x value equal to zero.
Write the product (2x - 5)² in the form az²+bx+c and identify the values of a, b, and c. Type numerical values only. No variables
ANSWER FAST FOR LOTS OF POINTS
The product of (2x - 5)² can be written in the form az²+bx+c as follows:
(2x - 5)² = (2x)² - 2(2x)(5) + (5)²
= 4x² - 20x + 25
In this expression, a = 4, b = -20, and c = 25.
1
Select the correct answer.
Which expression in factored form is equivalent to this expression? 4(x² - 2x)2(x² - 3)
O A. 2(x - 1)(x - 3)
O B.2(x + 1)(x + 3)
O C.(2x+3)(x + 1)
O D. (2x - 3)(x + 1)
This expression of is identical to the statement in factored form of
4 (x² - 2x) 2 (x² - 3) is 2 (x - 1) (x – 3)
What is meant by equivalent?When two or more fractions can be reduced to one another, they are said to be equivalent. As an illustration, the equivalent fractions of 1/5 are 5/25, 6/30, and 4/20, which when simplified yield the same fraction, i.e. 1/5. Equivalent expressions are those that, despite appearing differently, perform the same operations. When the same value of the variable is entered, the results of two algebraic expressions that are equal have the same value.When writing equivalent fractions, the numerator and denominator are both multiplied or divided by the same number. This explains why these fractions combine to equal one number.Given,
4(x² – 2x) – 2(x² – 3)
We would first of all expand the expression given
This becomes
= 4x² - 8x - 2x² + 6
Rearrange to enable us simplify
4x² - 2x² - 8x + 6
= 2x² - 8x + 6
Factorize
2 (x² - 4x + 3)
factorizing further using the factors of 3 that add up to -4
2(x² - x - 3x + 3)
pick out the common factors
2 (x (x - 1) - 3 (x - 1)
2 (x - 1) (x - 3)
Hence, the correct answer is A) 2(x - 1) (x – 3)
To learn more about equivalent, refer to:
https://brainly.com/question/24734894
#SPJ1
identify the ordered pair that has the same slope as the equation y = 1/2x
a. (4,3) and (1,2)
b. (1,2) and (2,4)
c. (1,1) and (2,2)
d. (2,1) and (4,2)
Answer:
d. (2,1) and (4,2)
Step-by-step explanation:
Slope is rise/run or (y2 - y1) / (x2 - x1)
We see option D, the y increase by 1, and the x increase by 2
So, the slope of this equation is 1/2, and it matches our question.
So, D is the answer!
If you can, please give me a Brainliest! Thank you, and have a good day!
Answer:
d. (2,1) and (4,2)
y = 1/2x
2y = x
x = 2y
CHECKING
x = 2y (substitute (2,1) and (4,2)
2 = 2(1)
2 = 2 correct
x = 2y
4 = 2(2)
4 = 4 correct
Therefore, d. (2,1) and (4,2) is the correct answer
Step-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-A real estate agent, working entirely on commission, makes a weekly average of $850 with da
standard deviation of $260 selling property in the city, and an average of $1340 with a standard
deviation of $390 selling property in the suburbs. Assuming independence, what are the mean and
standard deviation of her total weekly sales?
If the real estate agent's average weekly sales are $850 in the city and $1340 in the suburbs, then her total average weekly sales are $850 + $1340 = $2190.
The standard deviation of her weekly sales in the city is $260, and the standard deviation of her weekly sales in the suburbs is $390. If the sales in the city and the suburbs are independent, then the standard deviation of her total weekly sales is the square root of the sum of the squares of the standard deviations of her weekly sales in the city and the suburbs, which is the square root of $260^2 + $390^2 = $67600 + $152100 = $219100. This is equal to $468.
Therefore, the mean and standard deviation of the real estate agent's total weekly sales are $2190 and $468, respectively.
Write a point-slope equation for a line containing the points (-3,-1) and (5,2)
Answer:
Point-slope form: y + 2= 0.5(x + 3)
Slope-intercept form: y = 0.5x - 0.5
Step-by-step explanation:
Múltiple representación
The given data is used to obtain the equation as y = x - 2 and the table and graph has been drawn clearly.
How to write the equation of a straight line in slope-intercept form?A straight line can be written in the slope-intercept form as, y = mx + c.
In order to obtain the slope, the ratio of the difference of the coordinates are taken and c is the y-intercept which can be found by substituting x = 0 in the equation.
The data is given as below,
slope = 1 and y-intercept = -2.
The equation, graph and table for the given data can be formed as follows,
(a) The slope-intercept equation is y = mx + c, where m is the slope and c is the y-intercept.
Substitute the corresponding values to obtain,
y = 1 × x + (-2)
= x - 2
(b) The graph for the given equation is plotted below as,
(c) The table for the given equation can be formed with different values of x and y as below,
y = x - 2
Substitute x = 0 to get y = -2, x = 1, y = -1 and x = 2, y= 0.
Now, the table can be formed as below,
Hence, the equation is given as y = x - 2 and the graph and table for the equation is drawn properly.
To know more about slope click on,
https://brainly.com/question/3605446
#SPJ1
which inequality matches the graph 1,0,-1,-2,-3
3x+1>4
3x+1≥4
3x+1<4
3x+1≤4
In the inequality 3x + 1 ≤ 4 at x = {1, 0, -1, -2, -3} all the values are satisfied therefore, the graph matches with this inequality.
What is Linear Inequality?
A linear inequality is a mathematical inequality that incorporates a linear function. A linear inequality contains one of the inequality symbols. It displays data that is not equal in graph form.
Solution:
To find the answer we need to put the values of x in options
3x + 1 ≤ 4
Putting x = {1, 0, -1, -2, -3}
At x = 1
3*1 + 1 ≤ 4
At x = 0
3*0 + 1 ≤ 4
At x = -1
3*-1 + 1 ≤ 4
At x = -2
3 * -2 + 1 ≤ 4
At x = -3
3*-3 + 1 ≤ 4
To learn more about Linear Inequality from the given link:
https://brainly.com/question/24372553
#SPJ1
5. The value of
√33
is between what two numbers?
Obetween 3 and 4
between 5 and 6
Obetween 32 and 34
between 16 and 17
Answer: between 5 and 6
Step-by-step explanation: The square root of 5 is 25 and the square root of 6 is 36 . Hence, the square root of 33 lies between 5 and 6.
In 2005, Cory put $5,000 in an account that grows in value by 3.7% per year. what is the value of the account in 2020?
The value of the account in 2020 is given as $8622.86.
What is compound interest?The compound interest is a form of interest where the rate of interest is applied on the amount obtained after every interval of time.
The amount of one interval of time becomes the principal to the next one.
Suppose in 2005 t = 0.
Then, in the year 2020 t = 15.
Now, the given question can be solved by finding the Amount using the expression as below,
P(1 + r/100)ⁿ
Where p, r and n are the principal, rate and time for the compounding.
Substitute all the corresponding values to obtain,
A = 5000(1 + 3.7/100)¹⁵
= 5000 × 1.037¹⁵ = 8622.86
Hence, in 2020 the money invested in the account becomes as $8622.86.
To know more about compound interest click on,
https://brainly.com/question/14295570
#SPJ1
Marvin is playing a solitaire game with marbles. There are n bowls (for some positive integer n), and initially each bowl contains one marble. Each turn, Marvin may eitherremove a marble from a bowl, orchoose a bowl A with at least one marble and a different bowl B with at least as many marbles as bowl A, and move one marble from bowl A to bowl B.The game ends when there are no marbles left, but Marvin wants to make it last as long as possible.Prove that the game must end after at most n(n+1)/2 turns.Prove that for every positive integer k, there is an n such that Marvin can make the n-bowl game last for at least kn turns.Marisa comes along and asks to join the game. Marvin and Marisa revise the rules: they will alternate taking turns, starting with Marisa. When the game ends, whoever took the last turn is the winner.Prove that among any three consecutive values of n, there is at least one value for which Marvin has a winning strategy.
There is always at least one consecutive value of n for which Marvin has a successful approach.
By induction, we can demonstrate that the game must conclude after a maximum of n(n+1)/2 turns for the first half of the issue, Here Induction Means a method for demonstrating that a statement P(n) is true for every natural number n, and that the infinitely numerous examples P(0), P(1), P(2), P(3), etc. all hold, is mathematical induction. Informal analogies, such a ladder or a falling domino, are used to explain this technique .
Base case: If there is just one bowl, Marvin can get rid of it in one turn, ending the game after one.
Step one of induction: Assume that there are k bowls and that the game terminates after no more than k(k+1)/2 turns. When there are k+1 bowls, we shall demonstrate that the game terminates after at most (k+1)(k+2)/2 turns. The number of bowls lowers by one if Marvin takes a marble out of one, and the game must terminate after no more than k(k+1)/2 turns through induction hypothesis
When Marvin transfers a marble from bowl A to bowl B, where bowl A contains at least one marble and bowl B contains an equal number of marbles, bowl A loses one marble and bowl B gains one marble. The number of bowls stays the same because bowl B has at least as many marbles as bowl A. As a result, the game must still terminate after k(k+1)/2 turns at the latest.
In any scenario, the game ends after at most k(k+1)/2 turns, thus when there are k+1 bowls, the game must end after at most (k+1)(k+2)/2 turns. For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.
Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.
For the second component of the issue, we can demonstrate that for every positive integer k, there exists a n such that Marvin can devise a plan to extend the n-bowl game for at least kn turns.
Let's say Marvin wants the n-bowl match to go at least kn turns. He can accomplish this by ensuring that each bowl always contains at least k marbles. Marvin can start by putting k marbles in the first bowl, k-1 marbles in the second bowl, and so forth, down to 1 marble in the nth bowl, to accomplish this.
Marvin can select a bowl containing at least k marbles on each turn and transfer one marble to a bowl containing fewer than k marbles .Marvin can transfer one marble from the first bowl to the second bowl, for instance, if there are k+1 marbles in the first bowl and k marbles in the second bowl. This guarantees that there are at least k marbles in each bowl. The game will go for at least kn turns because Marvin can take at least one turn every bowl.
For the third component of the issue, we can demonstrate that there is always at least one value of n for which Marvin has a successful strategy.
Assume that n is a positive integer. We shall demonstrate that either Marvin or Marisa has a winning strategy for the game of n bowls or the game of (n+1) bowls.
If Marvin has a winning strategy for the N-bowl game, he can use that method to win the N-bowl game.
If Marisa has a winning strategy for the n-bowl game and Marvin does not, then Marisa does as well. In this scenario, Marvin can use Marisa's successful approach to the n-bowl game as his own successful approach to the (n+1)-bowl game.
To learn more about induction at: brainly.com/question/8419798
#SPJ4