Use Archimedes exhaustion method to estimate pi accurate to two decimal places
Answers
Step-by-step Answer:
Calculating Pi using Archimedes method of polygons.
We know that the definition of pi is the ratio of circumference of a circle divided by the diameter. Starting with Pythagorean Theorem, and proposition 3 of Euclid’s Elements, Archimedes was able to approximate pi to any precision arithmetically, without further resort to geometry!
He figured that the perimeter of any regular polygon (all sides and vertex angles equal) is an approximation to a circle. More sides will make closer approximations.
Starting with a hexagon, he bisects the central angles to make polygons 12-, 24-, 48- and 96-sides, whose perimeters approaches that of a circle, and hence the approximation to pi since the diameter remains known and constant.
Proposition 3 is also commonly referred to as the angle bisector theorem, which states that in a triangle, an angle bisector subdivides the opposite sides in the ratio of the two remaining sides.
Please refer to the attached image for the nomenclature of the geometry.
The accompanying diagram shows that the perimeter of a hexagon is 12 times the length of AB, or 12*(1.0/2) = 6. With the diameter equal to 2*1.0 = 2, the approximation to pi is 6/2=3.0.
Pi(6) = 3.0
If we divide the central angle by two, we end up with a 12-sided polygon (dodecagon), with the half central angle of 15 degrees (triangle A’BC). To calculate the new perimeter, we need to calculate the length A’B, which is given by the angle-bisector theorem as
A’B / A’A = BC / AC
All other sides are known in terms of A’B
A’B / (0.5-A’B) = sqrt(3)/2 / 1
Solve for A’B by cross-multiplication and solving for A’B, we get
A’B = sqrt(3)/(2sqrt(3)+4) = 0.2320508 (to 7 decimals)
At the same time, the radius has been reduced to
A’C = sqrt(A’B^2+BC^2) = 0.896575
That brings the approximation of pi as 12*A’B/A’C
P(12) = 3.1058285 (7 decimals)
Continuing bisecting, now using a polygon of 24 sides, we only have to replace
AB by A’B, AC by A’C, and 12 by 24 to get
Pi(24) = 3.132629 (7 decimals)
Repeating again for a polygon of 48 sides,
Pi(48) = 3.1393502
Pi(96) = 3.1410320
Pi(192) = 3.1414525
Pi(384) = 3.1415576
Etc.
The accurate value of pi to 10 digits is 3.1415926536
And we conclude that Pi(48) is the first approximation the provides 2 decimal places of accuracy.
Note: What was calculated was actually the lower bound value of pi.
We can obtain the upper bound value of pi using the length of BC as the radius, which gives the upper bound. The average of the two bounds for a 384-sided polygon gives P_mean(384) = 3.1416102, which is accurate to 2 units in the 5th decimal place.
The Archimedes exhaustion method is a geometric approach to estimate the value of pi. By inscribing and circumscribing regular polygons within and around a circle, Archimedes determined lower and upper bounds for the value of pi.
As the number of sides for the polygons increased, the approximation of pi became more accurate. With a 96-sided polygon, Archimedes found that pi was greater than 3.1408 and less than 3.1429. By taking the average of these bounds, a more precise estimation of pi accurate to two decimal places is achieved.
Thus, using the Archimedes exhaustion method, we can estimate pi to be approximately 3.14, making it precise to two decimal places.
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Related Questions
Find the slope of the line passing through the points (-7,-7) and (-3, 6)
Answers
Answer:
13/4
Step-by-step explanation:
The slope of the line between 2 points is found by
m = (y2-y1)/(x2-x1)
= (6--7)/(-3--7)
= (6+7)/(-3+7)
= 13/4
Answer:
The slope is 13/4.
Step-by-step explanation:
Slope formula:
[tex]\displaystyle \frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]\displaystyle \frac{6-(-7)}{(-3)-(-7)}=\frac{13}{4}[/tex]
[tex]\huge\boxed{\frac{13}{4}}[/tex], which is our answer.
Write equations for the horizontal and vertical lines passing through the point (-8,6)
Answers
Answer:
So since your point is (-8,6), then your vertical line is x=-8 and horizontal line is y=6.
Step-by-step explanation:
Horizontal lines are in the form y=b.
Vertical lines are in the form x=a.
a and b are just constant numbers.
So anyways, in general:
The horizontal line going through (a,b) is y=b.
The vertical line going through (a,b) is x=a.
So since your point is (-8,6), then your vertical line is x=-8 and horizontal line is y=6.
x=-8
y=6
good luck
In a study of cell phone use and brain hemispheric dominance, an Internet survey was e-mailed to 2455 subjects randomly selected from an online group involved with ears. 931 surveys were returned. Construct a 99% confidence interval for the proportion of returned surveys.
Answers
The 99% confidence interval for the proportion of returned surveys is between 35.378% and 40.4213%, based on 931 responses from 2455 surveyed subjects.
Explanation:To construct a 99% confidence interval for the proportion of returned surveys, we will use the formula for a proportion confidence interval which includes the sample proportion (π), the z-value that corresponds to the desired level of confidence, and the standard error of the proportion.
The sample proportion (π) can be calculated as the number of returned surveys divided by the total number of surveys sent:
π = 931 / 2455 = 0.379
The z-value for a 99% confidence interval is approximately 2.576. The standard error (SE) of π is calculated using the formula:
SE = √(π(1 - π) / n)
SE = √(0.379(1-0.379)/2455)
SE = √(0.379 * 0.621 / 2455)
SE = √(0.2353 / 2455)
SE = √(0.0000958)
SE = 0.009788
Now, we can calculate the margin of error (ME):
ME = z * SE
ME = 2.576 * 0.009788
ME = 0.025213
Finally, the 99% confidence interval (CI) is calculated as:
CI = π ± ME
CI = 0.379 ± 0.025213
CI = [0.353787, 0.404213]
We can be 99% confident that the true proportion of returned surveys falls between 35.378% and 40.4213%.
A rectangular swimming pool measures 14 feet by 30 feet. The pool is surrounded on all four sides by a path that is 3 feet wide. If the cost to resurface the path is $2 per square foot, what is the total cost of resurfacing the path?
Answers
To find the cost of resurfacing the path, we first calculate the area of the path which is 300 square feet. We then multiply this by the unit cost of resurfacing which comes out to be $600.
Explanation:This is a problem in area calculation and application of unit cost. Firstly, we need to calculate the area for the path surrounding the pool. The total area of the pool and the path is (14+2*3) feet by (30+2*3) feet = 20 feet by 36 feet, which equals 720 square feet. The area of the pool itself is 14 feet by 30 feet = 420 square feet. So, the area of just the path is 720-420 = 300 square feet. With a cost of $2 per square foot to resurface the path, the total cost would be 300*$2 = $600.
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The top of a ladder slides down a vertical wall at a rate of 0.675 m/s. At the moment when the bottom of the ladder is 6 m from the wall, it slides away from the wall at a rate of 0.9 m/s. How long is the ladder?
Answers
Answer:
The length of the ladder is 10 m.
Step-by-step explanation:
Let x shows the distance of the top of ladder from the bottom of base of the wall, y shows the distance of the bottom of ladder from the base of the wall and l is the length of the ladder,
Given,
[tex]\frac{dx}{dt}=-0.675\text{ m/s}[/tex]
[tex]\frac{dy}{dt}=0.9\text{ m/s}[/tex]
y = 6 m,
Since, the wall is assumed perpendicular to the ground,
By the pythagoras theorem,
[tex]l^2=x^2+y^2[/tex]
Differentiating with respect to t ( time ),
[tex]0=2x\frac{dx}{dt}+2y\frac{dy}{dt}[/tex] ( the length of wall would be constant )
By substituting the value,
[tex]0=2x(-0.675)+2(6)(0.9)[/tex]
[tex]0=-1.35x+10.8[/tex]
[tex]\implies x=\frac{10.8}{1.35}=8[/tex]
Hence, the length of the ladder is,
[tex]L=\sqrt{x^2+y^2}=\sqrt{8^2+6^2}=\sqrt{64+36}=\sqrt{100}=10\text{ m}[/tex]
Answer:
The length of ladder=8m.
Step-by-step explanation:
Given
The rate at which the top of a ladder slides down a vertical wall,[tex]\frac{\mathrm{d}z}{\mathrm{d}t}[/tex]= 0.675m/s
The distance of bottom of ladder from the wall,x=6m
The rate at which it slides away from the wall ,[tex]\frac{\mathrm{d}x}{\mathrm{d}t}[/tex]=0.9m/s
Let length of ladder =z
Length of wall=y
Distance between foot of ladder and wall=x
By using pythogorous theorem
[tex]x^2+y^2=z^2[/tex]
Differentiate w.r.t time
[tex]x\frac{\mathrm{d}x}{\mathrm{d}t}=z\frac{\mathrm{d}z}{\mathrm{d}t}[/tex]
y does not change hence, [tex]\frac{\mathrm{d}y}{\mathrm{d}t}=0[/tex]
[tex]6\times 0.9=z\times 0.675[/tex]
[tex]z=\frac{5.4}{0.675}[/tex]
z=8 m
Hence, the length of ladder=8m.
Six distinct integers are picked from the set {1, 2, 3,…, 10}. How many selections are there, in which the second smallest integer in the group is 3?
Answers
Answer:
1680 ways
Step-by-step explanation:
Total number of integers = 10
Number of integers to be selected = 6
Second smallest integer must be 3. This means the smallest integer can be either 1 or 2. So, there are 2 ways to select the smallest integer and only 1 way to select the second smallest integer.
2 ways 1 way
Each of the line represent the digit in the integer.
After selecting the two digits, we have 4 places which can be filled by 7 integers. Number of ways to select 4 digits from 7 will be 7P4 = 840
Therefore, the total number of ways to form 6 distinct integers according to the given criteria will be = 1 x 2 x 840 = 1680 ways
Therefore, there are 1680 ways to pick six distinct integers.
Answer:
70 total selections
Step-by-step explanation:
The set: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
You know that that 3 is definitly a part of the set, so you can ignore it. If 3 is the second smallest, the smallest number in the set is either 1 or 2, not both.
The number of ways to choose between 1 and 2 is [tex]2^{C}1[/tex] ways which is equal to 2, so all that's left is choosing from the group of the set between 4 and 10.
Since you've already chosen 2 numbers (3 and 1 or 2) you need to find out how many ways can you choose 4 out of the numbers between 4 and 10. Since there are 7 numbers from 4 to 10, you need to figure out [tex]7^{C}4[/tex] which is equal to 35.
Since you are looking to find the cross between the 2, multiply 2 by 35 = 70, the answer.
(1 pt) In a study of red/green color blindness, 950 men and 2050 women are randomly selected and tested. Among the men, 89 have red/green color blindness. Among the women, 6 have red/green color blindness. Construct the 99% confidence interval for the difference between the color blindness rates of men and women.
Answers
Answer: (0.066,0.116)
Step-by-step explanation:
The confidence interval for proportion is given by :-
[tex]p_1-p_2\pm z_{\alpha/2}\sqrt{\dfrac{p_1(1-p_1)}{n_1}+\dfrac{p_2(1-p_2)}{n_2}}[/tex]
Given : The proportion of men have red/green color blindness = [tex]p_1=\dfrac{89}{950}\approx0.094[/tex]
The proportion of women have red/green color blindness = [tex]p_2=\dfrac{6}{2050}\approx0.003[/tex]
Significance level : [tex]\alpha=1-0.99=0.01[/tex]
Critical value : [tex]z_{\alpha/2}=z_{0.005}=\pm2.576[/tex]
Now, the 99% confidence interval for the difference between the color blindness rates of men and women will be:-
[tex](0.094-0.003)\pm (2.576)\sqrt{\dfrac{0.094(1-0.094)}{950}+\dfrac{0.003(1-0.003)}{2050}}\approx0.091\pm 0.025\\\\=(0.09-0.025,0.09+0.025)=(0.066,\ 0.116)[/tex]
Hence, the 99% confidence interval for the difference between the color blindness rates of men and women= (0.066,0.116)
Determine the value of g(4), g(3 / 2), g (2c) and g(c+3) then simplify as much as possible.
g(r) = 2 [tex]\pi[/tex] r h
Answers
Answer:
[tex]g(4) = 8 \pi h\\\\g(\frac{3}{2}) = 3 \pi h\\\\ g(2c) = 4 \pi ch\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]
Step-by-step explanation:
You need to substitute [tex]r=4[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(4) = 2 \pi(4)h\\\\g(4) = 8 \pi h[/tex]
Substitute [tex]r=\frac{3}{2}[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(\frac{3}{2}) = 2 \pi(\frac{3}{2})h\\\\g(\frac{3}{2}) = 3 \pi h[/tex]
Substitute [tex]r=2c[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(2c) = 2 \pi(2c))h\\\\g(2c) = 4 \pi ch[/tex]
Substitute [tex]r=c+3[/tex] into [tex]g(r) = 2 \pi r h[/tex]. Then:
[tex]g(c+3) = 2 \pi (c+3)h\\\\g(c+3) = 2 \pi hc+6\pi h[/tex]
For this case we have the following function:
[tex]g (r) = 2 \pi * r * h[/tex]
We must evaluate the function for different values:
[tex]g (4) = 2 \pi * (4) * h = 8 \pi*h\\g (\frac {3} {2}) = 2 \pi * (\frac {3} {2}) * h = 3 \pi*h\\g (2c) = 2 \pi * (2c) * h = 4 \pi * c * h\\g (c + 3) = 2 \pi * (c + 3) * h = 2 \pi * c * h + 6 \pi * h[/tex]
Answer:
[tex]g (4) = 8 \pi*h\\g (\frac {3} {2}) =3 \pi*h\\g (2c) = 4 \pi * c * h\\g (c + 3) = 2 \pi * c * h + 6 \pi * h[/tex]
A professor has recorded exam grades for 10 students in his class, but one of the grades is no longer readable. If the mean score on the exam was 82 and the mean of the 9 readable scores is 84, what is the value of the unreadable score?
Answers
Answer:
64
Step-by-step explanation:
[tex]mean=\frac{sum\ of\ total\ number\ of\ score}{total\ number\ of\ students}[/tex]
we have given that mean of 9 students is 84
so total score of 9 students = mean×9
=84×9=756
and we have given mean score of exam is 82 and there is total 10 students so the total score of 10 students =10×82
=820
so the unreadable score = score of 10 students -score of 9 students =820-756=64
Your friend, Isabel, has a credit card with an APR of 19.9%! How many dollars would she pay as a finance charge for just 1 month on a $1000 charge?
Express your answer rounded to the nearest cent.
Answers
Answer:
$16.583
Step-by-step explanation:
Given :Your friend, Isabel, has a credit card with an APR of 19.9%!
To Find : How many dollars would she pay as a finance charge for just 1 month on a $1000 charge?
Solution:
We are given that finance charge for just 1 month on a $1000 charge.
So, Finance charge = [tex]\frac{19.9\% \times 1000}{12}[/tex]
Finance charge = [tex]\frac{\frac{199}{1000}\times 1000}{12}[/tex]
Finance charge = [tex]\frac{199\times 1000}{12}[/tex]
Finance charge = [tex]16.583[/tex]
Hence she pay $16.583 as a finance charge for just 1 month on a $1000 charge.
Convert 185 to base three
Answers
Answer:
20212
Step-by-step explanation:
Divide 185 by [tex]3^4[/tex] as [tex]3^4[/tex] is closest to 185
After dividing we get remainder 23 and quotient 2
Divide the remainder 23 by [tex]3^3[/tex] we get remainder 23 and quotient 0
Divide the remainder 23 by [tex]3^2[/tex] we get remainder 5 and quotient 2
Divide the remainder 5 by [tex]3^1[/tex] we get remainder 2 and quotient 1
Divide the remainder 2 by [tex]3^0[/tex] we get remainder 0 and quotient 2
Taking all the quotient values we get 20212
Hence, 185₁₀=20212₃
Homewood Middle School has 1200 students, and 730 of these students attend a summer picnic. If two-thirds of the girls in the school and one-half of the boys in the school attend the picnic, how many girls attend the picnic?
Answers
Answer:
520 girls attended the picnic
Step-by-step explanation:
Hello
step 1
Let
Homewood middle school total students=1200
A=unknown=total of girls
(2/3)A=total girls attended the picnic
B=unknown= total of boys
(1/2)B=total boys attended the picnic
step2
replace
[tex]A+B=1200\ equation (1) \\\frac{2A}{3}+\frac{B}{2}=730\ equation(2)\\[/tex]
Step 3
find A and B
from equation (1)
[tex]A=1200-B\ equation\ (3)\\\\[/tex]
from equation (2)
[tex]\frac{2A}{3}+\frac{B}{2}=730\\\frac{2A}{3}=730-\frac{B}{2}\\A=(730-\frac{B}{2})*\frac{3}{2} \\1200-B=1095-\frac{3B}{4}\\A=A\\-B+\frac{3B}{4}=1095-1200\\-\frac{B}{4}=-105\\ B=420\\\\hence\\\\A=1200-B\\A=1200-420\\A=780[/tex]
total girls attended the picnic=(2/3)A=(2/3)780=520
520 girls attended the picnic
Answer:
520
Step-by-step explanation:
i also got the problem and got it wrong - i found the answer is 520
John Smith made a one year investment that generated a nominal return of 6% or $3000. The real return was $2000. What was the original investment amount? what was the annual inflation rate? Macroeconomic
Answers
The nominal value - without discounting the inflation rate - of income was $ 3000.
If the interest rate was 6%, a rule of three is enough to find the value of the original investment.
3000 - 6%
x - 100%
x = 50,000
The value of the investment was $ 50,000
In this case, the inflation rate also requires a simple calculation.
Inflation corroded $ 1000 dollars of income of $ 3000
Therefore the inflation rate will be 1000/3000 = 33.3%
4. Find the general solution to 4y"+20y'+25y = 0
Answers
Answer:
[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]
Step-by-step explanation:
The given differential equation is 4y"+20y'+25y = 0
The characteristics equation is given by
[tex]4r^2+20r+25=0[/tex]
Now, solve the equation for r
Factor by middle term splitting
[tex]4r^2+10r+10r+25=0\\\\2r(2r+5)+5(2r+5)=0[/tex]
Factored out the common term
[tex](2r+5)(2r+5)=0[/tex]
Use Zero product property
[tex](2r+5)=0,(2r+5)=0[/tex]
Solve for r
[tex]r_{1,2}=-\frac{5}{2}[/tex]
We got the repeated roots.
Hence, the general equation for the differential equation is
[tex]y(x)=c_1e^{-\frac{5}{2}x}+c_2xe^{-\frac{5}{2}x}[/tex]
Final answer:
The general solution to the differential equation 4y"+20y'+25y = 0 is y(x) = (A + Bx)e^(-5/2x), where A and B are constants determined by initial conditions.
Explanation:
The general solution to the differential equation 4y"+20y'+25y = 0 can be found by looking for solutions in the form of y = ekx, where k is a constant. Substituting y into the differential equation, we get a characteristic equation of (ak² +bk+c)y= 0, which simplifies to (4k² + 20k + 25)y = 0. This is a quadratic equation in k that can be factored as (2k + 5)². Therefore, the two values of k that satisfy this equation are both -5/2, giving us a repeated root.
The general solution for a second-order linear homogeneous differential equation with repeated roots is y = (A + Bx)ekx, where A and B are constants determined by the initial conditions. In this case, k = -5/2, hence the general solution is y(x) = (A + Bx)e-5/2x.
A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. Imagine you stick your hand into the refrigerator and pull out a piece of fruit at random. What is the chance you don't get an apple? 10/4410/44 6/446/44 38/44
Answers
The chance you don't get an apple is:
[tex]\dfrac{38}{44}[/tex]
Step-by-step explanation:We know that the probability of an outcome is the chance of getting an outcome and it is calculated by:
Taking the ratio of number of favorable outcomes to total number of outcomes.
A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangoes.
Total number of fruits in the refrigerator i.e. total number of outcomes are: 6+5+10+3+7+11+2
= 44
Also the number of favorable outcomes i.e. number of fruits which are not apples in the refrigerator are: 44-6=38
This means that the probability of not getting an apple is:
[tex]\text{Probability(not\ getting\ an\ apple)}=\dfrac{38}{44}[/tex]
Final answer:
The chance you do not get an apple is38/44, or roughly86.36.
Explanation:
The chance you do not get an apple is38/44.
To calculate this, add up the total number of fruits in the refrigerator banning apples, which is 5 oranges 10 bananas 3 pears 7 peaches 11 catches 2 mangos = 38 fruits. The total number of fruits in the refrigerator is 6 apples 5 oranges 10 bananas 3 pears 7 peaches 11 catches 2 mangos = 44 fruits.
simplifies to19/22 or roughly0.8636, so the chance you do not get an apple is0.8636 or86.36.
The cost (in hundreds of dollars) of tuition at the community college is given by T = 1.25c + 3, where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits, use the model to determine how much money he should borrow.
Answers
Answer:
The amount he should borrow is 19.25 hundreds dollars.
Step-by-step explanation:
Given : The cost (in hundreds of dollars) of tuition at the community college is given by [tex]T = 1.25c + 3[/tex], where c is the number of credits the student has registered for. If a student is planning to take out a loan to cover the cost of 13 credits.
To find : Use the model to determine how much money he should borrow?
Solution :
The model is given by [tex]T = 1.25c + 3[/tex]
Where,
c is the number of credits the student has registered.
T is the cost of tuition at the community college.
If a student is planning to take out a loan to cover the cost of 13 credits.
The amount he should borrow will get by putting the value of c in the model,
[tex]T = 1.25c + 3[/tex]
[tex]T = 1.25(13) + 3[/tex]
[tex]T = 16.25 + 3[/tex]
[tex]T =19.25[/tex]
Therefore, The amount he should borrow is 19.25 hundreds dollars.
1.) Given P(A) = 0.28, P(B) = 0.01, P(B|A) = 0.47, what is P(A and B)?
Answers
Answer:
p(A and B) =0.1316
Step-by-step explanation:
We know that [tex]p(B|A)=\frac{p(A\cap B)}{p(A)}\\\\p(B\cap A )=p(B|A)\times p(A)[/tex]
Applying values we get
[tex]p(A and B)=0.47\times 0.28\\\\p(AandB)=0.1316[/tex]
Find and simplify the expression if
Answers
Answer:
[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]
Step-by-step explanation:
We are given the following expression and we are to simplify the given function:
[tex] f ( x ) = x ^ 2 - 8 [/tex]
Applying the function [tex]f(2x)[/tex] on [tex] f ( x ) = x ^ 2 - 8 [/tex] to get:
[tex] f ( 2 x ) = ( 2 x ) ^ 2 - 8 [/tex]
[tex] f ( 2 x ) = 4 x ^ 2 - 8 [/tex]
Question 10 (7 points) d Elizabeth borrowed $20,000 for five years at a 5% annual interest rate, what is her monthly payment, to the nearest dollar? A. $252 B. $334 C. $377 D. $4050 E. None of these Save
Answers
Answer:
monthly payment is $377
C is the correct option.
Step-by-step explanation:
The formula for the monthly payment is given by
[tex]C=\frac{Prt(1+r)^n}{(1+r)^n-1}[/tex]
Given that,
P = $20,000
n = 5 years = 60 months
r = 0.05
Substituting these values in the formula
[tex]C=\frac{20000\cdot \frac{0.05}{12}(1+\frac{0.05}{12})^{60}}{(1+\frac{0.05}{12})^{60}-1}[/tex]
On simplifying, we get
[tex]C=\$377.425\\\\C\approx \$377[/tex]
Therefore, the monthly payment is $377
C is the correct option.
Use the power series for 1 1−x to find a power series representation of f(x) = ln(1−x). What is the radius of convergence? (Note: you don’t need to use the ratio test here because we know the radius of convergence of the series P∞ n=0 x n .) (b) Use part (a) to find a power series for f(x) = x ln(1 − x). (c) By putting x = 1 2 in your result from part (a), express ln 2 as the sum of an infinite series
Answers
a. Recall that
[tex]\displaystyle\int\frac{\mathrm dx}{1-x}=-\ln|1-x|+C[/tex]
For [tex]|x|<1[/tex], we have
[tex]\displaystyle\frac1{1-x}=\sum_{n=0}^\infty x^n[/tex]
By integrating both sides, we get
[tex]\displaystyle-\ln(1-x)=C+\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]
If [tex]x=0[/tex], then
[tex]\displaystyle-\ln1=C+\sum_{n=0}^\infty\frac{0^{n+1}}{n+1}\implies 0=C+0\implies C=0[/tex]
so that
[tex]\displaystyle\ln(1-x)=-\sum_{n=0}^\infty\frac{x^{n+1}}{n+1}[/tex]
We can shift the index to simplify the sum slightly.
[tex]\displaystyle\ln(1-x)=-\sum_{n=1}^\infty\frac{x^n}n[/tex]
b. The power series for [tex]x\ln(1-x)[/tex] can be obtained simply by multiplying both sides of the series above by [tex]x[/tex].
[tex]\displaystyle x\ln(1-x)=-\sum_{n=1}^\infty\frac{x^{n+1}}n[/tex]
c. We have
[tex]\ln2=-\dfrac\ln12=-\ln\left(1-\dfrac12\right)[/tex]
[tex]\displaystyle\implies\ln2=\sum_{n=1}^\infty\frac1{n2^n}[/tex]
The power series of f(x) = ln(1 - x) is [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex], the power series of xln(1 - x) is [tex]\rm -\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex] and the value of ln(2) is [tex]\rm \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex].
Given :
f(x) = ln (1−x)
a) The integration of 1/(1 - x) is given by:
[tex]\rm \int \dfrac{1}{1-x}dx=-ln|1-x| + C[/tex]
When |x| >1 :
[tex]\dfrac{1}{1-x} = \sum^{\infty}_{n=0} x^n[/tex]
Now, integrate on both sides in the above equation.
[tex]\rm -ln(1-x) = C+\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex] --- (1)
Now, at (x = 0) the above expression becomes:
[tex]\rm -ln(1-0) = C+\sum^{\infty}_{n=0}\dfrac{0^{n+1} }{n+1}[/tex]
By simplifying the above expression in order to get the value of C.
C = 0
Now, substitute the value of C in expression (1).
[tex]\rm ln(1-x) =-\sum^{\infty}_{n=0}\dfrac{x^{n+1} }{n+1}[/tex]
Now, by shifting the index the above expression becomes:
[tex]\rm ln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n} }{n}[/tex]
b) Now, multiply by 'x' in the above expression in order to get the power series of (x ln(1 - x)).
[tex]\rm xln(1-x) =-\sum^{\infty}_{n=1}\dfrac{x^{n+1} }{n}[/tex]
c) Now, substitute the value x = 1/2 in the expression (1).
[tex]\rm ln2 = \sum^{\infty}_{n=0}\dfrac{1}{n2^n}[/tex]
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A youth basketball coach has 12 kids on his team and he selects 5 kids to start each game. To be fair he wants to start a different group of 5 kids each game. How many different ways can he start 5 of the 12 players? If there is 20 games will he be able to start a different group of 5 kids for each game?
Answers
[tex]_{12}C_5=\dfrac{12!}{5!7!}=\dfrac{8\cdot9\cdot10\cdot11\cdot12}{120}=792[/tex]
792>20, so yeah, he will be able.
Answer:
792>20, so yeah, he will be able.
Step-by-step explanation:
Sixty percent of the students at a certain school wear neither a ring nor a necklace. Twenty percent wear a ring and 30 percent wear a necklace. If one of the students is chosen randomly, what is the probability that this student is wearing (a) a ring or a necklace?
Answers
Answer:
0.4
Step-by-step explanation:
Given
60 % wear neither ring nor a necklace
20 % wear a ring
30 % wear necklace
This question can be Solved by using Venn diagram
If one person is choosen randomly among the given student the probability that this student is wearing a ring or necklace is
[tex]P\left ( wear \ ring\ or \ necklace )+P\left ( neither\ ring\ or\ necklace)=1[/tex]
[tex]P\left ( wear \ ring\ or\ necklace )=1-0.6=0.4[/tex]
The sum of probabilty is equal to 1 because it completes the set
Therefore the required probabilty is 0.4
Final answer:
The probability that a randomly chosen student is wearing either a ring or a necklace is 40%. This conclusion is based on the complement of the given percentage of students who wear neither, assuming that there is no overlap in the 20% and 30% who wear rings and necklaces respectively.
Explanation:
To find the probability that a randomly chosen student is wearing a ring or a necklace, we need to understand that 'or' in probability means either one or the other, or both. According to the question, 60% of the students wear neither, which means 40% of the students wear either a ring, a necklace, or both. Since 20% wear a ring and 30% wear a necklace, we might be tempted to add these percentages to get 50%. However, doing so could potentially double-count students who wear both a ring and a necklace.
Without additional information, we can simply state that the probability that a student is wearing a ring or a necklace is the complement of the probability of a student wearing neither, which is 40%. Here we're assuming that students either wear a ring or a necklace or both, as there is no mention of wearing neither in the probabilities given.
A round trip takes 3.5 hours going one way and 2 hours to return, if the trip back is at a speed 15 mph faster than the speed of the first trip. Find the speeds each way and the distance between the places.
Answers
Answer:
first trip 20 mph, trip back 35 mph, distance = 70 miles.
Step-by-step explanation:
A round trip takes 3.5 hours going one way and 2 hour to return.
Let the speed of first trip = v mph
Given that the trip back is at a speed 15 mph faster than the speed of first trip.
Therefore, the speed of trip back = ( v+15 ) mph
Let the Distance between places = d miles
Now we use the formula
[tex]Time=\frac{Distance}{Velocity}[/tex]
[tex]\frac{Distance}{Velocity}[/tex] = 3.5
and [tex]\frac{D}{V+15}[/tex] = 2
dividing these two
[tex]\frac{v+15}{v}[/tex] = [tex]\frac{3.5}{2}[/tex]
2v + 30 = 3.5v
v = 20 mph
d = 3.5 × 20 = 70 miles
The speed of the first trip is 20 mph and the speed of trip back (20+15) 35 mph. and the distance is 70 miles.
A and B and n x n matrices such that AB = 0. Prove that if A is invertible then B is not invertible.
Answers
Answer and Step-by-step explanation:
Since we have given that
AB = 0
where A and B are n x n matrices.
Consider determinant on both sides,
[tex]\mid AB\mid=\mid 0\mid\\\\\mid A\mid \mid B\mid =0\\\\either\ \mid A\mid =0\ or\ \mid B\mid =0[/tex]
since A is invertible, then |A| ≠ 0
so, it means |B| = 0.
Hence, B is not invertible.
Hence proved.
According to insurance records, a car with a certain protection system will be recovered 95% of the time. If 800 stolen cars are randomly selected, what is the mean and standard deviation of the number of cars recovered after being stolen?
Answers
Answer: Mean = 760
Standard deviation = 6.16
Step-by-step explanation:
Given : The number of trials: [tex]n=800[/tex]
The probability that a certain protection system will be recovered :[tex]p=0.95[/tex]
We know that the mean and standard deviation of binomial distribution is given by :_
[tex]\text{Mean}=np[/tex]
[tex]\text{Standard deviation}=\sqrt{np(1-p)}[/tex], where n is the number of trials and p is the probability of success.
Now, the mean and standard deviation of the number of cars recovered after being stolen is given by :-
[tex]\text{Mean}=800\times0.95=760[/tex]
[tex]\text{Standard deviation}=\sqrt{800\times0.95(1-0.95)}\\\\=6.164414002\approx6.16[/tex]
Hence, the mean is 760 and standard deviation is 6.16 .
The mean and standard deviation of the number of cars recovered after being stolen can be found using the properties of the binomial distribution.
Explanation:To find the mean and standard deviation of the number of cars recovered after being stolen, we can use the properties of the binomial distribution. In this case, the probability of recovering a car is 0.95, and the number of stolen cars is 800.
The mean can be calculated by multiplying the number of trials (800) by the probability of success (0.95), giving us a mean of 760 cars.
The standard deviation can be calculated using the formula:
standard deviation = sqrt(n * p * (1 - p))
Substituting in the values, we get:
standard deviation = sqrt(800 * 0.95 * (1 - 0.95))
standard deviation ≈ 8.72 cars.
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Write parametric equations for a circle of radius 2, centered at the origin that is traced out once in the clockwise direction for 0 ≤ t ≤ 4π. Use the module to verify your result. (Enter your answer as a comma-separated list of equations. Let x and y be in terms of t.)
Answers
Answer:
We know that equation of a circle with origin as it's center is given by
[tex]x^{2}+y^{2}=r^{2}\\\\\therefore x^{2}+y^{2}=2^{2}\\\\(\frac{x}{2})^{2}+(\frac{y}{2})^{2}=1\\\\[/tex]
Comparing with [tex]sin^{2}(\theta )+cos^{2}(\theta )=1[/tex] we get
[tex]\frac{x}{2}=sin(\theta )\\\\\therefore x=2sin(\theta )\\\\\frac{y}{2}=cos(\theta )\\\\\therefore y=cos(\theta )[/tex]
Since [tex]sin(\theta ),cos(\theta )[/tex] have a period of 2π but in the given question we need to increase the period to 4π thus we reduce the argument by 2
[tex]\therefore x=2sin(\frac{\theta }{2})\\\\y=2cos(\frac{\theta }{2})[/tex]
The parametric equations for a clockwise circle of radius 2 centered at origin traced out once for a full revolution (0 ≤ t ≤ 4π) are x = 2 cos(-t/2), y = -2 sin(t/2). This can be confirmed by substituting -t/2 for t in Pythagorean Identity sin²(t) + cos²(t) = 1 which results in 1, proving the correctness of the equations.
Explanation:The parametric equations for a circle of radius 2, centered at the origin, traced out once in the clockwise direction for 0 ≤ t ≤ 2π are x = 2 cos(t), y = 2 sin(t). However, as your question indicates the path traced out in the clockwise direction, the equations would be changed to x = 2 cos(-t) and y = 2 sin(-t). This is achieved by replacing t with -t in the original equations.
In the context of the question, parametric equations which are traced out once for a full revolution (0 ≤ t ≤ 4π in the negative or clockwise direction) are x = 2 cos(-t/2), y = -2 sin(t/2). This is because time is needed twice as much to make a full revolution, so 2t is replaced with t/2.
To verify these equations, you can use the Pythagorean Identity sin²(t) + cos²(t) = 1, substituting -t/2 for t in this identity equation, you will find that the result indeed equals 1, confirming these are the correct parametric equations.
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Solve the Differential equation 2(y-4x^2) dx + x dy = 0
Answers
The solution of the given differential equation involves rearranging it into the standard form of a first-order linear differential equation, determining the integrating factor, and subsequently solving for the dependent variable y(x) via integration.
Explanation:To solve the given differential equation, we can rewrite it in the form of dy/dx = f(x, y). That gives us (2(y-4x^2))/x = dy/dx. The resulting equation is a first-order linear differential equation, which can be solved using an integrating factor.
Here, the standard form of the differential equation is dy/dx + P(x)y = Q(x). Comparing this with our equation, we find P(x) = -2/x and Q(x) = -8x. We know that μ(x) = exp(∫P(x) dx) is the integrating factor. On solving we get μ(x) = 1/x2. We then multiply through our differential equation by μ(x) and integrate both sides to solve for y(x).
These steps on how to solve the differential equation involve certain knowledge in differential equation theory, namely about first-order linear differential equations, integrating factors, and the process of integration.
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A sample of 100 wood and 100 graphite tennis rackets are taken from the warehouse. If 1212 wood and 2020 graphite are defective and one racket is randomly selected from the sample, find the probability that the racket is wood or defective.
Answers
Answer: The probability that the racket is wood or defective is 0.6.
Step-by-step explanation:
Since we have given that
Number of wood tennis rackets = 100
Number of graphite tennis rackets = 100
Total number of rackets = 200
Number of wood are defective = 12
Number of graphite are defective = 20
Total number of defectives = 32
We need to find the probability that the racket is wood or defective.
Let A be the event of wood tennis rackets.
Let B be the event of defective.
So, it becomes,
[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)\\\\P(A\cup B)=\dfrac{100}{200}+\dfrac{32}{200}-\dfrac{12}{200}\\\\P(A\cup B)=\dfrac{100+32-12}{200}=\dfrac{120}{200}=0.6[/tex]
Hence, the probability that the racket is wood or defective is 0.6.
The sum of four consecutive natural numbers is 598. Identify any variables and write an equation to find the numbers. What are they?
Answers
Answer:
Equation is 4x + 6 = 598, where x represents smaller number.
Numbers are 148, 152, 156 and 160
Step-by-step explanation:
Let x be the smaller natural number,
So, the other consecutive natural numbers are x+1, x+2, x+3,
According to the question,
Sum of x, x+1, x+2 and x+3 is 598,
⇒ x + x + 1 + x + 2 + x + 3 = 598
⇒ 4x + 6 = 598
Which are the required equation,
Subtract 6 on both sides,
4x = 592
Divide both sides by 4,
x = 148
Hence, the numbers are 148, 152, 156 and 160
Final answer:
The equation to find four consecutive natural numbers with a sum of 598 is 4x + 6 = 598. Solving for x gives the first number as 148, which leads to the sequence: 148, 149, 150, and 151.
Explanation:
The student is tasked with finding four consecutive natural numbers whose sum is 598.
To solve this problem, we introduce a variable to represent the first number in the sequence, and then express the following three numbers in terms of this variable.
Let's denote the first number as x. Then the next three numbers will be x+1, x+2, and x+3, respectively. Our equation to find the numbers is:
x + (x+1) + (x+2) + (x+3) = 598
Combining like terms, we get:
4x + 6 = 598
We then solve for x:
4x = 598 - 6
4x = 592
x = 592 / 4
x = 148
So the four consecutive numbers are 148, 149, 150, and 151.
Assuming that it is known from previous studies that σ = 4.5 grams, how many mice should be included in our sample if we wish to be 95% confident that the mean weight of the sam- ple will be within 3 grams of the population mean for all mice subjected to this protein diet?
Answers
Answer:
The required amount of mice is 11.82 gram.
Step-by-step explanation:
Given : Assuming that it is known from previous studies that σ = 4.5 grams. If we wish to be 95% confident that the mean weight of the sample will be within 3 grams of the population mean for all mice subjected to this protein diet.
To find : How many mice should be included in our sample?
Solution :
The formula used in the situation is
[tex]z=\frac{x-\mu}{\sigma}[/tex]
Where, z value at 95% confidence interval is z=1.96
[tex]\mu=3[/tex] gram is the mean of the population
[tex]\sigma=4.5[/tex] gram is the standard deviation of the sample
Substituting the value, to find x sample mean
[tex]1.96=\frac{x-3}{4.5}[/tex]
[tex]1.96\times 4.5=x-3[/tex]
[tex]8.82=x-3[/tex]
[tex]x=8.82+3[/tex]
[tex]x=11.82[/tex]
Therefore, The required amount of mice is 11.82 gram.