A Ferris wheel has a radius of 10 feet and is boarded in the 6 o’clock position from a platform that is 3 feet above the ground. The wheel turns counterclockwise and completes a revolution every 3 minutes. At t = 0 the person is at the 6 o’clock position.

1. Draw a diagram and impose coordinates

2. Find a function, F(t), for the height of the person above the ground after t minutes

3. Find two times when a passenger is at a height of 18.0 feet.

4. How far does a rider travel along the circle if they travel from the boarding point on the bottom of the Ferris wheel to the top of the Ferris wheel?

Assigned Exercise IX.1. IX.1. (a) Suppose that f : [a, b] → R is continuous. Define A := 1/b−a integral of f from a to b, and B := 1/b-a integral of f² from a to b . Show that 1/b − a integral from a to b of (f(x) − A)² dx = B − A ² . Conclude that A² ≤ B. (b) Assume the Cauchy–Schwarz Inequality for Integrals of Exercise 6.3 #2, which we state here for continuous functions f : [a, b] → R and g : [a, b] → R: integral from a to b of (fg)² ≤ integral from a to b  (f ² ) integral from a to b  (g ² ) . How does this Cauchy–Schwarz inequality imply the inequality A² ≤ B of part (a)?

2a. Find the orthogonal projection of [9,40,-29,4] onto the subspace of R4 spanned by [1,6,5,6] and [5,1,5,5].

Answer choices: [2,14,-15,7] [-32,13,-10,7] [0,9,12,6] [-5,-2,3,2] [-12,0,-9,-9] [-16,20,0,4] [27,29,29,21] [-3,1,2,7] [-23,7,-3,-9] [-15,5,-15,30]

2b. Find the orthogonal projection of [17,18,-10,24] onto the subspace of R4 spanned by [2,7,1,6] and [3,7,3,4].

Answer choices: [-34,-22,-29,-34] [-6,4,-2,0] [-12,36,21,33] [3,21,-3,24] [7,-14,-12,1] [5,3,32,45] [14,32,12,11] [9,13,18,11] [20,2,-3,19] [-2,-6,1,-7]

Give examples–a formula and an illustration–of two-dimensional vector fields F⃗(x,y) with each of the following properties. You could do the illustrations by hand.

a) The direction of F⃗ is constant but the magnitude is not constant.

b) The magnitude |F⃗| is constant but the direction is not constant.

c) All the vectors F⃗ along a horizontal line are equal, but F⃗ is not constant overall.

d) F⃗ (x, y) is perpendicular to xˆi + yˆj at every point (x, y).

e) F⃗ is a force field which repels from the origin. It is strongest near the origin, and weaker farther

away.

After studying the material in this module, please solve and submit the following problems from the module reading assignment on Sets from Math in Society, by D. Lippman (v. 2.4): #6, 14, 16, 20, 22, 26, 30, 34, 36, 44

q) a)Parametrize the following paths from (−2, 0) to (2, 0) in the xy-plane:
• A straight line path.
• A path consisting of three lines: (−2, 0) to (−2, 2), from (−2, 2) to (2, 2), and from (2, 2) to (2, 0). (Give three parametrizations.)
• A path counterclockwise along a circle centered at the origin.

b) Compute the integral F⃗ · d⃗r along each of these paths for F⃗ (x, y) = (−y, x).

c) Compute the integral G⃗ · d⃗r along each of these paths for G⃗ (x, y) = (y2, 2xy).

d)What do you notice about your answers to parts (b) and (c)? Are either F⃗ or G⃗ a gradient field?

The Weemow Lawn Service wants to start doing snow removal in winter when there are no lawns to maintain. Jeff and Julie Weems, who own the service, are trying to determine how much equipment they need to purchase, based on various job types they have. They plan to work themselves and hire some local college students on a per-job basis. Based on historical weather data, they estimate that there will be 6 major snowfalls next winter. Virtually all customers want their snow removed no more than 2 days after the snow stops falling. Working 10 hours per Day (into the night), Jeff and Julie can remove the snow from a normal driveway in about 1 hour, and it take about 4 hours to remove snow from a business parking lot and sidewalk. The variable cost (mainly for labor and gas) per job is $12 for a driveway and $47 for a parking lot. Using their lawn service customer base as a guideline, they believe they will have demand of no more than 40 homeowners and 25 businesses. They plan to charge $35 for home driveway and $120 for a business parking lot, which is slightly less than the going rate. They want to know how many jobs of each type will maximize their profit.

Problem 2

A: Alternatively, hiring additional people on a per job basis will increase Jeff and Julie’s variable cost to $16 for a driveway and $53 for a parking lot, but it will lower the time it takes to clear a driveway to 40 mins and a parking lot to 3 hours. Will this affect their profit?

B: If Jeff and Julie combine the two alternatives (b) and (c), will this affect their profit?

Only problem 2...Specifically, why does the time constraint change to be less than 100? The correct answers for 2a is 2435 and for 2b is 2073

Please prove the following formally and clearly:

Let X₁ = 1. Define X_n+1 = sqrt(3 + X_n). Show that (X_n) is convergent and find its limit.

A coffeehouse sells a pound of coffee for ​$8.25. Expenses are ​$3 comma 500 each​ month, plus ​$3.50 for each pound of coffee sold. ​(a) Write a function​ R(x) for the total monthly revenue as a function of the number of pounds of coffee sold. ​(b) Write a function​ E(x) for the total monthly expenses as a function of the number of pounds of coffee sold. ​(c) Write a function ​(Rminus​E)(x) for the total monthly profit as a function of the number of pounds of coffee sold. ​(a) The revenue function is ​R(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.) ​(b) The expense function is ​E(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.) ​(c) The profit function is ​(Rminus​E)(x)equals nothing. ​(Simplify your answer. Use integers or decimals for any numbers in the expression. Do not include the​ $ symbol in your​ answer.)

Prove that the number of partitions of n into parts of size 1 and 2 is equal to the number of partitions of n + 3 into exactly two distinct parts

x^2 y ′′ + xy′ + λy = 0 with y(1) = y(2) and y ′ (1) = y ′ (2)

Please find the eigenvalues and eigenfunctions and eigenfunction expansion of f(x) = 6.

4. Let a < b and f be monotone on [a, b]. Prove that f is Riemann integrable on [a, b].

A cohort study was undertaken to examine the association between high lipid level and coronary heart disease (CHD). Participants were classified as having either a high lipid level (exposed) or a low or normal lipid level (unexposed). Because age is associated with both lipid level and risk of heart disease, age was considered a potential confounder or effect modifier and the age of each subject was recorded. The following data describes the study participants: Overall, there were 11,000 young participants and 9,000 old participants. Of the 4,000 young participants with high lipid levels, 20 of them developed CHD. Of the 6,000 old participants with high lipid levels, 200 of them developed CHD. In the unexposed, 18 young and 65 old participants developed CHD.. Calculate the appropriate crude ratio measure of association combining the data for young and old individuals. 3. Now, perform a stratified analysis and calculate the appropriate stratum-specific ratio measures of association. What are they? 4. Do the data provide evidence of effect measure modification on the ratio scale? Justify your answer.

discrete mathematics

1. 

Show that if a | b and b | a, where a and b are integers, then a = b or a = -b.  

//Ex. 5, Page 208.

2.

Show that if a, b, c, and d are integers such that a | c and b | d, then ab | cd.

//Ex. 6, Page 208. 

3.

What are the quotient and remainder when

a) 44 is divided by 8?

b) 777 is divided by 21?

f) 0 is divided by 17?

g) 1,234,567 is divided by 1001?   

//(a), (b), (f), and (g) Ex. 10, Page 209.

4.

Determine whether each of these integers is a prime.

a) 19

b) 27

e) 107

f) 113

//(a), (b), (e), and (f) Ex. 2, Page 217.

5.

Find the prime factorization of each of these integers.

a) 39

c) 101

d) 143

f) 899

//(a), (c), (d), and (f) Ex. 4, Page 217.

6.

What are the greatest common divisors of these pairs of integers?

a) 23 * 33 * 55 and 25 * 33 * 52

d) 22 * 7 and 53 * 13 

//(a) and (d) Ex. 20, Page 218.

7.

Find gcd(1000, 625) and lcm(1000, 625) and verify that gcd(1000, 625)*lcm(1000, 625) = 1000*625 

//Ex. 24, Page 218.