Stats project

Important sections

1. data collection 3%
2. data entry in excel 3%
3. 3 different graphs using excel 3 %
4. demonstration of mode, mean, median 3%
5. report and data collection forms and excel file and minute meetings 3%

std dev questions

1. find the standard deviation of 1, 5, 4, 2, 6, 2, 1,1 5,3
2. find the standard deviation of 20, 17, 17, 14, 12, 11, 7, 8, 9, 16, 19, 22

A sample of data can have a lot of different numbers and it will require grouping to make any sense e.g 3-5 hrs, 6-8 hrs, 9-11 hrs, 12-14 hrs, 15-17 hrs, 18-20 hrs can represent the difefrnt ranges to group the numbers

Find the standard deviation for

0-4 2

5-9 7

10-14 12

15 - 19 17

20-24 22

25 - 29 27

Standard deviation

A conclusion is drawn from a sample of data. e.g. a conclusion can be drawn from the time usage of 50 internet users. these 50 users are a sample of all internet users.
if a conclsion has to be drawn from the population thatwill be impractical. So a sample of the population is selected an used. This means that the deviation from the mean is important.
Steps for calculating standard deviation

1. find the mean
2. subtract the mean from each member
3. square the difference
4. find the sum of these squares
5. divide this sum by n-1
6. find the square root of this result

Statistics

Statistics is all about numbers, is this true
why does one keep a set of numbers
do you know anything or place with a set of numbers
Why does the play whe outlet keep a set of previous play for the day?

Think of the grocery bill when you are buying items for a graduation or party?
A lot of common items are bought.
Information of interest will be which item cost the most money?
Which item do you have the most of?
How many different items do you have?
What item cost the least?
What item do you have the least of?


23 12 16 13 12 23 23 12 27 16 13 23 43 13 12 23 23 12 23 12 23 43 16 23 26 23 26

What information can you create or obtain from the numbers?

The different numbers are
27 numbers
23 12 16 13 27 43 (just 6 different numbers)
23 occurred the most (10 times)
27 occured the least ( 1 time)
23 occurred 10 times
12 occurred 6 times
16 occurred 3 times
13 occurred 3 times
etc

23 which occured the most is called the mode

median
1st arrange is a descending or ascending order
2nd remove one from both sides and the last one is the median. if two are left the average of the two numbers are found
the mean is the average of all the numbers

12 12 12 12 12 12 13 13 13 16 16 16 23 23 23 23 23 23 23 23 23 23 26 26 27 43 43

Trapezoidal Rule Questions

A lake has the following lenghts, in order: 100m, 150m, 200m, 175m, 250m, 150m, 100m, 50m. The lengths were measured at intervals of 50m, what is the surface area of the lake?

A cieling need to be covered. Measurements are taken every 10 inches, and the lengths are recorded as 100in, 150in, 200in, 150in, 200in, 200in, 150in, 200in and 50in. What is the area of the ceiling?

A floor needs to be covered in vinyl carpet. The floor however is irregularly shaped and measures, from one end to the other, in order, 10m, 30m, 50m, 60m, 20m, 20m, 30m, 10m and 5m. How large is the floor if teh measurements are taken every 2m?

A cover is built for a reservoir located in a natural landform. Surveyors measure the landform's width every 10 metres and record the following distances: 125m, 130m, 190m, 230m, 200m, 180m, 150m.

The first and last distnaces across a plot of land are 60 feet and 100 feet respectively. The other distances are 80ft, 100ft, 120ft, 100ft, 90ft and 110ft. How big is the plot of land?

What is the area of a figure that measures 1, 2, 4, 5, 3, 2, 1 and 8 inches across, respectively, if the measurements are taken every 0.5 inches?

A satellite photograph covers an island. The key on the island shows that the distances across the island are, from left to right, 100km, 120km, 300km, 200km, 180km, 100km, 150km and 175km. If the measurements are 50km apart, how much surface area is there on the island?

A hole was broken on the side of a metal fence and it needs to be covered over with sheeting. If the hole measures, from top to bottom, at intervals of 15 cm, 1m, 50cm, 1.5m, 150cm, and 75cm, what area needs to be filled?

A painter paints an outline on the road, to be covered in white paint for an advertising spot. The outline is measured every 10 cm and is 50, 40, 30, 50, 45, 40 and 80cm wide, and 10 and 30 cm wide at the ends. How much area needs to be painted?

One end of a field is 100m wide, and the other end is 110m wide. How much area is there on the field for sports if every 15m the field is measured and the distances (excluding the other two already mentioned) are 105m, 110m, 108m, 103m, 97m, 98m, 105m and 103m?

Expand and Factorize

(27x + 56y) (3 - t)

(52x - 12w) (23x + 3w)

(ax + bx) (x + bx)

(3x - 5) (10x + 4)

(12y - 3x) (20x - 20y)

(4a + 2a) (2x - 4)

(30x - 2y) (30x + 2y)

w (23a + 33x)

(10x - 3) (25y + 11x)

(x - y) (25a)

p (54 + p)

(11x + 11y) (px - 33y)

24x (5x - 5y)

(3r + ab) (3ab + r)

(2xy - x) (z - 20)




Factorize:

7xy^2 + 21xy + 14x

x^2 + 13x + 42

z^2 + 6z - 16

-w^2 + 15w - 50

a^2 + 25a + 150

3x^2 - 5x + 2

6y^2 + 14y + 12

y^2 - 9y - 36

AP Question Set

If the interval is 15, and each row increases by 6, given that the 6th row has 700 seats, how many seats are there in the first row?

If there are 100 rows and there are 600 seats in the final row, how many are there in the first row if there are 8 more in each row?

If a sequence has 34 terms, the first row has 12, and the final row has 65, what is the common difference?

What is the common difference in a series that begins at 10 and ends at -50, if there are 14 terms?

If a sequence goes 41, 49, 57, 65, et cetera, taking 57 to be the first term, what is the value of the 4th and 14th terms?

In a sequence of 18 items, if the final item is 44 and the difference is 8, what is the first item?

The first item in a sequence of 38 items is 6 and the final item is 86, what is the common difference?

How many numbers are divisible by 5 between 42 and 193?

The 23rd term in a sequence that increases by 8 is 134. What is the 1st term?

A park has thirty-three trees in the 5th row of trees. If the trees increase by 4 every row, how many are there in the first row?


If the third term in a sequence is 6.5, and the fifth is 13.5 and the seventh is 20.5, what is the first term?

In 46 numbers, the last term is 12, and the difference is 3.4. What is the first term?

The 15th number in a sequence is 54, and the first is 30. What is the common difference?

How many numbers leave a remainder of 2 when divided by 7 in the range 4 to 104?

After counting by threes, the 54th number is 431. What was the 22nd number?

A sequence begins at 92, goes to 103, then to 114, and then 125 and continues along this line. If a sequence begins at 114 and ends at 499, how many terms are in the sequence?

The common difference in a sequence is -13. If the final term is -65 and the sequence is 12 items long, what is the 1st, 5th and 6th terms?

Beginning at 5, a sequence ends at 65. How many terms are there in the sequence if the common difference is 12?

20 is the 5th number in an evenly progressing sequence and 80 is the 10th. What is the first?

In 65 numbers, the final number is 64.1. If the sequence progresses by 2.1, what is the 10th term?

Express the following as a single logarithm:
1. log [3] 7 + log [3] 5

2. log [2] 16 + 3log [2] 4 - log [2] 8
3log [5] 9 + log [5] 2
4. log [5] 5 + log [5] 10 - log [5] 3 = 0
3.
4.

Log set 3

1. Solve log[7](x^2) = log[7](2x – 1).
2. Solve 2log[3](x) = log[3](4) + log[3](x – 1)
3. Solve log2(x) + log2(x – 2) = 3

1. A closed box has a fixed surface area A and a square base with side x .
   (a) Find a formula for the volume V of the box, as a function of x .
   (b) Find the rate of change of V with respect to x.
2. The revenue from selling q items is given by the formula R(q) = 500q - q^2
   and the total cost is given by C(q) = 150 + 10q . Write down a function that gives the total
   profit earned. Find the rate of change of total profit with respect to q.
3. For positive constants A and B, the force between two atoms in a molecule is given by
   f(r) = A/r^2 + B/r^3 where r > 0 is the distance between the atoms.
   What is the expression for the instantenous rate of change of force between the atoms with respect to distance?

Log set 2

1. Find the value of the unknown variable in each of these:
   (i) 2^x = 128 , (ii) 3^y = 1/9 , (iii) 5^x = 625 , (iv) 4^s = 164 , (v) 16^t = 4 ,
   (vi) 8^ = 1/4 .
2. Solve each of the following equations for the value of x (i) log 3x = 6

ii) log[3]x + 3log[3](3x) = 3 iii) log (5x - 1) = 2 + log (x - 2) iv) 2logx = log(7x - 12)

3. Write each of the following as a single logarithmic expression:
(i) log[10](x + 5) + 2 log[10] x (ii) log[4](x^3 - y^3) - log[4] (x - y)

iii) 1/2 (3 log[5] (4x) + log [5] (x + 3) - log[5] 9)

Expand and Factorize

Expand:

(x + 13) (2x + 5)

(7y + 6) (1y + 50)

(y - 5) (y + 5)

(65t + 1) (3t + 15)

(-7y - 7) (-11y - 21)

(3z - z) (6z + z)

(8 + 2w) (-5 - 2r)

(-r + 12) (-5r - 12)

(x + b) (a - b)

(22i - 10) (4i - 3)



Factorize:

x^2 +5x +6

y^2 - y - 6

m^2 - 5m - 24

x^2 + 13x + 42

z^2 + 18z + 80

r^2 + 36 + 15r

e^2 + 10e + 28

54 + 15c + c^2

t^2 + 2t - 15

x^2 + x - 2

AP question set 2

Explain all steps and strategy not answers

1. If d = 3 and n= 30 and S30 = 1875, find a and S5.
2. Write the first 7 terms if a = 4 and d = 2.
3. Write the first 7 terms if a = 6 and d = -1/2.
4. If the 2nd term = -2 and the 5th = 43, write the first 7 terms.
5. If the 3rd term = 5/2 and the 5th = -3/2, write the first 7 terms.
6. A beach now has an area of 9500m2 but is eroding such that it loses 100m2 more of its area each year than during the previous year. If it lost 400m2 during the last year, what will be its area 8 years from now?

AP questions

1. Find the common difference between successive terms of the arithmetic sequence for ehich the first rem is 5 and the 32nd trem is -119.
2. How many numbers between 10 and 1000 are divisible by 6?
3. Find the 4th and 14th terms for 2,5,8,.......
4. A package delivery company uses a metal ramp to slide packages from the sorting area to the loading. If a package is pushed to start it down the damp at 25 cm/s and the package accelerates as it slides such that it gains 35cm/s during each second, after how many seconds is the velocity 305cm/s?
5. Find the sum of the first 1000 positive integers.
6. Find the sum of the first ten terms of the arithmetic progression in which thr first term is 4 and the common difference is -5.
7. If a = 2, d = 2/3 and Sn = 72, find n. Find S4.
8. THe voltage across a resistor increases such that during each second the increase is 0.002mV less than during the previous second. Given thath the increase during the first second is 0.350 mV, what is the total voltage increade during the first 10 s?
9. If the 6th term of an AP is 56 and the 10th term is 72. Find the 1st term, the common difference and the sum of the first 10 terms.
10. If the 17th term of an AP is -91 and the 2nd term is -73. Find the 1st term, the common difference and the sum of the first 40 terms.

AP /GP

What is a series or progression?

Why is this usefull in life?

Does anything you do remind you of AP?
Does anything around you remind you of AP?

What are the important things in a series or progression?

Things that are changing must have a starting position.
Your age had a starting point. Your height had a starting point.
Everything you know bankaccount, friendship, a movie, etc

Now if it is changing, then it means by what amount it is changing?
If these amount is the same then we have an AP.

If you need the 7th position or term then

• out of the 7 terms, the 1st must be the start
• then the rest will be the consistant changes. In this case it will be (7-1) which is 6
• So the 7th term 1st + (7 -1) change i.e. a + (n-1)d is the nth term

Give some examples of terms e.g. 27th term, 32nd term, etc

Arithmetic progression questions

A Stadium is built with 200 seats in the first row. After that, each row has 50 more seats than the previous row. How many seats are in the 25th row?

The 15th value in a set that increases by 6 is 250, what is the first value?

What is the sum of the values of a set that contains 23 numbers, begins at 5 and increases by 4?

A housing estate is built with five homes on the first row, and there are 3 more homes on each row. How many homes are there on the 15th row?

If there are 50 boxes on the bottom of a stack which is 15 rows tall, and as the stack grows in height, there are 2 less boxes in each row, how many rows of boxes are there in the first row?


Please post other questions for the class if you do have any to solve.

Volume and Area Questions II

NOTE: Use the Trapezoidal rule for all irregular shaped questions. Assume measurements are taken in order (first to last) unless otherwise stated.

1. An artificial green needs to be covered in turf, and the gardener in an effort to calculate exactly how much is needed, measures the lengths across the green at 1 metre intervals. The gardener measures lengths of 1, 3, 9, 4, 5, 6 and 8 metres respectively. What area needs to be covered?

2. A parade float is in the shape of an island. The designer decides to cover the island in green cloth to make it look more realistic, and measures the width of the float every 50 centimetres, and records the first length as 1.5 metres, and the others as 2.5, 3, 4, 3, 2.5, 3 and 1 metre respectively. How much cloth will be placed on the float?

3. A section of a roller coaster's tracks needs to have a cover installed to put up a sign. The track on this section has no curves, but does go up and down. The engineers measure each of the nine supports, which are spaced 3 metres apart, and record the following lenghts in order: 4, 5, 6, 3, 2, 3, 4, 5, 5 (in metres). What area will the sign cover?

4. An artist designed abstract pool needs to be covered in case of rain, and other bad weather. The pool is measured in intervals of 30 centimetres and has lengths of 0m, 2m, 1m, 2m, 3m, 4m, 3m and 1 m, where m is metres. What is the surface area of the cover?

5. A satellite photo of a parcel of land shows that the distances from one side of the resort, which is 500 metres long, to the other, is 0 metres, 100 metres, 320 metres, 200 metres, 150 metres, 200 metres, 400 metres, 350 metres, 250 metres, 200 metres and 10 metres. How large is the parcel of land, given that the distances are measured at even intervals?

6. A hole in the side of a blimp needs to be patched. How much material is needed if the gap, measured at 1 foot intervals, is 3 feet, 2 feet, 4 feet, 4 feet, 5 feet, 3 feet, 2 feet and 1 foot?

7. Trinidad Asphalt is loading asphalt to fill a pot hole that is 1 foot, 6 feet, 3 feet, 4 feet, and 2 feet wide at points in the pothole, which are 1 foot apart from each other. What surface area does the pothole have?

8. A man man decides to paint the town red, but because of the global economic recession, wants to make sure he is as efficient as possible in the purchasing of his paint. If the town is measured at 100 metre intervals and the following distances (in metres) are found respectively: 500, 900, 850, 450, 700, 800, 850, 900, 850, 900, 1000, 1250, 1100, 900, 800, 750. What surface area would he need to paint?

9. A tee-shirt printer needs to purcahse just the right amount of ink to print a splash on a tee shirt. The splash is measured at 1cm intervals and is 1cm high at one end, 2cm at the other, and the lengths in between are 2cm, 3cm, 4cm, 2cm, 3cm, 5cm, 3cm and 2cm.

10. A footprint is seen at a crime scene and investigators want to find out the surface area to link it to another suspicious footprint they have found. They measure the following lengths, at 1 inch intervals, respectively. and find 2.5 inches, 2.8 inches, 2.5 inches, 2.4 inches, 2.2 inches, 2.2 inches, 2.0 inches, 2.4 inches, 2.5 inches, 2.6 inches and 2.7 inches. What is the area of the footprint?

11. An oil barrel measures 4 feet in diameter and is 6 feet high, how much oil can it hold?

12. A grain silo is in the shape of a cylinder and is placed on a square base that fits it perfectly with no overlap of any kind. The base measures 6 metres along one side. If the silo is 20 metres tall, how much grain can it hold?

13. You think a can of Juice is rather small for it to hold the 2000 centimetres cubed that it says on the label. You measure the can and get the following dimensions: Height: 15 centimetres, Diameter: 5 centimetres, Circumference: 15.714 cm. How much juice does the can actually hold?

14. How much water can your dog's bowl hold if it has a radius of 6 inches and a height of the same?

15. A programmer needs to tell the computers how much grease to fill in a cylindrical tube of height 30 centimetres and diameter 6 centimetres. How much will he need to tell the computers to fill the tubes?

16. If a coin measures 2cm in diameter and .1 cm in thickness, and ten of these are placed in a tub of water that is full to the brim, how much cm³ will the coins cause to spill?

17. An absorbant rope is used to trace the edge of an oil spill. Measuring the rope at 1.5 metre intervals, the distances across the spill, in order, is found to be 1m, 3m, 10m, 20m, 15m, 10m, 15m, 20m, 10m, 5m, 10m, and 2m. How much oil has spilled?

18. A cupboard door needs to be lined with insulating material, and the door measures 30cm, 40cm, 45cm, 45cm, 45cm, 40cm, 35cm, 30cm and 20cm across, at points which are 12 cm apart. How much insulation is needed?

19. A patch of grass in a field is dead, determine how much grass died if you are told that the area was measured in gaps of 32 inches and the distances across were found to be 40 inches, 4 feet, 5 feet, 53 inches, 6 feet, 50 inches and 23 inches.

20. A hole needs to be patched. The following measurements are taken at 1.25 cm intervals: 1cm, 2.20cm, 2.10cm, 1.75cm, 1.44cm, 1.20cm, 1.0cm, 1cm, .55cm and 0cm. How much area needs to be patched?

Expansion

Expand the following.

(5x + 3) (x + 7)

(10x + 2) (-1x + 20)

(y - 5) (y + 5)

(3w + 1) (3x + 5)

(11u - 11) (6u - 1)

(xy - x) (3x + xy)

(8 + 30r) (5 + 2r)

(100t + 12) (52t - 12)

(a + b) (a - b)

(22i - 10) (4i - 3)

Simpson's Rule questions

1. The widths of a plane wing were measured as 0.30m intervals. Calculate the surface area of the pool given the respective lengths were 0, .16, .23, .32, .35, .3 and .2. Draw the diagram on paper then write and explain your answer.
2. The ariel view of a lake with 1.5km intervals had lengths respectfully of 0, 4.8, 5.7, 10.5, 15.2, 18.5, 18.8, 17.9, 11.3, 8.8 and 3.1 Draw the diagram on paper then write and explain your answer.
3. The widths of a kidney-shaped swimming pool were measured as 20m intervals. Calculate the surface area of the pool given the respective lengths were 0, 6, 7, 8, 6, 5, 4, 5 and 0. Draw the diagram on paper then write and explain your answer.

Simpson Rule

Another approach for measuring area of irregular shapes is the Simpson 's Rule. This also uses equal intervals. Unlike drawing trapeziums, arcs of parabola are drawn somewhat getting closer to the real area. The formula is

A = h/3 {yo + 4y1 + 2y2 +4y3 + ....+ 2yn-2 + 4yn-1 + yn)

Trapezoidal Rule

1. A plate cam for opening and closing a valve has an irregular shape. The widths of the face of the cam are 2cm intervals. Find the area of the face of the cam if the length respectfully are 3, 4, 3, 3, 2 and 0. Draw a diagram in a paper then explain your answer.
2. From a satellite photograph of a lake, the widths were 26 kmm intervals. The lengths were 0, 45, 50, 60, 61, 66, 74, 87, 76, 66, 86, 77, 0 respectfully. Draw a diagram in a paper then explain your answer.
3. The widths of a kidney-shaped swimming pool were measured as 20m intervals. Calculate the surface area of the pool given the respective lengths were 0, 6, 7, 8, 6, 5, 4, 5 and 0.
4. The widths of a plane wing swimming pool were measured as 0.30m intervals. Calculate the surface area of the pool given the respective lengths were 0, .16, .23, .32, .35, .3 and .2.

Irregular shapes

In our last tutorial, approaches were investigated to find the area of an irregular shape. Suggestions

1. Make the best rectangle out of the irregular shape
2. Make many rectangles out of the irregular shape
3. Make many rectangles with equal widths out of the irregular shape
4. Make many trapeziums with equal widths out of the irregular shape

This was named Trapezoidal rule.

A hand was used to show that many trapeziums meant that the 2 outer lengths were the only lengths never repeated

Assignment due Monday March 9th. Trigonometry.

You need to draw the diagrams for each of the following questions and submit them in class on Monday March 9th.


1. A bird is standing on the ground, 5 metres away from a tree, which is seven metres tall. The bird wants to build a nest at the top of the tree, what is the shortest distance the bird must fly to get from its point on the ground to the top of the tree?

2. Your friend climbs up a flight of stairs, travelling 10 metres diagonally, when she gets to the top of the stairs she accidentally kicks over your cell phone which you left charging. To make sure your phone isn’t damaged, you need to calculate how far it fell from the top of the stairs straight down to the floor, and you know the stairs are at an angle of 53 degrees.

3. You are 6 metres away from a target with a canon that rests on the ground. The target is 5m off the ground. What angle would the canon need to be at to hit the target?
4. A friend is sleeping on the street across the road from your building, and you are standing on the balcony of the 5th floor. The road is 20m wide, and each floor is 5m high. What angle would your friend need to look up to see you if you called to him?

5. The same friend from question 4 also happens to make very good paper planes, and writes a note in one and sends it to you. How far does the plane have to travel to get to you if it flies straight from him to you?

6. A projector sits on a stand 1 metre high, and is 8 metres away from the screen. If the top of the image is projected at an angle of 45 degrees from the projector, how high does the screen need to be?

7. You are 15 metres away from your friend, who is standing next to a wall. You know your friend is 1.1 metres tall, and he is the same height as the wall, but on top of that wall is a flag pole. You only know that from the ground where you stand to the top of the flagpole is elevated 30 degrees, how high is the flagpole itself?

8. A boat is tied to a pier using 10 metres of rope between the boat and pier. The point on the pier where the rope is tied to the top of the boat gives an angle of 20 degrees. If the pier itself is 2 metres above the water, how much of the boat is above water?

9. A pulley is used to lift a bucket from one end of a construction site to another, across a 15 metre gap. At each end of the gap there are poles, with a rope to carry the bucket. The first pole is 5 metres high, and the second is 10 metres high. What is the actual distance the bucket travels between the two poles along the rope?

10. The engineers moving the bucket in question 9 want to ensure that it is being done as efficiently as possible, and want to know what angle the bucket is ascending with. What is that angle?

Area and Volume Questions

1. The corner of a square room needs to be covered in carpet. A triangular piece is needed. If one side has to be 5 feet and the other needs to be 7 feet, what area needs to be covered?

2. You are making a flag with two colours. The flag is sewn so the cloth meets a the middle of the flag going diagonally downwards. If you can get exactly the cloth cut exactly as you need it, how much cloth of each colour, in square feet, is needed if the flag’s dimensions are 4 feet by 3 feet.

3. A diamond window is made from four panes of glass, which meet at a “t” in the middle. The T is 1 meter high and 50 centimetres wide. If the bottom left pane of glass breaks, how much glass is needed to replace it?