A diving vessel descends below the surface of the water at a constant rate so that the depth of the vessel after 1 minute, 2 minutes and 3 minutes is 50 metres, 100 metres and 150 metres respectively.
By how much is the depth increasing each minute?
What will the depth of the vessel be after 4 minutes?
Continuing at this rate, what will be the depth of the vessel after 10 minutes?
A paver needs to pave a floor with an area of 800 square metres. He can pave 50 square metres a day.
Complete the table showing the area left to pave at the start of each day:
Is this linear or exponential decay?
| \text{Day} | \text{Area of floor} \\ \text{left to pave }(m^{2}) |
|---|---|
| 1 | 800 |
| 2 | |
| 3 | |
| 4 | |
| 5 |
For a fibre-optic cable service, Christa pays a one off amount of \$200 for installation costs and then a monthly fee of \$30.
Complete the table of values for the total cost \left(T\right) of Christa's service over n months.
By how much are consecutive terms in the sequence increasing?
From the table of values, plot the first four points on a cartesian plane.
If the points on the graph were joined, would they form a straight line or a curve?
| n | 1 | 2 | 3 | 4 | 18 |
|---|---|---|---|---|---|
| T |
A scuba diver descends below the surface of the water at a constant rate so that his depth after 4 minutes, 8 minutes and 12 minutes is 15 metres, 30 metres and 45 metres respectively.
Create a table of values for the scuba diver's time underwater and depth underwater.
Identify whether the sequence is arithmetic or geometric.
Write an explicit rule for the depth D of the diver after m minutes.
Find the number of minutes it takes for the diver to reach a depth of 120 metres.
Luigi starts his career with a monthly wage of \$5400. At the beginning of each year that follows he receives a raise and his monthly wage for that year will be \$120 greater than the previous year.
What will be his yearly salary in the second year of his service?
Determine the number of years that it would take for him to earn a total of \$792\,000.
A telecommunications company sells 1700 mobile phones in the first month of its operation. The company plans to increase its sales by 150 mobile phones each month.
How many phones does the company plan to sell in the last month of the 4th year of its operation?
How many phones does the company plan to sell in the entire 4 year period?
A ball starts rolling down a slope. It has rolled 23\text{ cm} after the first second, 44\text{ cm} after the second, 65\text{ cm} after the third second and so on.
Create a table of values for the time and distance of the ball rolling.
Determine what type of sequence the ball's distance can be modelled by and state the common difference or ratio for the sequence.
Write an explicit rule for the distance the ball has rolled, d, after t seconds.
Determine how far the ball will roll after 12 seconds.
A termite treatment will cost \$260 for the first half hour, \$250 for the second half hour, \$240 for the third half hour and so on.
What will be the cost of a treatment that takes 5 hours?
A roof restoration company charges \$250 for the first half hour, \$245 for the second half hour, \$240 for the third half hour and so on.
What will be the cost of a restoration that takes 6 hours?
A worker at a factory is stacking cylindrical-shaped pipes which are stacked in layers. Each layer contains one pipe less than the layer below it. There are 6 pipes in the topmost layer, 7 pipes in the next layer, and so on.
Determine what type of sequence the number of pipes in a layer can be modelled by and state the common difference or ratio for the sequence.
Write an expression for the number of pipes in the nth layer.
How many pipes are in the 10th layer?
For a sprint training exercise, a number of balls are placed in a straight line. The first ball is 2 metres from the start, and there is a 3 metre distance between each of the remaining consecutive balls. There are n balls placed out in the line. Pauline must run from the start, collect the nearest ball and run back to the start, depositing the ball into a box. She must run back to collect the next ball, returning with it to the start. She continues this until all n balls have been collected.
Write an expression for how far Pauline runs to collect and deposit the kth ball.
How far does she runs to collect and deposit the 12th ball?
Eileen starts training for a 3.75\text{ km} charity trail run by running every week for 25 weeks. She runs 1\text{ km} of the course in the first week and each week after that she runs 250\text{ m} more than the previous week. She continues this until she runs the same distance as the trail run. She then continues to run this distance each week.
How far does she run in the 12th week?
What is the total distance that Eileen runs in 25 weeks? Round your answer to two decimal places.
A fishing trawler spends several days netting crabs. On the first day, it nets 530\text{ kg} of crabs, on the next day it nets 512\text{ kg}, and the amount netted continues to decrease by the same amount each day.
How many kilograms are netted on the 10th day?
What is the total weight netted in the first 10 days?
The trawler returns to port when it has netted a total weight of 7992 kg. Find the number of days the trawler spent at sea.
A car bought at the beginning of 2009 is worth \$1500 at the beginning of 2015. The value of the car has depreciated by a constant amount of \$50 each year since it was purchased.
What was the car purchased for in 2009?
Plot the value of the car, V_n, on the graph from 2009 (n = 0) to 2015 (n = 6).
Write an explicit rule for the value of the car after n years.
Find the year, n, at the end of which the car will be worth half the price it was bought for.
A rare figurine was purchased for \$60 and ten years later it is worth \$460.
The figurine appreciated in value by a constant amount each year.
How much did it appreciate each year?
What will the value of the figurine be in another 10 years time?
The balance of a savings account earning simple interest each year is given by the explicit rule V_n = 2100 + 400 \left(n - 1\right), where V_n is the balance after n years.
How much interest is the account earning each year?
How much is in the account after 1 year?
What was the original investment amount?
A mobile phone depreciates in value by a constant amount per month and its value is given by the explicit rule V_n = 1200 - 25 n, where V_n is the balance after n months.
By how much does the value of the phone depreciate each month?
What was the purchase price of the phone?
When a new school first opened, a students started at the school. Each year, the number of students increases by the same amount, d.
At the beginning of its 7th year, it had 361 students. Form an equation for a in terms of d.
At the end of the 11th year, the school had 536 students. Use this to form another equation relating a and d.
Hence, find the value of d.
How many students started at the school when it first opened?
At the end of the nth year, the school has reached its capacity at 921 students. Find the value of n.
Due to favourable market conditions, a company's annual costs are expected to decrease by 5\% per year for the next few years. Their annual costs for the current financial year are \$322\,000.
Write a function, y, to represent the company's annual costs in m years.
Calculate the company's annual costs in 5 years.
The local rodent population, numbering approximately 840, is decreasing at a rate of 4\% per year.
Write a function for the population, y, after m years.
Determine the size of the population after 8 years, rounding your answer to the nearest whole number.
The number of taekwondo clubs worldwide was approximately 2310 in 1988 and has been decreasing at a rate of 2.6\% per year since.
Write a function, y, to represent the number of taekwondo clubs n years after 1988.
Determine the number of taekwondo clubs in 2009. Round your answer to the nearest whole number.
Justin purchased a piece of sports memorabilia for \$2900, and it is expected to increase in value by 9\% per year.
Write a function, y, to represent the value of the piece of sports memorabilia after v years.
Determine the value of the piece of sports memorabilia after 8 years.
A car enthusiast purchases a vintage car for \$220\,000. Each year, its value increases at a rate of 12 percent of its value at the beginning of the year. Find its value after 7 years, to two decimal places.
A ball dropped from a height of 21 metres will bounce back off the ground to 50\% of the height of the previous bounce (or the height from which it is dropped when considering the first bounce).
Write a function, y, to represent the height of the nth bounce.
Calculate the height of the fifth bounce. Round your answer correct to two decimal places.
Average annual salaries are expected to increase by 5 percent each year. If the average annual salary this year is found to be \$49\,000:
Calculate the expected average annual salary in 4 years.
This year, Ray starts at a new job in which he will receive the average annual salary for each year of his employment. Over the coming 4 years (including this year) he plans to save half of each year’s annual salary.
What will be his total savings over these 4 years?
Suppose you save \$1 the first day of a month, \$2 the second day, \$4 the third day, \$8 the fourth day, and so on. That is, each day you save twice as much as you did the day before.
What will your total savings be for the first 12 days?
What will your total savings be for the first 28 days?
Suppose you save \$1 the first day of a month, \$2 the second day, \$4 the third day, \$8 the fourth day, and so on. That is, each day you save twice as much as you did the day before.
What will you put aside for savings on the 17th day of the month?
What will you put aside for savings on the 29th day of the month?
The number of members at the local club is expected to increase by 19\% per year from the current number of 160. The manager of the club is set to receive a bonus when the number of members rises above 727.
Write a function, y, to represent the number of members after t years.
Find n, the number of whole years it takes for the number of members at the club to first rise above 727.
A sample of 2600 bacteria was taken to see how rapidly the bacteria would spread. After 1 day, the number of bacteria was found to be 2912.
By what percentage had the number of bacteria increased over a period of one day?
If the bacteria continue to multiply at this rate each day, what will the number of bacteria grow to eighteen days after the sample was taken? Round your answer to the nearest whole number.
A conveyor belt is being used to remove materials from a quarry. Every thirty minutes, the conveyor belt empties out \dfrac{1}{5} of whatever material remains in the quarry. The quarry initially holds 14\,000 \text{ m}^3 of materials.
How much material has been pumped out after 90 minutes? Round your answer to two decimal places.
How much material is left in the dam after 90 minutes?
A scientist observed that a certain species of bacteria grew by 13\% each hour. How many bacteria will be present after 17 hours if there were 79 initially? Round your answer to the nearest whole number.
A country's population of 18 million people is expected to increase by 2.1\% p.a. over the next 38 years.
Calculate the expected population in 38 years time. Round your answer to the nearest whole number.
The population of a town is expected to double in the next 20 years. At what rate, r, will it grow each year, assuming growth is constant? Round your answer as a percentage rounded to two decimal places.
State whether the following scenarios involve a limiting value:
The amount of radiation in the air after a nuclear disaster if the radiation decreases by half every 100 years.
The number of employees in a company over the next 10 years.
Saving \$100 each week until the savings target is reached.
The value of a piece of machinery if it decreases by 30\% each year.
The zoom function in a camera multiplies the dimensions of an image. In an image, the height of waterfall is 30 mm. After the zoom function is applied once, the height of the waterfall in the image is 36 mm. After a second application, its height is 43.2 mm.
Each time the zoom function is applied, by what factor is the image enlarged?
If the zoom function is applied a third time, what will be the exact height of the waterfall in the image?
The zoom function in a camera multiplies the dimensions of an image by a constant amount. In an image, the height of waterfall is 35\text{ mm} . After the zoom function is applied once, the height of the waterfall in the image is 66.5 mm. After a second application, its height is 126.35\text{ mm}.
Each time the zoom function is applied, by what factor is the image enlarged?
If the zoom function is applied a third time, what will be the height of the waterfall in the image? Round your answer to three decimal places.
The annual output of a coal mine decreases by twenty percent each year. The output in the first year is 274\,000 cubic metres.
Find the total amount of output in the first 8 years, rounded to two decimal places.
Assuming there is always enough coal in the mine for the plant's operations, what is the limit to this plant's total output production? Round your answer to the nearest cubic metre.
To test the effectiveness of a new antibiotic, a certain bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are 14 bacteria in the body, and after four hours the number is found to quadruple.
If the bacterial population continues to quadruple every four hours, how many bacteria will there be in the body after 24 hours?
The antibiotic is applied after 24 hours, and is found to kill half of the germs every four hours. How many bacteria will there be left in the body 24 hours after applying the antibiotic? Assume the bacteria stops multiplying and round your answer to the nearest integer.
The perimeter of a triangle is x cm. A second triangle is formed by joining the midpoints of the three sides. A third, fourth and fifth triangle are formed in the same way (ie by joining the midpoints of the three sides of the preceding triangle). The perimeter of the 4th triangle is 3 cm. Determine:
The ratio of the perimeter of the second triangle to that of the first triangle.
The value of x.
The sum of the perimeters of the first five triangles.
The sum of an infinite number of triangles created in this way.
A car’s brakes failed and the driver immediately turned the engine of the car off. In the first second after the engine was shut off, the car travelled 20 metres. Every successive second, the car travelled eighty percent of the distance covered in the previous second.
Find an expression for the total distance the car travelled in the first n seconds
Hence, find the total distance the car travelled in the first four seconds to two decimal places.
Traffic had built up 109 metres away from where the driver turned off the engine. How far from the traffic build-up did the car come to rest?
The first blow of a hammer drives a post a distance of 64\text{ cm} into the ground. Each successive blow drives the post \dfrac{3}{4} as far as the preceding blow.
In order for the post to become stable, it needs to be driven \dfrac{781}{4}\text{ cm} into the ground. If n is the number of hammer strikes needed for the pole to become stable, find n.
The population of some native bees is declining at a rate of 10\% per year.
Identify whether the population can be modeled by an arithmetic or geometric sequence.
Write an explicit rule for the population P_n after n years. Let P_0 be the starting population.
If there are 23\,700 bees in a hive now, how many will there be in 6 years time?
The government wants to decrease its spending on job creation. Currently it is spending \$11 million and will decrease it by 6.85\% each year over the next 6 years.
Determine what type of sequence the government's spending can be modelled by and state the common difference or ratio for the sequence.
Calculate the government's spending in 6 years' time, rounding to the nearest dollar.
An online business that sells shoes online has been growing very quickly over the last year. They have asked you to model their growth over the next three quarters, and further into the future.
Assuming the number of customers changed in the same way each quarter, complete the table to predict the number of shoes sold over the next three quarters:
| \text{Quarter }(N) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| \text{Online Customers }(C) | 200 | 800 | 3200 | 12,800 |
Find an equation for the number of sales after the nth quarter.
Use the formula to predict how many sales they will have 2.25 years after they started the business.
Explain whether your answer to part (c) is realistic.
The population of cows at a free range farm has a growth rate of 17\% each year. If the farm originally has 140 cows, how many will there be after 4 years?
Tina currently follows 174 people on Twitter. The number of people she follows is growing at a rate of 5\% per month. How many people will she be following in 6 months?
The population of Joanne's home town is 27\,400 and has a growth rate of 6\% each year. What is the estimated population after 7 years?
The area covered by an ice shelf was measured over some warmer months. At the beginning of the first month the ice shelf was spread over an area of 1437 square kilometres, and it was found that this area decreased by 2\% each month over the recording period.
What area was covered by the ice shelf after 13 months of recording? Round your answer to the nearest square kilometre.
Using the rounded value from the previous part, what was the loss in the ice shelf's area over 13 months?
This year, 640 people are expected to enter the workforce as registered nurses. This number is expected to increase by 2 percent next year, and increase by the same percentage every year after that.
What is the explicit rule to determine the number of nurses entering the workforce P_n in the nth year?
Calculate the number of nurses expected to enter the workforce between six and seven years from now. Round your answer to the nearest whole number.
Calculate the number of nurses expected to enter the workforce over the next six years. Round your answer to the nearest whole number.
Cinema ticket sales for a new movie increased by 5\% per day for the first 10 days after its release date, and then decreased by 7\% per day for the next 5 days. \$275\,000 in ticket sales were made on the release date.
Using a geometric sequence, find the total sales for the first 10 days.
Using a geometric sequence, find the total sales for the next 5 days.
Hence or otherwise, find the total sales over the 15day period.
A gym trainer posts Monday's training program on the board, along with how you should progress each day that follows based on your level of fitness. The information is below:
MONDAY TRAINING PROGRAM
9 single rope skips
Weight lift 6\dfrac{1}{2} kg
Rest 2 minutes
Row \dfrac{1}{4} mile
BEGINNER LEVEL: Each day, increase the numbers and time by \dfrac{1}{3} of the first day.
INTERMEDIATE LEVEL: Each day, increase the numbers and time by \dfrac{1}{3} of the previous day.
Using the Intermediate Level training program:
Find the number of single rope skips you would need to complete on Wednesday.
Find the weight you would need to weight lift on Wednesday. Express the weight as a mixed number.
Find the rest time on Wednesday.
Find the distance to be rowed on Wednesday.
Using the Beginner Level training program:
Find the number of single rope skips that will need to be done on Wednesday.
Find the distance you would need to row on Wednesday.
Which level training plan is the most realistic in the long term, Beginner or Intermediate?
Consider the following sequences:
1, x and y are the first three terms of an arithmetic sequence.
1, y and x are also the first three terms in a geometric sequence.
Using the arithmetic sequence, form an equation for y in terms of x.
Using the geometric sequence, form an equation for x in terms of y.
Hence find the value(s) of y.
When y = 1 and x = 1, it produces the sequence 1, 1, 1.
Find the first three values of the arithmetic sequence for the other solution for x and y, along with the common difference.
Find the first three values of the geometric sequence for the other solution for x and y, along with their common ratio.
A piece of machinery depreciated at a constant rate per hour of use. After 170 hours of use, it was worth \$23\,980. After 230 hours of use, it was worth \$23\,620.
What was the amount of depreciation each hour?
Find V_0, the initial value of the machinery.
Write a recursive rule for V_n in terms of V_{n - 1}, and an initial condition V_0.
Write an explicit rule, V_n, for the value of the machinery after n hours of use.
The average annual rate of inflation in Kazakhstan is 2.6\%. Bread cost \$3.65 in 2015.
What would the bread cost in 2016?
At this rate, what would bread cost in 2018?
Write a recursive rule, V_n, defining the cost of bread n years after 2015.
Each day, Namika withdraws \$5 from her bank account to spend on lunch. Before she withdraws this amount on January 1, she has \$3000 in her bank account.
After Namika withdraws \$5 on January 7, how much does she have left in her bank account?
How much does Namika have left in her bank account at the end of February 7?
Write a recursive rule for t_{n+1} in terms of t_n which defines the amount Namika has in her account at the end of day n, and an initial condition for t_0.
Zuber is a taxi service that charges a \$1.50 pick-up fee and \$1.25 per kilometre of travel.
What is the total charge for a 10 km journey?
Write a recursive rule for T_n in terms of T_{n - 1} which defines the price of a n km trip, and an initial condition for T_0.
The value of Beirut Bank shares is decreasing by \$2.05 each day. At the beginning of today's trading, the shares are worth \$42.54.
Today is March 8. How much are they worth at the start of March 17?
Write a recursive rule for V_{n+1} in terms of V_n which defines the the value of the shares at the end of day n, and an initial condition for V_0.
Tabitha is a salesperson and is paid a travel allowance of \$45 for each business trip she takes to demonstrate the use of the product she sells. Each week she is also paid a retainer of \$230.
Calculate the total amount she is paid in a week where she does three business trips.
Write a recursive rule for u_n in terms of u_{n - 1} which defines how much Tabitha is paid in a week where she makes n sales, and an initial condition for u_0.
Write a recursive rule for v_n in terms of v_{n - 1} which defines how much Tabitha is paid in two week where she makes n sales, and an initial condition for v_0.
Jack is learning to drive. His first lesson is 28 minutes long, and each subsequent lesson is 4 minutes longer than the lesson before.
Write a recursive rule with which describes the length of the next lesson, t_{n + 1} based on the previous one, t_n.
How long will his 14th lesson be?
If Jack reaches 10 total hours of lessons during his nth lesson, find the value of n.
What is the total number of hours Yuri has had lessons after his 6th lesson?
A mobile phone depreciates in value by a constant amount per month and its value is given by the explicit rule V_n = 1200 - 20 n, where V_n is the balance after n months.
By how much does the value of the phone depreciate each month?
What was the purchase price of the phone?
Write a recursive rule for V_{n+1} in terms of V_n , and an initial condition V_0.
A motorbike depreciates in value by a constant amount each year and its value is modelled by the recurrence relation:
V_{n+1} = V_n - 1200, \text{ } V_0 = 15000
where V_{n+1} is the value of the motorbike, in dollars, after n+1 years.
State the purchase price of the motorbike.
Identify how much the value is decreasing each year.
Write an explicit rule for V_n that gives the value of the motorbike after n years.
What will the motorbike be worth after 9 years?
A piece of jewellery appreciates in value by a constant amount each year and its value is modelled by the recurrence relation:
V_n = V_{n - 1} + 320, \text{ } V_0 = 2000
where V_n is the value of the jewellery, in dollars, after n years.
State the initial value of the piece of jewellery.
By how much does it appreciate each year?
Write an explicit rule for V_n that gives the value of the piece of jewellery after n years.
What will the investment be worth after 11 years?
Wildlife authorities are trying to control the spread of cane toads in a reserve. Each year there is a 22\% increase in the population size of the cane toads due to breeding and the wildlife authorities aim to capture and remove 500 cane toads each year. At the start of 2015 the cane toad population in the reserve was 2200.
Write a recurrence relation to model the size of the cane toad population, P_n.
How many cane toads will be in the reserve by the end of 2018? Round your answer to the nearest whole number.
Will the removal of 500 cane toads per year ever completely eradicate the cane toad population from this reserve?
Anna's garden has 5000 weeds in it whose population is increasing at a rate of 0.4\% per month. At the end of each month Anna kills 750 weeds with herbicide.
How many weeds are there in Anna's garden at the end of the first month?
Write down the recursive rule for the number of weeds, W_{n+1} in terms of W_n , and an initial condition W_0.
After how many months will Anna have a weed free garden?
The volume of water in a pond is being monitored to ensure that it does not dry out. The pond initially contained 70\,000\text{ L} but it appears that 20\%of water is lost through evaporation each day. This is compensated for by 2000\text{ L} being pumped into the pond each day from a nearby dam.
Using V_{n+1}, write the recurrence relation to model this situation in terms of V_n , and an initial condition V_0.
Determine how many litres of water will be contained in the pond after 20 days. Round your answer to the nearest litre.
How many litres of water will be contained in the pond in the long term?