Radioactive element D loses half its mass every day.
Complete the table for element D:
| \text{Day} | \text{Mass of element }D\text{ (g)} |
|---|---|
| 0 | 800 |
| 1 | |
| 2 | |
| 3 | |
| 4 |
Is this type of decay linear or exponential?
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.
How much will you put aside for savings on the 17th day of the month?
How much will you put aside for savings on the 29th day of the month?
How much will your total savings be for the first 13 days?
How much will your total savings be for the first 29 days?
A car enthusiast purchases a vintage car for \$220\,000. Each year, its value increases by 12 percent of its value at the beginning of the year.
Find the value of the car after 7 years.
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:
Monday training program
| Single rope skips | 9 |
|---|---|
| Weight lift | 6 \dfrac{1}{2} \text{ kg} |
| Rest | 2 \text{ minutes} |
| Row | \dfrac{1}{4} \text{ km} |
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 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? Explain your answer.
Valerie invests \$24\,000 in an account that earns 6\% p.a. with interest calculated at the end of each year.
Complete the table, calculating the value of the investment, A, at the end of each year for the first four years. Round all values to the nearest dollar.
| n | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| A | 24\,000 |
State the common ratio between consecutive values of the investment from year to year.
Find the equation that models the value of the investment, A, after n years.
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 to the nearest whole number.
The zoom function in a camera multiplies the dimensions of an image. In an image, the height of waterfall is 30\text{ mm}. After the zoom function is applied once, the height of the waterfall in the image is 36\text{ mm}. After a second application, its height is 43.2\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, find the exact height of the waterfall in the image.
The zoom function in a camera multiplies the dimensions of an image. In an image, the height of building is 35\text{ mm}. After the zoom function is applied once, the height of the building in the image is 66.5\text{ 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, find the exact height of the building in the image.
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?
How much material is left in the dam after 90 minutes?
The average rate of depreciation of the value of a Ferrari is 14\% per year. A new Ferrari is bought for \$60\,000.
Find the Ferrari's value after 1 year.
Find the Ferrari's value after 3 years.
Write a recursive rule, V_{n + 1}, defining the value of the Ferrari after \left(n + 1\right) years with intial value V_0.
The average daily growth of a seedling is 6\% per day. A seedling measuring 8\text{ cm} in height is planted.
Find the height of the seedling at the end of day 1.
Find the height of the seedling, to the nearest hundredth, 4 days after it is planted.
Write a recursive rule, H_{n+1}, defining the height of the seedling n+1 days after it is planted with initial height H_0.
The average annual rate of inflation in Azerbaijan is 2.3\%. Bread costs \$4.15 in 2015.
Find the cost of bread in 2016.
At this rate, find the cost of bread in 2018.
Write a recursive rule, V_{n + 1}, defining the cost of bread \left(n + 1\right) years after 2015 with initial condition V_0.
A ball is dropped from a height of 9 metres. After each bounce, it rebounds back up to 70\% of its previous height.
Let h_n be the height of the ball after the nth bounce.
Find h_1.
Write a recursive rule for h_{n + 1} in terms of h_n that describes the height of the ball.
Write a general formula for h_n in terms of n.
Hence, find the height of the ball after the 4th bounce.
A new car purchased for \$38\,200 depreciates at a rate, r , each year. The value of the car over the first three years is shown in the following table:
| \text{Years passed } (n) | 0 | 1 | 2 |
|---|---|---|---|
| \text{Value of car } (A) | 38\,200 | 37\,818 | 37\,439.82 |
Use the table of values to determine the value of r.
Determine the rule for A, the value of the car, n years after it is purchased.
Assuming the rate of depreciation remains constant, how much can the car be sold for after 6 years?
A new motorbike purchased for the same amount depreciates according to the model
V = 38200 \times 0.97^{n}. Which vehicle depreciates more rapidly?
A bouncy ball is dropped onto the ground from a height of 13 metres. On each bounce, the ball reaches a maximum height of 50\% of its previous maximum height.
Write a recursive rule for a_{n + 1}, the height of the ball on the \left(n + 1\right)th bounce, in terms of a_n and an initial condition a_0.
Write a general formula for a_n, for the height reached on the nth bounce in terms of n.
How high does the bouncy ball reach after the 4th bounce? Round your answer to two decimal places.
In a new email marketing campaign, a company monitors how many of their emails to customers were opened each day.
-
On the first day, 4000 were opened.
On the second day, 3600 were opened.
On the third day, 3240 were opened.
By what percentage is the number of emails opened each day decreasing?
How many emails are opened on the 6th day? Round your answer to the nearest whole number.
On day 12, 90\% of emails that were sent were opened. How many emails were sent on that day? Round your answer to the nearest whole number.
Mato's parents offer him two ways to be paid his pocket money:
-
Option A: They credit \$25 to his account each week
Option B: They credit \$1 to his account the first week, \$1.20 the second week, \$1.44 the third week and so on.
Write a recursive rule for B_{n+1}, the amount received in the (n+1)th week if Mato chooses Option B, in terms of B_{n}.
Under Option B, how much money would he receive in the 10th week?
Write a recursive rule for A_{n+1}, the total amount received after (n+1) weeks if Mato chooses Option A, in terms of A_{n}.
After a year, how much money will Mato have received if he chooses Option A?
After a year, how much money will Mato have received if he chooses Option B? Round your answer to the nearest dollar.
A rectangular poster originally measures 81 centimetres in width and 256 centimetres in length. To edit the poster once, the length of the rectangle is decreased by \dfrac {1}{4} and the width is increased by \dfrac {1}{3}.
If the poster is edited once, find the ratio of the original area of the rectangle to the new area.
If the edit is repeated 3 times, calculate the new area of the poster to the nearest square centimetre.
Find the number of times, n, that the process must be repeated to produce a square poster.
To test the effectiveness of a new antibiotic, first a certain bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are 19 bacteria in the body, and after four hours the number is found to double.
If the bacterial population continues to double 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 one third of the germs every two 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 if necessary.
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.
Dakota colours her hair with a non-permanent colour that gradually washes out over a series of washes. Every time she washes her hair, the colour diminishes by 20\%.
Find the proportion of colour left after 1 wash.
Write a recursive rule which defines a_{n+1}, the proportion of colour left in her hair after (n+1) washes, in terms of a_{n} with initial condition a_0.
Calculate the percentage of colour remaining after 6 washes. Round your answer to two decimal places.
Calculate the percentage of colour lost after 7 washes. Round your answer to two decimal places.
Dakota will colour her hair again once the proportion of colour remaining falls below 20\%. Find n, the number of times she can wash her hair before recolouring.
A real estate agent values a block of land at \$130\,000. In line with market trends, this block of land will increase in value by 6\% each year.
Find the value of the land after 1 year.
Write a recursive rule for A_{n + 1}, the value of the block of land after \left(n + 1\right) years, in terms of A_n with initial condition A_0.
By how much has the block of land increased in value after 7 years? Round your answer to the nearest dollar.
An investor wants to buy the land and sell it once it doubles in value. Find n, the whole number of years she will have to wait before she can sell it.