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Recursion, GEN2 2024 NHT 8

Cleo owns equipment that was purchased for $50 000.

She depreciates the value of the equipment using the unit cost method.

Let \(V_n\) be the value of the equipment, in dollars, after \(n\) units of use.

A recurrence relation that can model this value from one unit of use to the next is given by

\(V_0=50\,000, \quad V_{n+1}=V_n-k\)

  1. What does \(k\) represent in this recurrence relation?   (1 mark)

  2. If  \(k=12.50\), determine the value of the equipment after one year if it is used twice per day on all 365 days of the year.   (1 mark)

Another option for Cleo is to depreciate the value of the $50 000 equipment using the reducing balance method.

The value of the equipment, in dollars, after \(n\) months, \(V_n\), can be modelled by a recurrence relation of the form

\(V_0=50\,000, \quad V_{n+1}=R V_n\)

  1. If the depreciation rate per month was 1.5%, what would be the value of \(R\) in this recurrence relation?   (1 mark)

  2. For what value of \(R\) would the equipment be valued at $42 868.75 after three months?   (1 mark)

Show Answers Only

a.   \(k\ \text{represents the depreciation per unit of use.}\)

b.   \(V_{730} =\$40\,875\)

c.   \(R=0.985\)

d.   \(R=0.95\)

Show Worked Solution

a.   \(k\ \text{represents the depreciation per unit of use.}\)
 

b.   \(\text{Units of use}\ = 365 \times 2=730\)

\(V_{730} = 50\,000-(730 \times 12.5)=\$40\,875\)
 

c.   \(R=1-0.015=0.985\)
 

d.   \(\text{Find \(R\) such that:}\)

\(R^{3} \times 50\,000\) \(= 42\,868.75 \)  
\(R\) \(=\sqrt[3]{\dfrac{42\,868.75}{50\,000}}\)  
  \(=0.95\)  

Recursion, GEN1 2022 VCAA 20 MC

Nidhi owns equipment that is used for 10 hours per day for all 365 days of the year.

The value of the equipment is depreciated by Nidhi using the unit cost method.

The value of the equipment, \(E_n\), in dollars, after \(n\) years can be modelled by the recurrence relation

\(E_0=100\ 000,\ \ \ E_{n+1}=E_n-5475\)

The value of the equipment is depreciated by

  1. $1.50 per hour.
  2. $10 per hour.
  3. $15 per hour.
  4. $1.50 per day.
  5. $10 per day.
Show Answers Only

\(A\)

Show Worked Solution

\(\text{1st year depreciation = \$5475}\)

\(\text{Hourly depreciation =\dfrac{5475}{365 \times 10}=$1.50\)

\(\Rightarrow A\)

CORE, FUR2 2021 VCAA 7

Sienna owns a coffee shop.

A coffee machine, purchased for $12 000, is depreciated in value using the unit cost method.

The rate of depreciation is $0.05 per cup of coffee made.

The recurrence relation that models the year-to-year value, in dollars, of the coffee machine is

`M_0 = 12 \ 000,`     `M_{n+1} = M_n - 1440`

  1. Calculate the number of cups of coffee that the machine produces per year.   (1 mark)

  2. The recurrence relation above could also represent the value of the coffee machine depreciating at a flat rate.
  3. What annual flat rate percentage of depreciation is represented?   (1 mark)

  4. Complete the rule below that gives the value of the coffee machine, `M_n`, in dollars, after `n` cups have been produced. Write your answers in the boxes provided.   (1 mark)

     
       `M_n =`
     
    		  `+`  
     
    		 `xx n`  

Show Answers Only

  1. `2880`
  2. `12text{% p.a.}`
  3. `12 \ 000 + (-0.05) xx n`

Show Worked Solution

a.   `text{Cups of coffee}` `= 1440/0.05`
    `= 28 \ 800`

 

b.   `text{Flat rate (%)}` `= {1440}/{12 \000}`
    `= 12text{% p.a.}`

   
c.  `M_n = 12 \ 000 + (-0.05) xx n`

CORE, FUR2 2020 VCAA 7

Samuel owns a printing machine.

The printing machine is depreciated in value by Samuel using flat rate depreciation.

The value of the machine, in dollars, after `n` years, `Vn` , can be modelled by the recurrence relation

`V_0 = 120\ 000, qquad V_(n+1) = V_n-15\ 000`

  1. By what amount, in dollars, does the value of the machine decrease each year?   (1 mark)

  2. Showing recursive calculations, determine the value of the machine, in dollars, after two years.   (1 mark)

  3. What annual flat rate percentage of depreciation is used by Samuel?   (1 mark)

  4. The value of the machine, in dollars, after `n` years, `V_n`, could also be determined using a rule of the form `V_n = a + bn`.

     

    Write down this rule for `V_n`.   (1 mark)

Show Answers Only
  1. `$15\ 000`
  2. `$90\ 000`
  3. `12.5%`
  4. `V_n = 120\ 000-15\ 000n, n = 0, 1, 2, …`
Show Worked Solution

a. `$15\ 000`
  

b.   `V_1` `= 120\ 000-15\ 000 = $105\ 000`
  `V_2` `= 105\ 000-15\ 000 = $90\ 000`

 

c.   `text(Flat rate percentage` `= (15\ 000)/ (120\ 000) xx 100`
    `= 12.5 text(%)`

 

♦ Mean mark part d. 44%.

d.  `V_n = 120\ 000-15\ 000n, \ n = 0, 1, 2, …`

CORE, FUR1 2019 VCAA 22 MC

A machine is purchased for $30 000

It produces 24 000 items each year.

The value of the machine is depreciated using a unit cost method of depreciation.

After three years, the value of the machine is $18 480.

A rule for the value of the machine after `n` units are produced, `V_n`, is

  1. `V_n = 0.872 n`
  2. `V_n = 24\ 000 n - 3840`
  3. `V_n = 30\ 000 - 24\ 000 n`
  4. `V_n = 30\ 000 - 0.872 n`
  5. `V_n = 30\ 000 - 0.16 n`
Show Answers Only

`E`

Show Worked Solution
`text(Unit cost)` `= (30\ 000 – 18\ 480)/(3 xx 24\ 000)`
  `= 0.16`

 
`:. V_n = 30\ 000 – 0.16 n`

`=>  E`

CORE, FUR2 2017 VCAA 5

Alex is a mobile mechanic.

He uses a van to travel to his customers to repair their cars.

The value of Alex’s van is depreciated using the flat rate method of depreciation.

The value of the van, in dollars, after `n` years, `V_n`, can be modelled by the recurrence relation shown below.

`V_0 = 75\ 000 qquad V_(n + 1) = V_n - 3375`

  1. Recursion can be used to calculate the value of the van after two years.

     

    Complete the calculations below by writing the appropriate numbers in the boxes provided.   (2 marks)


    1. By how many dollars is the value of the van depreciated each year?   (1 mark)
    2. Calculate the annual flat rate of depreciation in the value of the van.
    3. Write your answer as a percentage.   (1 mark)
  3. The value of Alex’s van could also be depreciated using the reducing balance method of depreciation.
  4. The value of the van, in dollars, after `n` years, `R_n`, can be modelled by the recurrence relation shown below.

     

            `R_0 = 75\ 000 qquad R_(n + 1) = 0.943R_n`

    At what annual percentage rate is the value of the van depreciated each year?   (1 mark)

Show Answers Only

a.

b.i.  `$3375`

b.ii. `4.5text(%)`

c.  `5.7text(%)`

Show Worked Solution

a.   

  
b.i.   `$3375`
  

b.ii.    `text(Annual Rate)` `= 3375/(75\ 000) xx 100`
    `= 4.5text(%)`

 

c.    `text(Annual Rate)` `= (1-0.943) xx 100text(%)`
    `= 5.7text(%)`

CORE, FUR2 SM-Bank 4

Damon runs a swim school.

The value of his pool pump is depreciated over time using flat rate depreciation.

Damon purchased the pool pump for $28 000 and its value in dollars after `n` years, `P_n`, is modelled by the recursion equation below:

`P_0 = 28\ 000,qquad P_(n + 1) = P_n - 3500`

  1. Write down calculations, using the recurrence relation, to find the pool pump's value after 3 years.   (1 mark)

  2. After how many years will the pump's depreciated value reduce to $7000?   (1 mark)

The reducing balance depreciation method can also be used by Damon.

Using this method, the value of the pump is depreciated by 15% each year.

A recursion relation that models its value in dollars after `n` years, `P_n`, is:

`P_0 = 28\ 000, qquad P_(n + 1) = 0.85P_n`

  1. After how many years does the reducing balance method first give the pump a higher valuation than the flat rate method in part (a)?   (2 marks)

Show Answers Only
  1. `$17\ 500`
  2. `6\ text(years)`
  3. `4\ text(years)`
Show Worked Solution
a.    `P_1` `= 28\ 000-3500 = 24\ 500`
  `P_2` `= 24\ 500-3500 = 21\ 000`
  `P_3` `= 21\ 000-3500 = 17\ 500`

  
`:.\ text(After 3 years, the pump’s value is $17 500.)`
  

b.   `text(Find)\ n\ text(such that:)`

`7000` `= 28\ 000-3500n`
`3500n` `= 21\ 000`
`n` `= (21\ 000)/3500`
  `= 6\ text(years)`

  
c.   `text(Using the reducing balance method)`

`P_1` `= 0.85 xx 28\ 000 = 23\ 800`
`P_2` `= 0.85 xx 23\ 800 = 20\ 230`
`P_3` `= 0.85 xx 20\ 230 = 17\ 195`
`P_4` `= 0.85 xx 17\ 195 = 14\ 615.75`

  
`text{Using the flat rate method (see part (a))}`

`P_4 = 17\ 500-3500 = 14\ 000`

`14\ 615.75 > 14\ 000`
  

`:.\ text(After 4 years, the reducing balance method)`

`text(first values the pump higher.)`

CORE, FUR2 SM-Bank 1

Joe buys a tractor under a buy-back scheme. This scheme gives Joe the right to sell the tractor back to the dealer through either a flat rate depreciation or unit cost depreciation.

  1. The recurrence relation below can be used to calculate the price Joe sells the tractor back to the dealer `(P_n)`, based on the flat rate depreciation, after `n` years
     
    `qquadP_0 = 56\ 000,qquadP_n = P_(n-1)-7000`
     

    1. Write the general rule to find the value of `P_n` in terms of `n`.?   (1 mark)
    2. Hence or otherwise, find the time it will take Joe's tractor to lose half of its value.   (1 mark)
  2. Joe uses the unit cost method to depreciate his tractor, he depreciates $2.75 per kilometre travelled.
    1. How many kilometres does Joe's tractor need to travel for half its value to be depreciated? Round your answer to the nearest kilometre?   (1 mark)
    2. Joe's tractor travels, on average, 2500 kilometres per year. Which method, flat rate depreciation or unit cost depreciation, will result in the greater annual depreciation? Write down the greater depreciation amount correct to the nearest dollar.   (1 mark)

Show Answers Only

  1. i.  `P_n = 56\ 000-7000n`
    ii. `4\ text(years)`
  2. i. `10\ 182\ text{km  (nearest km)}`
    ii. `text(The flat rate depreciation results in an extra)`
         `text($125 depreciation each year.)`

Show Worked Solution

a.i.    `P_1` `= P_0-7000`
  `P_2` `= P_0-7000-7000`
    `= 56\ 000-7000 xx 2`
  `vdots`  
  `P_n`  `= 56\ 000-7000n` 

 

a.ii.    `text(Depreciated value)` `= 56\ 000 ÷ 2=$28\ 000`

`text(Find)\ n,`

`28\ 000` `= 56\ 000-7000n`
`7000n` `= 28\ 000`
`:. n` `= 4\ text(years)`

 

b.i.    `text(Distance travelled)` `= ((56\ 000-28\ 000))/2.75`
    `= 10\ 181.81…`
    `= 10\ 182\ text{km  (nearest km)}`

  
b.ii.   `text(Annual depreciation of unit cost)`

`= 2500 xx $2.75`

`= $6875`

`text(Annual flat rate depreciation = $7000)`

`text(Difference)\ = 7000-6875 = $125`
 

`:.\ text(The flat rate depreciation results in an extra)`

 `text($125 depreciation each year.)`

CORE*, FUR2 2016 VCAA 6

Ken’s first caravan had a purchase price of $38 000.

After eight years, the value of the caravan was $16 000.

  1. Show that the average depreciation in the value of the caravan per year was $2750.   (1 mark)

  2. Let `C_n` be the value of the caravan `n` years after it was purchased.

     

    Assume that the value of the caravan has been depreciated using the flat rate method of depreciation.

     

    Write down a recurrence relation, in terms of `C_(n +1)` and `C_n`, that models the value of the caravan.   (1 mark)

  3. The caravan has travelled an average of 5000 km in each of the eight years since it was purchased.

     

    Assume that the value of the caravan has been depreciated using the unit cost method of depreciation.

     

    By how much is the value of the caravan reduced per kilometre travelled?   (1 mark)

Show Answers Only
  1. `text(Proof)\ text{(See Worked Solutions)}`
  2. `text(See Worked Solutions)`
  3. `$0.55`
Show Worked Solution
a.    `text(Average depreciation)`

`= {(38\ 000-16\ 000)}/8`

`= $2750`


♦♦ Mean mark (a) 39%.
MARKER’S COMMENT: A “show that” question should include an equation
  

b.    `C_0` `= 38\ 000,`
  `C_(n+1)` `= C_n-2750`

♦♦ Mean mark (b) 33%.
MARKER’S COMMENT: A lack of attention to detail and careless errors were common!
  

c.    `text(Total kms travelled)` `= 8 xx 5000`
    `= 40\ 000`

 
`:.\ text(Depreciation per km)`

`= {(38\ 000-16\ 000)}/(40\ 000)`

`= $0.55`


♦♦ Mean mark (c) 29%.