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A new high-end coffee machine is purchased for $25 000 in January 2020.
At the end of each year, starting in 2021, the machine depreciates in value by 15% per annum, using the declining balance method of depreciation.
In which year will the value of the machine first fall below $15 000? (2 marks)
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`text(The value falls below $15 000 in the fourth year)`
`text{which will be during 2024.}`
`text(Using the formula:)\ \ S = V_0(1-r)^n`
`text(where) \ \ V_0 = 25\ 000,\ \ r = 0.15`
`text(If)\ n = 1\ \text{(2021)}`
| `S` | `= 25\ 000(0.85)^1` |
| `=21\ 250` |
`text(If)\ n = 2\ \text{(2022)}`
| `S` | `=25\ 000(0.85)^2` |
| `=25\ 000(0.7225)` | |
| `=18\ 062.50` |
`text(If)\ n = 3\ \text{(2023)}`
| `S` | `=25\ 000(0.85)^3` |
| `=25\ 000(0.614125)` | |
| `=15\ 353.13` |
`text(If)\ n = 4\ \text{(2024)}`
| `S` | `=25\ 000(0.85)^4` |
| `=25\ 000(0.52200625)` | |
| `=13\ 050.16` |
`:.\ \text{The value first falls below $15 000 in the fourth year}`
`text{which will be during 2024.}`
Peter installed his new pool fence on 1 January 2013 at a cost of $12 000.
On 1 January of each year after 2013 its value is depreciated by 15% using the reducing balance method.
The value of the pool fence will be below $4000 for the first time on 1 January of what year? Show all working. (3 marks)
--- 5 WORK AREA LINES (style=lined) ---
`text{Test the values of different years using:}\ \ S=V_0(1-r)^n`
`S \text{(Jan17)} = 12\ 000(1-0.15)^4= 5000 xx 0.85^4= $2560`
`S \text{(Jan19)} = 12\ 000 xx 0.85^6= $4525.79`
`S \text{(Jan20)} = 12\ 000 xx 0.85^7= $3846.93`
`\text{2020 is the 1st year its value is below $4000 on 1 January.}`
`r=15% = 0.15`
`text{Test the values of different years using:}\ \ S=V_0(1-r)^n`
`S \text{(Jan17)} = 12\ 000(1-0.15)^4= 5000 xx 0.85^4= $2560`
`S \text{(Jan19)} = 12\ 000 xx 0.85^6= $4525.79`
`S \text{(Jan20)} = 12\ 000 xx 0.85^7= $3846.93`
`\text{2020 is the 1st year its value is below $4000 on 1 January.}`
A sports car worth $150 000 is bought in December 2005.
In December each year, beginning in 2006, the value of the sports car is depreciated by 10% using the declining balance method of depreciation.
In which year will the depreciated value first fall below $120 000? (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
`text(The value falls below $120 000 in the third year)`
`text{which will be during 2008.}`
`text(Using)\ \ S = V_0(1-r)^n`
`text(where)\ \ V_0 = 150\ 000, r = text(10%)`
`text(If)\ \ n = 2,`
| `S` | `= 150\ 000(1-0.1)^2` |
| `= 121\ 500` |
`text(If)\ \ n= 3,`
| `S` | `= 150\ 000(1-0.1)^3` |
| `= 109\ 350` |
`:.\ text(The value falls below $120 000 in the third year)`
`text{which will be during 2008.}`