| Date | May 2011 | Marks available | 2 | Reference code | 11M.1.sl.TZ2.14 |
| Level | SL only | Paper | 1 | Time zone | TZ2 |
| Command term | Calculate | Question number | 14 | Adapted from | N/A |
Question
Shiyun bought a car in 1999. The value of the car \(V\) , in USD, is depreciating according to the exponential model
\[V = 25000 \times {1.5^{ - 0.2t}}{\text{, }}t \geqslant 0\]
where \(t\) is the time, in years, that Shiyun has owned the car.
Write down the value of the car when Shiyun bought it.
Calculate the value of the car three years after Shiyun bought it. Give your answer correct to two decimal places.
Calculate the time for the car to depreciate to half of its value since Shiyun bought it.
Markscheme
\(25000{\text{ USD}}\) (A1) (C1)
[1 mark]
\(25000 \times {1.5^{ - 0.6}}\) (M1)
\(19601.32{\text{ USD}}\) (A1) (C2)
[2 marks]
\(12500 = 25000 \times {1.5^{ - 0.2t}}\) (A1)(ft)(M1)
Notes: Award (A1)(ft) for \(12 500\) seen. Follow through from their answer to part (a). Award (M1) for equating their half value to \(25000 \times {1.5^{ - 0.2t}}\) . Allow the use of an inequality.
OR
Graphical method (sketch):
(A1)(ft) for \(y =12 500\) seen on the sketch. Follow through from their answer to part (a). (A1)(ft)
(M1) for the exponent model and an indication of their intersection with their horizontal line. (M1)
\(8.55\) (A1)(ft) (C3)
[3 marks]
Examiners report
A substituted value of \(t = 1\) in part (a) saw many incorrect answers of \(23052.70\) for this part of the question. Part (b) was better attempted with many correct answers seen. Many candidates picked up the first two marks of part (c) equating a correct expression to half their answer found in part (a). Many though did not seem to know the correct process of using their GDC to find the required answer. Much trial and improvement was seen here with varying degrees of success.
A substituted value of \(t = 1\) in part (a) saw many incorrect answers of \(23052.70\) for this part of the question. Part (b) was better attempted with many correct answers seen. Many candidates picked up the first two marks of part (c) equating a correct expression to half their answer found in part (a). Many though did not seem to know the correct process of using their GDC to find the required answer. Much trial and improvement was seen here with varying degrees of success.
A substituted value of \(t = 1\) in part (a) saw many incorrect answers of \(23052.70\) for this part of the question. Part (b) was better attempted with many correct answers seen. Many candidates picked up the first two marks of part (c) equating a correct expression to half their answer found in part (a). Many though did not seem to know the correct process of using their GDC to find the required answer. Much trial and improvement was seen here with varying degrees of success.
