11 The daily world production of oil can be modelled using
$$V = 10 + 100 \left( \frac { t } { 30 } \right) ^ { 3 } - 50 \left( \frac { t } { 30 } \right) ^ { 4 }$$
where \(V\) is volume of oil in millions of barrels, and \(t\) is time in years since 1 January 1980.
11
- The model is used to predict the time, \(T\), when oil production will fall to zero.
Show that \(T\) satisfies the equation
$$T = \sqrt [ 3 ] { 60 T ^ { 2 } + \frac { 162000 } { T } }$$
11
- (ii) Use the iterative formula \(T _ { n + 1 } = \sqrt [ 3 ] { 60 T _ { n } { } ^ { 2 } + \frac { 162000 } { T _ { n } } }\), with \(T _ { 0 } = 38\), to find the values of \(T _ { 1 } , T _ { 2 }\), and \(T _ { 3 }\), giving your answers to three decimal places.
11 - (iii) Explain the relevance of using \(T _ { 0 } = 38\)
11
- From 1 January 1980 the daily use of oil by one technologically developing country can be modelled as
$$V = 4.5 \times 1.063 ^ { t }$$
Use the models to show that the country's use of oil and the world production of oil will be equal during the year 2029.
[0pt]
[4 marks]
\(12 \quad \mathrm { p } ( x ) = 30 x ^ { 3 } - 7 x ^ { 2 } - 7 x + 2\)