4 A painting was valued on 1 April 2001 at \(\pounds 5000\).
The value of this painting is modelled by
$$V = A p ^ { t }$$
where \(\pounds V\) is the value \(t\) years after 1 April 2001, and \(A\) and \(p\) are constants.
- Write down the value of \(A\).
- According to the model, the value of this painting on 1 April 2011 was \(\pounds 25000\).
Using this model:
- show that \(p ^ { 10 } = 5\);
- use logarithms to find the year in which the painting will be valued at \(\pounds 75000\).
- A painting by another artist was valued at \(\pounds 2500\) on 1 April 1991. The value of this painting is modelled by
$$W = 2500 q ^ { t }$$
where \(\pounds W\) is the value \(t\) years after 1 April 1991, and \(q\) is a constant.
- Show that, according to the two models, the value of the two paintings will be the same \(T\) years after 1 April 1991,
$$\text { where } T = \frac { \ln \left( \frac { 5 } { 2 } \right) } { \ln \left( \frac { p } { q } \right) }$$
- Given that \(p = 1.029 q\), find the year in which the two paintings will have the same value.
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