1.02z Models in context: use functions in modelling

\includegraphics{figure_5} A horizontal path connects an island to the mainland. On a particular morning, the height of the sea relative to the path, \(H\) m, is modelled by the equation $$H = 0.8 + k \cos(30t - 70)°$$ where \(k\) is a constant and \(t\) is number of hours after midnight. Figure 5 shows a sketch of the graph of \(H\) against \(t\). Use the equation of the model to answer parts (a), (b) and (c).

1. Find the time of day at which the height of the sea is at its maximum. [2] Given that the maximum height of the sea relative to the path is 2 m,
2. 1. find a complete equation for the model,
   2. state the minimum height of the sea relative to the path.
   [2] It is safe to use the path when the sea is 10 centimetres or more below the path.
3. Find the times between which it is safe to use the path. (Solutions relying entirely on calculator technology are not acceptable.) [4]

\includegraphics{figure_10} The figure above shows solid right prism of height \(h\) cm. The cross section of the prism is a circular sector of radius \(r\) cm, subtending an angle of 2 radians at the centre.

1. Given that the volume of the prism is 1000 cm\(^3\), show clearly that $$S = 2r^2 + \frac{4000}{r},$$ where \(S\) cm\(^2\) is the total surface area of the prism. [5]
2. Hence determine the value of \(r\) and the value of \(h\) which make \(S\) least, fully justifying your answer. [7]

A publisher has to choose the price at which to sell a certain new book. The total profit, \(£t\), that the publisher will make depends on the price, \(£p\). He decides to use a model that includes the following assumptions. • If the price is low, many copies will be sold, but the profit on each copy sold will be small, and the total profit will be small. • If the price is high, the profit on each copy sold will be high, but few copies will be sold, and the total profit will be small. The graphs below show two possible models. \includegraphics{figure_3}

1. Explain how model A is inconsistent with one of the assumptions given above. [1]
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k(12p - p^2)\), and find the value of the constant \(k\). [2]
3. The publisher needs to make a total profit of at least £6400. Use the equation found in part (b) to find the range of values within which model B suggests that the price of the book must lie. [4]
4. Comment briefly on how realistic model B may be in the following cases. • \(p = 0\) • \(p = 12.1\) [2]