Applied quadratic optimization

A piece of wire of length 24 cm is bent to form the perimeter of a sector of a circle of radius \(r\) cm.

1. Show that the area of the sector, \(A\) cm\(^2\), is given by \(A = 12r - r^2\). [3]
2. Express \(A\) in the form \(a - (r - b)^2\), where \(a\) and \(b\) are constants. [2]
3. Given that \(r\) can vary, state the greatest value of \(A\) and find the corresponding angle of the sector. [2]

\includegraphics{figure_1} A stone is thrown over level ground from the top of a tower, \(X\). The height, \(h\), in meters, of the stone above the ground level after \(t\) seconds is modelled by the function. $$h(t) = 7 + 21t - 4.9t^2, \quad t \geq 0$$ A sketch of \(h\) against \(t\) is shown in Figure 1. Using the model,

1. give a physical interpretation of the meaning of the constant term 7 in the model. [1]
2. find the time taken after the stone is thrown for it to reach ground level. [3]
3. Rearrange \(h(t)\) into the form \(A - B(t - C)^2\), where \(A\), \(B\) and \(C\) are constants to be found. [3]
4. Using your answer to part c or otherwise, find the maximum height of the stone above the ground, and the time after which this maximum height is reached. [2]

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]