Optimisation via quadratic model

5 \includegraphics[max width=\textwidth, alt={}, center]{b6ae63ce-a8a8-45ef-9c75-2fab30de8ad9-2_364_625_1873_762} A farmer divides a rectangular piece of land into 8 equal-sized rectangular sheep pens as shown in the diagram. Each sheep pen measures \(x \mathrm {~m}\) by \(y \mathrm {~m}\) and is fully enclosed by metal fencing. The farmer uses 480 m of fencing.

1. Show that the total area of land used for the sheep pens, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 384 x - 9.6 x ^ { 2 }$$
2. Given that \(x\) and \(y\) can vary, find the dimensions of each sheep pen for which the value of \(A\) is a maximum. (There is no need to verify that the value of \(A\) is a maximum.)

7 \includegraphics[max width=\textwidth, alt={}, center]{56d376c5-b91f-488d-89e2-18edcb14052d-3_534_895_255_625} The diagram shows the dimensions in metres of an L-shaped garden. The perimeter of the garden is 48 m .

1. Find an expression for \(y\) in terms of \(x\).
2. Given that the area of the garden is \(A \mathrm {~m} ^ { 2 }\), show that \(A = 48 x - 8 x ^ { 2 }\).
3. Given that \(x\) can vary, find the maximum area of the garden, showing that this is a maximum value rather than a minimum value.

3 \includegraphics[max width=\textwidth, alt={}, center]{11bfe5bd-604c-43e5-81e7-4c1f5676bcbb-2_485_755_751_696} The diagram shows a plan for a rectangular park \(A B C D\), in which \(A B = 40 \mathrm {~m}\) and \(A D = 60 \mathrm {~m}\). Points \(X\) and \(Y\) lie on \(B C\) and \(C D\) respectively and \(A X , X Y\) and \(Y A\) are paths that surround a triangular playground. The length of \(D Y\) is \(x \mathrm {~m}\) and the length of \(X C\) is \(2 x \mathrm {~m}\).

1. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the playground is given by $$A = x ^ { 2 } - 30 x + 1200$$
2. Given that \(x\) can vary, find the minimum area of the playground.

3 A publisher has to choose the price at which to sell a certain new book. The total profit, \(\pounds t\), that the publisher will make depends on the price, \(\pounds p\). 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. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Explain how model A is inconsistent with one of the assumptions given above.
2. Given that the equation of the curve in model B is quadratic, show that this equation is of the form \(t = k \left( 12 p - p ^ { 2 } \right)\), and find the value of the constant \(k\).
3. The publisher needs to make a total profit of at least \(\pounds 6400\). 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. Comment briefly on how realistic model B may be in the following cases.

6. \begin{figure}[h] \end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,

1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
   State, according to the model,
3. 1. the maximum possible annual profit,
   2. the selling price of the toy that maximises the annual profit.

1. A company started mining tin in Riverdale on 1st January 2019.

A model to find the total mass of tin that will be mined by the company in Riverdale is given by the equation $$T = 1200 - 3 ( n - 20 ) ^ { 2 }$$ where \(T\) tonnes is the total mass of tin mined in the \(n\) years after the start of mining.
Using this model,

1. calculate the mass of tin that will be mined up to 1st January 2020,
2. deduce the maximum total mass of tin that could be mined,
3. calculate the mass of tin that will be mined in 2023.
4. State, giving reasons, the limitation on the values of \(n\).

1. A small factory makes bars of soap.

On any day, the total cost to the factory, \(\pounds y\), of making \(x\) bars of soap is modelled to be the sum of two separate elements:

• a fixed cost
• a cost that is proportional to the number of bars of soap that are made that day
  1. Write down a general equation linking \(y\) with \(x\), for this model.

The bars of soap are sold for \(\pounds 2\) each.
On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of \(\pounds 80\) Using the above information,

show that \(y = 0.84 x + 428\)

With reference to the model, interpret the significance of the value 0.84 in the equation. Assuming that each bar of soap is sold on the day it is made,

find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.