Perimeter or area constraint inequality

3. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

1. show that \(x > 1.5 \mathrm {~m}\) Given also that the area of the garden is less than \(72 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of possible values of \(x\).
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1. A rectangular room has a width of \(x \mathrm {~m}\).

The length of the room is 4 m longer than its width. Given that the perimeter of the room is greater than 19.2 m ,

1. show that \(x > 2.8\) Given also that the area of the room is less than \(21 \mathrm {~m} ^ { 2 }\),
2. 1. write down an inequality, in terms of \(x\), for the area of the room.
   2. Solve this inequality.
3. Hence find the range of possible values for \(x\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres.
Given that the perimeter of the garden is greater than 40 m ,

1. show that \(x > 1.7\) Given that the area of the garden is less than \(120 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of the possible values of \(x\).

8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.

1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
3. By solving the inequalities in parts (i) and (ii), determine the set of possible values of \(x\).

11 A lawn is to be made in the shape shown below. The units are metres. \includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}

1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).

9

1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).

4. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. by solving your inequalities, find the set of possible values of \(x\).

3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:

• the length of the flower bed is to be 3 m longer than the width,
• the length of the flower bed must be at least 14.5 m ,
• the area of the flower bed must be less than \(180 \mathrm {~m} ^ { 2 }\).

The width of the flower bed is \(x \mathrm {~m}\).
By writing down and solving appropriate inequalities in \(x\), determine the set of possible values for the width of the flower bed.

2 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least \(45 \mathrm {~m} ^ { 2 }\).

1. Write down, in terms of \(x\), an inequality satisfied by
   1. the perimeter of the patio,
   2. the area of the patio.
2. Hence determine the set of possible values of \(x\).