1.03c Straight line models: in variety of contexts

5. \begin{figure}[h] \end{figure} Figure 2 shows the plan view of a garden.
The shape of the garden \(A B C D E A\) consists of a triangle \(A B E\) and a right-angled triangle \(B C D\) joined to a sector \(B D E\) of a circle with radius 6 m and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(A B = 10.8 \mathrm {~m}\) Angle \(B C D = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.3\) radians and \(A E = 12.2 \mathrm {~m}\)

1. Find the area of the sector \(B D E\), giving your answer in \(\mathrm { m } ^ { 2 }\)
2. Find the size of angle \(A B E\), giving your answer in radians to 2 decimal places.
3. Find the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\) to 3 significant figures.

1. Given that

• the point \(A\) has coordinates \(( 4,2 )\)
• the point \(B\) has coordinates \(( 15,7 )\)
• the line \(l _ { 1 }\) passes through \(A\) and \(B\)
  1. find an equation for \(l _ { 1 }\), giving your answer in the form \(p x + q y + r = 0\) where \(p , q\) and \(r\) are integers to be found.

The line \(l _ { 2 }\) passes through \(A\) and is parallel to the \(x\)-axis.
The point \(C\) lies on \(l _ { 2 }\) so that the length of \(B C\) is \(5 \sqrt { 5 }\)

Find both possible pairs of coordinates of the point \(C\).

Hence find the minimum possible area of triangle \(A B C\).

1. Given that

$$( x - 5 ) ( 2 x + 1 ) ( x + 3 ) \equiv a x ^ { 3 } + b x ^ { 2 } - 32 x - 15$$ where \(a\) and \(b\) are constants,

1. find the value of \(a\) and the value of \(b\).
2. Hence find $$\int \frac { ( x - 5 ) ( 2 x + 1 ) ( x + 3 ) } { 5 \sqrt { x } } \mathrm {~d} x$$ writing each term in simplest form.

1. A tree was planted in the ground.

Its height, \(H\) metres, was measured \(t\) years after planting.
Exactly 3 years after planting, the height of the tree was 2.35 metres.
Exactly 6 years after planting, the height of the tree was 3.28 metres.
Using a linear model,

1. find an equation linking \(H\) with \(t\). The height of the tree was approximately 140 cm when it was planted.
2. Explain whether or not this fact supports the use of the linear model in part (a).

1. In 1997 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(190 \mathrm {~g} / \mathrm { km }\).

In 2005 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK had fallen to \(169 \mathrm {~g} / \mathrm { km }\).
Given \(\mathrm { Ag } / \mathrm { km }\) is the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK \(n\) years after 1997 and using a linear model,

1. form an equation linking \(A\) with \(n\). In 2016 the average \(\mathrm { CO } _ { 2 }\) emissions of new cars in the UK was \(120 \mathrm {~g} / \mathrm { km }\).
2. Comment on the suitability of your model in light of this information.

1. A tank, which contained water, started to leak from a hole in its base.

The volume of water in the tank 24 minutes after the leak started was \(4 \mathrm {~m} ^ { 3 }\) The volume of water in the tank 60 minutes after the leak started was \(2.8 \mathrm {~m} ^ { 3 }\) The volume of water, \(V \mathrm {~m} ^ { 3 }\), in the tank \(t\) minutes after the leak started, can be described by a linear model between \(V\) and \(t\).

1. Find an equation linking \(V\) with \(t\). Use this model to find
2. 1. the initial volume of water in the tank,
   2. the time taken for the tank to empty.
3. Suggest a reason why this linear model may not be suitable.

3. \begin{figure}[h] \end{figure} The points \(A ( 3,0 )\) and \(B ( 0,4 )\) are two vertices of the rectangle \(A B C D\), as shown in Fig. 2.

1. Write down the gradient of \(A B\) and hence the gradient of \(B C\). The point \(C\) has coordinates \(( 8 , k )\), where \(k\) is a positive constant.
2. Find the length of \(B C\) in terms of \(k\). Given that the length of \(B C\) is 10 and using your answer to part (b),
3. find the value of \(k\),
4. find the coordinates of \(D\).

1. The circle \(C\) has equation

$$x ^ { 2 } + y ^ { 2 } + 4 x - 30 y + 209 = 0$$

1. Find
   1. the coordinates of the centre of \(C\),
   2. the exact value of the radius of \(C\). The line \(L\) has equation \(y = m x + 1\), where \(m\) is a constant.
      Given that \(L\) is the tangent to \(C\) at the point \(P\),
2. show that $$2 m ^ { 2 } - 7 m - 22 = 0$$
3. Hence find the possible pairs of coordinates of \(P\).