3.3 Applications of trigonometryIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   A vertical tower stands on level ground. A surveyor at point $A$, 40 m from the base of the tower, measures the angle of elevation of the top of the tower as $35^\circ$. Use your GDC where needed.

   (a)

   Find the height of the tower.

   [1 mark]

   • A$28.0$ m
   • B$57.1$ m
   • C$22.9$ m
   • D$32.8$ m

   (b)

   Find the distance from $A$ to the top of the tower.

   [1 mark]

   • A$32.8$ m
   • B$48.8$ m
   • C$69.7$ m
   • D$28.0$ m

   (c)

   A flagpole of height 5 m stands on top of the tower. Find the angle of elevation of the top of the flagpole from $A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ship leaves port $P$ and sails 15 km on a bearing of $070^\circ$ to a point $Q$.

   (a)

   How far east of $P$ is $Q$?

   [1 mark]

   • A$5.13$ km
   • B$41.2$ km
   • C$14.1$ km
   • D$16.0$ km

   (b)

   How far north of $P$ is $Q$?

   [1 mark]

   • A$14.1$ km
   • B$43.9$ km
   • C$41.2$ km
   • D$5.13$ km

   (c)

   Find the bearing of $P$ from $Q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two lifeguard towers $A$ and $B$ stand on a straight, level beach, 200 m apart. A swimmer is at a point $S$ in the sea. From $A$, $S\hat{A}B=52^\circ$ and from $B$, $S\hat{B}A=71^\circ$. Use your GDC where needed.

   (a)

   Find the size of $A\hat{S}B$ and hence find the distance $AS$.

   [3 marks]

   (b)

   Find the shortest distance from $S$ to the beach, and hence find the area of triangle $ABS$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A helicopter flies from its base $A$ on a bearing of $040^\circ$ for 30 km to a point $B$, then on a bearing of $110^\circ$ for 40 km to a point $C$. A radar station at $B$ has a range of 25 km. Use your GDC where needed.

   (a)

   (i) Show that $A\hat{B}C=110^\circ$.
   (ii) Find the distance $AC$.
   (iii) Find the bearing of $C$ from $A$, to the nearest degree.

   [6 marks]

   (b)

   The helicopter later flies directly from $A$ to $C$.
   (i) Find the area of triangle $ABC$.
   (ii) Hence find the shortest distance from $B$ to the line $AC$.
   (iii) Find the length of $AC$ that lies within the range of the radar at $B$.

   [6 marks]

   Total for question 4: 12 marks

End of questions