TrigonometryIB MYP Maths Extended: Topic test

1. 1
   A wheelchair ramp at a library in Accra rises $0.45$ m vertically over a horizontal distance of $5.4$ m. The ramp is a straight, sloping surface on level ground.

   (a)

   Find the angle the ramp makes with the horizontal, correct to $1$ decimal place.

   [1 mark]

   • A$85.2^\circ$
   • B$0.083^\circ$
   • C$4.8^\circ$
   • D$12^\circ$

   (b)

   Which expression gives the length, in metres, of the sloping surface of the ramp?

   [1 mark]

   • A$\sqrt{5.4^2+0.45^2}$
   • B$\sqrt{5.4^2-0.45^2}$
   • C$5.4+0.45$
   • D$5.4\times0.45$

   (c)

   A second ramp at the café next door has the same angle and rises $0.30$ m. Find its horizontal length.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ferry leaves Piraeus harbour $P$ and sails on a bearing of $125^\circ$ to an island $I$. A second boat leaves $P$ at the same time and sails on a bearing of $215^\circ$ to an island $J$.

   (a)

   What is the bearing of $P$ from $I$?

   [1 mark]

   • A$055^\circ$
   • B$125^\circ$
   • C$235^\circ$
   • D$305^\circ$

   (b)

   What is the angle $I\hat PJ$ between the two paths at $P$?

   [1 mark]

   • A$35^\circ$
   • B$90^\circ$
   • C$270^\circ$
   • D$340^\circ$

   (c)

   $PI=24$ km and $PJ=18$ km. Find the distance $IJ$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Mountain rescue teams stand at points $A$ and $B$ on a straight, level path, $450$ m apart. A stranded climber is seen at $C$ on the hillside, with $C\hat AB=58^\circ$ and $C\hat BA=71^\circ$. Treat $A$, $B$ and $C$ as lying in one flat plane.

   (a)

   Find the distance $AC$.

   [3 marks]

   (b)

   Find the area of triangle $ABC$ and hence the shortest distance from $C$ to the path $AB$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A search aircraft takes off from airport $A$ and flies $80$ km on a bearing of $060^\circ$ to a point $B$. It then turns and flies $100$ km on a bearing of $128^\circ$ to a point $C$.

   (a)

   (i) Show that angle $A\hat BC=112^\circ$.
   (ii) Calculate the distance $AC$.

   [6 marks]

   (b)

   (i) Use the sine rule to find angle $B\hat AC$.
   (ii) Find the bearing of $C$ from $A$.
   (iii) Find the bearing of $A$ from $C$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A cylindrical pencil pot in a stationery shop in Taipei has an inside diameter of $8$ cm and an inside height of $12$ cm. A straight pencil rests with its lower end at the bottom edge of the pot on one side and touches the rim on the opposite side.

   (a)

   How long is the part of the pencil inside the pot, to $1$ decimal place?

   [1 mark]

   • A$20$ cm
   • B$14.4$ cm
   • C$8.9$ cm
   • D$208$ cm

   (b)

   What angle does the pencil make with the base of the pot?

   [1 mark]

   • A$33.7^\circ$
   • B$36.9^\circ$
   • C$53.1^\circ$
   • D$56.3^\circ$

   (c)

   The pencil is actually $20$ cm long and stays in the same line. Find the height of its top end above the rim of the pot.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The displacement $d$ cm of a mass hanging from a spring in a school laboratory is modelled by $d=5\cos(90t)+2$, where $t$ is the time in seconds and angles are measured in degrees.

   (a)

   What is the amplitude of this function?

   [1 mark]

   • A$5$
   • B$7$
   • C$2$
   • D$90$

   (b)

   What is the period of the motion, in seconds?

   [1 mark]

   • A$90$
   • B$0.25$
   • C$4$
   • D$360$

   (c)

   A stiffer spring changes the model to $d=5\cos(bt)+2$ and the period becomes $1.5$ seconds. Find $b$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A triangular community garden $PQR$ in Medellín has $PQ=14$ m and $PR=22$ m, and the angle between these two sides at $P$ is $68^\circ$.

   (a)

   Find the length of the side $QR$.

   [3 marks]

   (b)

   Find the area of the garden and hence the shortest distance from $P$ to $QR$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A giant wheel in a Singapore park turns steadily. The height $h$ metres of a rider above the ground is modelled by $h=65-60\cos(6t)$, where $t$ is the time in seconds after the rider leaves the lowest point and angles are in degrees, for $0\le t\le120$.

   (a)

   (i) State the amplitude, the period and the greatest height of the rider.
   (ii) Find the height of the rider after $10$ seconds.
   (iii) Find the height of the rider after $45$ seconds.

   [6 marks]

   (b)

   (i) Solve $h=95$ for $0\le t\le60$.
   (ii) For how many seconds in each turn of the wheel is the rider higher than $95$ m?

   [6 marks]

   Total for question 8: 12 marks

End of questions