Applications of right-angled trigonometryIB MYP Maths Extended: Subtopic test

1. 1
   A cuboid $ABCDEFGH$ has a rectangular base $ABCD$ with $AB=8$ cm and $BC=6$ cm. The vertical edges are $AE$, $BF$, $CG$ and $DH$, and $AE=5$ cm.

   (a)

   Find the length of the diagonal $AC$ of the base.

   [1 mark]

   • A$14$ cm
   • B$10$ cm
   • C$28$ cm
   • D$100$ cm

   (b)

   Find the length of the space diagonal $AG$.

   [1 mark]

   • A$11.2$ cm
   • B$19$ cm
   • C$10$ cm
   • D$125$ cm

   (c)

   Find the angle between $AG$ and the base $ABCD$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A pyramid has a square base $PQRS$ of side $10$ cm. Its apex $V$ is directly above the centre $M$ of the base, and $VM=12$ cm.

   (a)

   Find the length of $PM$.

   [1 mark]

   • A$5$ cm
   • B$10$ cm
   • C$14.1$ cm
   • D$7.07$ cm

   (b)

   Find the length of the sloping edge $VP$.

   [1 mark]

   • A$15.6$ cm
   • B$13.0$ cm
   • C$13.9$ cm
   • D$19.1$ cm

   (c)

   Find the angle between $VP$ and the base.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ship leaves port $P$ and sails $40$ km on a bearing of $035^\circ$ to a point $Q$. A lighthouse $L$ is $50$ km due north of $P$.

   (a)

   Find how far $Q$ is north of $P$ and how far $Q$ is east of $P$. Give both answers to $3$ significant figures.

   [3 marks]

   (b)

   Find the distance $QL$ and the bearing of $L$ from $Q$. Give the distance to $3$ significant figures and the bearing to the nearest degree.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A cable car runs in a straight line from a bottom station $A$ to a top station $B$. The horizontal distance between $A$ and $B$ is $1200$ m, and $B$ is $350$ m higher than $A$. Safety rules say the cable must not rise at more than $20^\circ$ to the horizontal. The cable car moves at $4$ m/s.

   (a)

   (i) Find the length of the cable $AB$.
   (ii) Find the angle at which the cable rises, to $1$ decimal place.
   (iii) Find the time for one journey from $A$ to $B$, in seconds.

   [6 marks]

   (b)

   An engineer proposes moving $B$ so that it is $500$ m higher than $A$, with the horizontal distance still $1200$ m. Show whether the new cable meets the safety rule. Find the greatest whole number of metres by which $B$ can be higher than $A$ and still meet the rule, and explain why a real cable may not follow your calculation exactly.

   [6 marks]

   Total for question 4: 12 marks

End of questions