1. 1
   A ladder leans against a vertical wall, forming a right-angled triangle with the ground. The base of the ladder is 6 m from the wall, and it reaches 8 m up the wall.

   (a)

   A right-angled triangle has legs of length 6 cm and 8 cm. Find the length of the hypotenuse.

   [1 mark]

   • A$10$ cm
   • B$14$ cm
   • C$48$ cm
   • D$7$ cm

   (b)

   Using the ladder's base distance of 6 m and height reached of 8 m, find the angle the ladder makes with the ground, to the nearest degree.

   [1 mark]

   • A$48°$
   • B$37°$
   • C$53°$
   • D$42°$

   (c)

   Which trigonometric ratio directly relates the angle at the base of the ladder to the opposite side (height) and adjacent side (base distance)?

   [1 mark]

   • ASine
   • BTangent
   • CCosine
   • DPythagoras' theorem

   Total for question 1: 3 marks

2. 2
   A farmer walks diagonally across a rectangular field measuring 24 m by 7 m, and wants to know the angle this diagonal path makes with the longer side.

   (a)

   A rectangular field has length 24 m and width 7 m. Find the length of the diagonal path crossing the field.

   [2 marks]

   (b)

   Find the angle the diagonal path makes with the longer side (24 m) of the field, to 1 decimal place.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A surveyor stands 50 m from the base of a tower, measuring angles of elevation and depression to determine its height and the distance to a nearby object on the ground.

   (a)

   A surveyor stands 50 m from the base of a tower and measures the angle of elevation to the top of the tower as $34°$. Calculate the height of the tower, giving your answer to 1 decimal place.

   [3 marks]

   (b)

   From the top of the tower (height found in part (a)), the angle of depression to a small boat on the ground is $22°$. Calculate the horizontal distance from the base of the tower to the boat, giving your answer to 1 decimal place.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A construction team is designing a wheelchair ramp that rises 1.5 m over a 9 m horizontal run, and must check the ramp's angle of incline meets safety regulations.

   (a)

   A ramp is being designed to rise 1.5 m over a horizontal distance of 9 m. Using Pythagoras' theorem, find the length of the ramp's sloped surface, giving your answer to 1 decimal place.

   [4 marks]

   (b)

   Calculate the angle of incline the ramp makes with the horizontal ground, giving your answer correct to 1 decimal place.

   [4 marks]

   (c)

   Safety regulations require the ramp's angle of incline to be no more than $10°$. State, with a calculation-based reason, whether this ramp design meets the regulation. Then, if a stricter regulation required the angle to be no more than $8°$, calculate the minimum horizontal run needed (keeping the 1.5 m rise the same), to 1 decimal place.

   [5 marks]

   Total for question 4: 13 marks

End of questions