IB Mathematics SL 3.1 The distance between two points AI SL Paper 1- Exam Style Questions- New Syllabus
Question
Emma designs a doghouse. The front view of the doghouse is made up of a square with an isosceles triangle on top. The doghouse is \( 1.35 \text{ m} \) high and \( 0.9 \text{ m} \) wide, and sits on a square base.
The top of the rectangular surfaces of the roof of the doghouse are to be painted.
Find the area to be painted. [5]
▶️ Answer/Explanation
Markscheme
Height of triangle: \( 1.35 – 0.9 = 0.45 \text{ m} \). A1
Slant height:
\[ \begin{aligned} \text{slant height} &= \sqrt{0.45^2 + 0.45^2} \\ &= \sqrt{0.405} \approx 0.636396 \text{ m} \end{aligned} \] M1 A1
Area of one rectangle:
\[ \begin{aligned} \text{area} &= \sqrt{0.405} \times 0.9 \\ &\approx 0.572756 \text{ m}^2 \end{aligned} \] M1
Total area painted:
\[ \begin{aligned} \text{total area} &= 2 \times \sqrt{0.405} \times 0.9 \\ &= 2 \times 0.572756 \approx 1.15 \text{ m}^2 \end{aligned} \] A1
Area: \( 1.15 \text{ m}^2 \) (or \( 0.81 \sqrt{2} \text{ m}^2 \)).
[5 marks]
Slant height:
\[ \begin{aligned} \text{slant height} &= \sqrt{0.45^2 + 0.45^2} \\ &= \sqrt{0.405} \approx 0.636396 \text{ m} \end{aligned} \] M1 A1
Area of one rectangle:
\[ \begin{aligned} \text{area} &= \sqrt{0.405} \times 0.9 \\ &\approx 0.572756 \text{ m}^2 \end{aligned} \] M1
Total area painted:
\[ \begin{aligned} \text{total area} &= 2 \times \sqrt{0.405} \times 0.9 \\ &= 2 \times 0.572756 \approx 1.15 \text{ m}^2 \end{aligned} \] A1
Area: \( 1.15 \text{ m}^2 \) (or \( 0.81 \sqrt{2} \text{ m}^2 \)).
[5 marks]
Total Marks: 5
Question
A piece of candy is shaped as a solid hemisphere with a radius of 6 mm.
(a) Find the total surface area of one piece of candy. [3]
(b) The entire surface of the candy is coated in chocolate. Given that 1 gram of chocolate covers an area of 240 mm², find the weight of chocolate required to coat one piece of candy. [3]
▶️ Answer/Explanation
Markscheme
(a)
Total surface area: Curved area + base area. Radius \( r = 6 \, \text{mm} \). M1
Curved area: \( 2 \pi r^2 = 2 \pi \times 6^2 = 72 \pi \, \text{mm}^2 \). A1
Base area: \( \pi r^2 = \pi \times 6^2 = 36 \pi \, \text{mm}^2 \).
Total: \( 72 \pi + 36 \pi = 108 \pi \approx 339.292 \approx 339 \, \text{mm}^2 \). A1
[3 marks]
Total surface area: Curved area + base area. Radius \( r = 6 \, \text{mm} \). M1
Curved area: \( 2 \pi r^2 = 2 \pi \times 6^2 = 72 \pi \, \text{mm}^2 \). A1
Base area: \( \pi r^2 = \pi \times 6^2 = 36 \pi \, \text{mm}^2 \).
Total: \( 72 \pi + 36 \pi = 108 \pi \approx 339.292 \approx 339 \, \text{mm}^2 \). A1
[3 marks]
(b)
Surface area: \( 108 \pi \approx 339.292 \, \text{mm}^2 \). M1
Coverage: 1 g covers 240 mm².
Weight: \( \dfrac{339.292}{240} \approx 1.41371 \). A1
Rounded: 1.41 g (exact: \( \dfrac{9 \pi}{20} \)). A1
[3 marks]
Surface area: \( 108 \pi \approx 339.292 \, \text{mm}^2 \). M1
Coverage: 1 g covers 240 mm².
Weight: \( \dfrac{339.292}{240} \approx 1.41371 \). A1
Rounded: 1.41 g (exact: \( \dfrac{9 \pi}{20} \)). A1
[3 marks]
Total Marks: 6