IB DP Math MAA SL : IB Style Mock Exams – Set 5 Paper 2

(a)
Using the 3D distance formula:
\( AB = \sqrt{(6-4)^2 + (-1+5)^2 + (-3+9)^2} \)
\( = \sqrt{2^2 + 4^2 + 6^2} = \sqrt{4 + 16 + 36} = \sqrt{56} \)
\( AB = 2\sqrt{14} \) (exact) or \( \approx 7.48 \) (to 3 s.f.).

(b)
Total surface area of a solid hemisphere:
Curved surface area \( = 2\pi r^2 \), base area \( = \pi r^2 \)
Total \( = 3\pi r^2 \).
Using \( r = 2\sqrt{14} \):
\( r^2 = 56 \)
\( \text{Surface area} = 3\pi \times 56 = 168\pi \) (exact) \( \approx 528 \) (to 3 s.f.).