Question
Consider the infinite geometric sequence 4480, -3360, 2520, -1890, 000
(a) Find the common ratio, r.
(b) Find the 20th term.
(c) Find the exact sum of the infinite sequence.
Answer/Explanation
Ans:
(a) We have u1 = 4480 and u2 = -3360.
Hence the common ratio is
\(r = \frac{u_{2}}{u_{1}}\)
\(=-\frac{3360}{4480}\)
= -0.75
(b) Using the nth term formula un = u1rn-1 with n = 20, we get
u20 = u1r20-1
= (4480) (-0.75)20-1
≈ -18.9
(c) Using the sum of an infinite geometric sequence formula \(S_{\infty }=\frac{u_{1}}{1-r},\) we find
\(S_{\infty }=\frac{4480}{1-(-0.75)}\)
=2560
Question
A bronze sphere has a radius of 10.5 cm.
(a) Find the volume of the sphere, expressing your answer in the form a × 10k, 1 ≤ a ≤ 10 and k ∈ Z+.
The sphere is to be melted down and remoulded into the shape of a cone with a height of 11.9 cm.
(b) Find the radius of the base of the cone, giving your answer correct to 3 significant figures.
Answer/Explanation
Ans:
(a) Using the volume of a sphere formula, we get
\(V_{sphere}=\frac{4}{3}\pi r^{3}\)
\(=\frac{4}{3}\pi (10.5)^{3}\)
≈ 4849.05 cm3
≈ 4.85 × 103 cm3
(b) Using the volume of a cone formula, we obtain
\(V_{cone}=\frac{1}{3}\pi r^{2}h\)
\(4849.5=\frac{1}{3}\pi r^{2}(11.9)\) [since Vcone = V sphere]
r ≈ 19.73 cm [by using G.D.C.]
r ≈ 19.7 cm