(a) The first term of an Arithmetic Progression (A.P) is 8. The ratio of the...
(a) The first term of an Arithmetic Progression (A.P) is 8. The ratio of the 7th term to the 9th term is 5 : 8. Calculate the common difference of the progression.
(b) A sphere of radius 2 cm is of mass 11.2g. Find (i) the volume of the sphere ; (ii) the density of the sphere ; (iii) the mass of a sphere of the same material but with radius 3cm. [Take \(\pi = \frac{22}{7}\)].
Explanation
(a) \(T_{n} = a + (n - 1)d\) (terms of an AP)
Given a = -8;
\(T_{7} = a + 6d = -8 + 6d\)
\(T_{9} = a + 8d = -8 + 8d\)
\(\frac{-8 + 6d}{-8 + 8d} = \frac{5}{8}\)
\(5(-8 + 8d) = 8(-8 + 6d)\)
\(-40 + 40d = -64 + 48d\)
\(-40 + 64 = 48d - 40d \times 24 = 8d\)
\( d = 3\)
(b) Given r = 2 cm, m = 11.2g
(i) \(V = \frac{4}{3} \pi r^{3}\)
= \(\frac{4}{3} \times \frac{22}{7} \times 2^{3}\)
= \(\frac{704}{21}\)
= \(33.52 cm^{3} = 33.52 \times 10^{-6} m^{3}\)
= \(3.352 \times 10^{-5} m^{3}\)
(ii) \(Density = \frac{mass}{volume}\)
= \(\frac{11.2}{33.52}\)
= \(0.334 g/cm^{3}\)
(iii) \(V = \frac{4}{3} \pi r^{3}\)
= \(\frac{4}{3} \times \frac{22}{7} \times 3^{3}\)
= \(\frac{2376}{21}\)
= \(113.14 cm^{3}\)
\(mass = density \times volume\)
\(0.334 \times 113.14 = 37.789g\)