(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)