Home » Mathematics Theory (a) The first term of an Arithmetic Progression (A.P) is 8. The ratio of the…

Mathematics Theory (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)