(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 progr...
Question 1 Report
(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 \(\pi = \frac{22}{7}\)].
Answer Details
(a) Common difference. With \(a = 8\), the 7th and 9th terms are \(8 + 6d\) and \(8 + 8d\), and their ratio is \(5 : 8\):
\[\frac{8 + 6d}{8 + 8d} = \frac{5}{8}\Rightarrow 8(8 + 6d) = 5(8 + 8d).\]
\[64 + 48d = 40 + 40d \Rightarrow 8d = -24 \Rightarrow d = -3.\]
(b)(i) Volume of the sphere (\(r = 2\) cm):
\[V = \tfrac{4}{3}\pi r^3 = \tfrac{4}{3}\times\tfrac{22}{7}\times8 = \frac{704}{21} = 33.52\text{ cm}^3.\]
(ii) Density:
\[\rho = \frac{\text{mass}}{\text{volume}} = \frac{11.2}{33.52} = 0.334\text{ g/cm}^3.\]
(iii) Mass of a sphere of radius 3 cm (same material). Mass varies as \(r^3\):
\[\text{mass} = 11.2\times\left(\frac{3}{2}\right)^3 = 11.2\times\frac{27}{8} = 37.8\text{ g}.\]
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