Mathematics Questions and Answers

The required locus is angle bisector of the two lines

\(\frac{h_1}{h_2}\) = \(\frac{2}{3}\)

h₂ = \(\frac{2h_1}{3}\)

\(\frac{r_1}{r_2}\) = \(\frac{9}{8}\)

r₂ = \(\frac{9r_1}{8}\)

v₁ = \(\pi\)(\(\frac{9r_1}{8}\))₂(\(\frac{2h_1}{3}\))

= \(\pi\)r₁ 2h₁ x \(\frac{27}{32}\)

v = \(\frac{\pi r_1 2h_1 \times \frac{27}{32}}{\pi r_1 2h_1}\) = \(\frac{27}{32}\)

v₂ : v₁ = 27 : 32

Volume of a cylinder = πr\(^2\)h

First convert 3m to cm by multiplying by 100

Volume of External cylinder = π \times 13\(^2\) \times 300

Volume of Internal cylinder = π \times 10\(^2\) \times 300

Hence; Volume of External cylinder - Volume of Internal cylinder

Total volume (v) = π (169 - 100) \times 300

V = π \times 69 \times 300

V = 20700πcm\(^3\)

Cos \(\theta\) = \(\frac{12}{13}\)

x² + 12² = 13²

x² = 169- 144 = 25

x = 25

= 5

Hence, tan\(\theta\) = \(\frac{5}{12}\) and cos\(\theta\) = \(\frac{12}{13}\)

If cos²\(\theta\) = 1 + \(\frac{1}{tan^2\theta}\)

= 1 + \(\frac{1}{\frac{(5)^2}{12}}\)

= 1 + \(\frac{1}{\frac{25}{144}}\)

= 1 + \(\frac{144}{25}\)

= \(\frac{25 + 144}{25}\)

= \(\frac{169}{25}\)

For a regular polygon of n sides

n = \(\frac{360}{\text{Exterior angle}}\)

Exterior < = 180° - 140°

= 40°

n = \(\frac{360}{40}\)

= 9 sides