(a) Without using mathematical table or calculator, evaluate : (sqrt{frac{0.18 times 12.5}{0.05 times 0.2}}).
(b) Simplify : (frac{8 – 4sqrt{18}}{sqrt{50}}).
(c) x, y and z are related such that x varies directly as the cube of y and inversely as the square of z. If x = 108 when y = 3 and z = 4, find z when x = 4000 and y = 10.
Explanation
(a) (sqrt{frac{0.18 times 12.5}{0.05 times 0.2}})
(frac{0.18 times 12.5}{0.05 times 0.2} = frac{18 times 10^{-2} times 125 times 10^{-1}}{5 times 10^{-2} times 2 times 10^{-1}})
= (9 times 25 times 10^{-3 – (-3)})
= (9 times 25)
= (225)
(therefore sqrt{frac{0.18 times 12.5}{0.05 times 0.2}} = sqrt{225})
= (15).
(b) (frac{8 – 4sqrt{18}}{sqrt{50}})
(sqrt{18} = sqrt{9 times 2} = 3sqrt{2})
(sqrt{50} = sqrt{25 times 2} = 5sqrt{2})
(frac{8 – 4(3sqrt{2})}{5sqrt{2}} = frac{8 – 12sqrt{2}}{5sqrt{2}})
Rationalising, we have
= (frac{8 – 12sqrt{2}}{5sqrt{2}} times frac{sqrt{2}}{sqrt{2}})
= (frac{8sqrt{2} – 24}{10})
= (0.8sqrt{2} – 2.4)
= (-2.4 + 0.8sqrt{2})
(c) (x propto frac{y^{3}}{z^{2}})
(implies x = frac{ky^{3}}{z^{2}})
(108 = frac{k times 3^{3}}{4^{2}})
(k = frac{108 times 16}{27})
(k = 64)
(therefore x = frac{64y^{3}}{z^{2}})
(therefore 4000 = frac{64 times 10^{3}}{z^{2}})
(4 = frac{64}{z^{2}})
(z^{2} = frac{64}{4} = 16)
(z = sqrt{16} = pm 4)