(a) Without using mathematical table or calculator, evaluate : \(\sqrt{\frac{0.18 \times 12.5}{0.05 \times 0.2}}\). (b) Simplify : \(\frac{8 - 4\sqrt{18}}{\...
Question 1 Report
(a) Without using mathematical table or calculator, evaluate : \(\sqrt{\frac{0.18 \times 12.5}{0.05 \times 0.2}}\).
(b) Simplify : \(\frac{8 - 4\sqrt{18}}{\sqrt{50}}\).
(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.
Answer Details
(a) Simplify inside the root first:
\[\frac{0.18\times12.5}{0.05\times0.2} = \frac{2.25}{0.01} = 225,\qquad \sqrt{225} = 15.\]
(b) Write each surd in simplest form: \(\sqrt{18} = 3\sqrt2\) and \(\sqrt{50} = 5\sqrt2\).
\[\frac{8 - 4\sqrt{18}}{\sqrt{50}} = \frac{8 - 12\sqrt2}{5\sqrt2} = \frac{8}{5\sqrt2} - \frac{12\sqrt2}{5\sqrt2} = \frac{4\sqrt2}{5} - \frac{12}{5} = \frac{4\sqrt2 - 12}{5}.\]
(c) The relation is \(x = \dfrac{k y^3}{z^2}\). Use \(x=108,\,y=3,\,z=4\):
\[108 = \frac{k(27)}{16}\Rightarrow k = \frac{108\times16}{27} = 64.\]
Now find z when \(x=4000,\,y=10\):
\[4000 = \frac{64(1000)}{z^2}\Rightarrow z^2 = \frac{64000}{4000} = 16\Rightarrow z = 4.\]
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