(a) The probability that an athlete will not win any of three races is 1/4.If the athlete runs in all the races, what is the probability that the athlete will win;
(i) only the second race; (ii) all the three races; (ii) only two of the races?
(b) A cone with perpendicular height 24cm has a volume of 1200cm\(^3\). Find the volume of a cone with same base radius and height 84cm. [Take pi = \(\frac{22}{7}\)]
9(a)(i)
p = \(\frac{1}{4}\),
p = 1 - \(\frac{1}{4}\)
p = \(\frac{3}{4}\)
= \(\frac{3}{64}\)
= 0.04687 ≈ 0.05
(ii) p(winning all) = \(\frac{3}{4}\) * \(\frac{3}{4}\) * \(\frac{3}{4}\)
= \(\frac{27}{64}\) = 0.422
(iii) p(winning two)
= (\(\frac{3}{4}\) * \(\frac{3}{4}\) * \(\frac{1}{4}\)) + (\(\frac{3}{4}\) * \(\frac{1}{4}\) * \(\frac{3}{4}\)) + (\(\frac{1}{4}\) * \(\frac{3}{4}\) * \(\frac{3}{4}\))
= \(\frac{9}{64}\) + \(\frac{9}{64}\) + \(\frac{9}{64}\)
= \(\frac{27}{64}\) = 0.4218
= 0.42
(b) Volume of a cone = 1/3πr\(^2\)h
1200 = \(\frac{1}{3}\) * \(\frac{22}{7}\) * r\(^2\) * 24
r\(^2\) = \(\frac{1200 * 3 * 7}{22 * 24}\)
r\(^2\) = \(\frac{25200}{528}\)
r\(^2\) = 47.727
r = \(\sqrt 47.727\)
r = 6.9085cm
Volume of a cone with same radius and height 84cm:
v = \(\frac{1}{3}\) * \(\frac{22}{7}\) * (6.9085)\(^2\) * 84
= \(\frac{1}{3}\) * \(\frac{22}{7}\) * 47.727 * 84
volume = 4200cm\(^3\)
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