Re: In a certain bathtub, both the cold-water and the hot-water fixtures
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20 Nov 2021, 02:52
Let the capacity of bucket be \(L\)
rate of cold-water leak, \(r_c = \frac{L}{c}\)
rate of hot-water leak, \(r_h = \frac{L}{h}\)
Both filling bucket together -
\(t(r_c + r_h) = L\)
\(t(\frac{L}{c} + \frac{L}{h}) = L \Rightarrow t = \frac{1}{\frac{1}{c} + \frac{1}{h}} = \frac{ch}{c+h}\)
I - \(0 < t < h\)
This is so obviously true since both filling the bucket together is going to take less time than individual. Hence, true
II - \(c < t < h\)
Same point as stated in I, \(t < c,h\). Hence, false
III - \(\frac{c}{2} < t < \frac{h}{2}\)
\(\frac{c}{2} < t < \frac{h}{2}\)
\(\frac{c}{2} < \frac{ch}{c+h} < \frac{h}{2}\)
\(c\frac{1}{2} < c\frac{h}{c+h}\) and \(h\frac{c}{c+h} < h\frac{1}{2}\)
\(\frac{1}{2} < \frac{h}{c+h}\) and \(\frac{c}{c+h} < \frac{1}{2}\)
\(c+h < 2h\) and \(2c < c+h\)
\(c < h\) and \(c < h\) - this has already been given in question. Hence, true
Therefore, I and III are true
Hence, Answer is E