Algebra

Level 2

Zeno's tank holds 24 gallons of water. Zeno starts filling the tank, but when it is halfway full he decides to start draining water out. Once half of the water that he added is drained, he decides to add back half of the water that he just drained. He then drains half of the water that he just added, and continues alternately adding or draining half of the previous quantity of water. After 100 cycles of adding and draining water, how many gallons of water are in the tank? Express your answer to the nearest whole number.

9 5 3 None of these Insufficient information 0 4 8

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2 solutions

Chew-Seong Cheong
Sep 16, 2017

The volume in gallons after n n th cycle of adding and draining is given by:

V n = ( 12 + 3 + 3 4 + + 12 4 n 1 ) Adding ( 6 + 6 4 + 6 4 2 + + 6 4 n 1 ) Draining = ( 6 + 6 4 + 6 4 2 + + 6 4 n 1 ) Remaining after n th cycle = 6 k = 0 n 1 ( 1 4 ) k = 6 × 1 1 4 n 1 1 4 = 8 ( 1 1 4 n ) \begin{aligned} V_n & = \underbrace{\left(12 + 3 + \frac 34 + \cdots + \frac {12}{4^{n-1}}\right)}_{\text{Adding}} - \underbrace{\left(6 + \frac 64 + \frac 6{4^2} + \cdots + \frac 6{4^{n-1}}\right)}_{\text{Draining}} \\ & = \underbrace{\left(6 + \frac 64 + \frac 6{4^2} + \cdots + \frac 6{4^{n-1}}\right)}_{\text{Remaining after }n \text{th cycle}} \\ & = 6\sum_{k=0}^{n-1} \left(\frac 14\right)^k = 6 \times \frac {1-\frac 1{4^n}}{1-\frac 14} = 8 \left(1-\frac 1{4^n} \right) \end{aligned}

V 100 = 8 ( 1 1 4 100 ) 8 \begin{aligned} \implies V_{100} & = 8 \left(1-\frac 1{4^{100}} \right) \approx \boxed{8} \end{aligned}

Sumukh Bansal
Sep 16, 2017

This is an infinite geometric sequence. The first term is 12 and the ratio is -1:2 The few terms are 12,-6,3,... The sum of an infinite geometric sequence with first term a and ratio r is a/1-r Putting the values, you get 12/1-[-1/2]= 12/[3/2]=8

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