A water tank had 1778 liters of water. The volume of the water decreased by 14, 28, and 56 liters over the next three days, respectively. How long will it take to be empty given the water volume decreases following the same pattern?
We are told in the question that a tank contains 1778 liters of water. The amount of water decreases by 14, 28, and 56 liters over the next three days. And we are asked to calculate how long it will take for the tank to empty. We begin by noticing that the values 14, 28, 56 and so on form a geometric sequence. Any geometric sequence has a common ratio, or multiplier, between consecutive terms. Since 14 multiplied by two is 28 and 28 multiplied by two is 56, this sequence has a common ratio equal to two.
We recall that the sum of the first 𝑛 terms of a geometric sequence can be calculated using the formula 𝑎 sub one multiplied by 𝑟 to the power of 𝑛 minus one all divided by 𝑟 minus one. Substituting in the values of 𝑎 sub one and 𝑟 from our sequence, we have 14 multiplied by two to the power of 𝑛 minus one all divided by two minus one. The denominator simplifies to one. Therefore, this expression is equal to 14 multiplied by two to the power of 𝑛 minus one.
As we want the time taken for the tank to empty and the tank started with 1778 liters of water, we can set this expression equal to 1778. We can then divide through by 14 such that two to the power of 𝑛 minus one is equal to 127. Adding one to both sides of this equation, we have two to the power of 𝑛 is equal to 128. We know that 128 is a power of two. Therefore, 𝑛 is an integer value. In fact, two to the seventh power is 128. And this means that 𝑛 is equal to seven. As our units are days, we can therefore conclude that it will take seven days for the water tank to empty.
We can check this answer by calculating the amount of water in the tank at the end of each day. To calculate the amount of water in the tank at the end of day one, we subtract 14 from 1778. This gives us 1764. Next, we can subtract 28 from this as 28 is the second number in our sequence. This gives us 1736. We can then repeat this process for days three, four, five, six, and seven. This confirms that at the end of day seven, the water tank was empty.