Problem 7 - Algebra for College Students

Question

A tank contains 16,000 liters of water. Each day one-half of the water in the tank is removed and not replaced. How much water remains in the tank at the end of 7 days? 125 liters

Step-by-Step Solution

Answer

125 liters remain after 7 days.

Step 1: Understanding the Problem

The task is to find out how much water remains in a tank after removing half of it every day over a span of 7 days. The tank initially contains 16,000 liters of water.

Step 2: Day 1 Calculation

On the first day, half of the water in the tank is removed. So, we calculate the remaining water as follows: Remaining water after Day 1 = \( \frac{1}{2} \times 16000 \) liters = 8000 liters.

Step 3: Day 2 Calculation

On the second day, half of the remaining 8,000 liters of water is removed. Remaining water after Day 2 = \( \frac{1}{2} \times 8000 \) liters = 4000 liters.

Step 4: Day 3 Calculation

On the third day, half of the remaining 4,000 liters of water is removed. Remaining water after Day 3 = \( \frac{1}{2} \times 4000 \) liters = 2000 liters.

Step 5: Day 4 Calculation

On the fourth day, half of the remaining 2,000 liters of water is removed. Remaining water after Day 4 = \( \frac{1}{2} \times 2000 \) liters = 1000 liters.

Step 6: Day 5 Calculation

On the fifth day, half of the remaining 1,000 liters of water is removed. Remaining water after Day 5 = \( \frac{1}{2} \times 1000 \) liters = 500 liters.

Step 7: Day 6 Calculation

On the sixth day, half of the remaining 500 liters of water is removed. Remaining water after Day 6 = \( \frac{1}{2} \times 500 \) liters = 250 liters.

Step 8: Day 7 Calculation

On the seventh day, half of the remaining 250 liters of water is removed. Remaining water after Day 7 = \( \frac{1}{2} \times 250 \) liters = 125 liters.

Key Concepts

Water Removal Problem

The water removal problem is a practical example of exponential decay, a concept that describes how quantities decrease rapidly over time.
It involves a tank that contains a fixed amount of water, initially 16,000 liters.
Each day, half of the water is removed from the tank and is not replaced.
This scenario challenges us to calculate how much water remains after a certain number of days.

• The initial volume is crucial as our calculations depend on this starting point.
• The problem simplifies the real-world scenarios, such as evaporation or leakage, focusing only on removal by halving.
• Understanding this problem aids in grasping how exponential decay operates in other contexts, like radioactive decay and loan amortization.

By modelling this problem mathematically, we can directly pinpoint the amount of water remaining over each specific day, ultimately leading us to the final day's quantity.

Mathematical Modeling

Mathematical modeling in this context is about creating a representation of the water removal process using mathematical concepts and formulas.
This is achieved by applying the exponential decay formula, which allows us to determine the remaining quantity of water over consecutive days.

• The model begins with the initial amount of water: 16,000 liters.
• Each day, the quantity of the remaining water is multiplied by a factor of 0.5 (or one-half).
• This exponential model is expressed as \( R_n = R_0 \times (0.5)^n \), where \( R_n \) is the remaining water after \( n \) days, and \( R_0 \) is the initial amount of water.

By applying this model, we can solve the problem efficiently, predicting the remaining amount of water without needing multiple complex calculations.

Fractional Reduction

Fractional reduction refers to the continuous process of reducing the quantity of water in the tank by a specific fraction daily, which in this problem is one-half.
Understanding this concept is key when applying it as a recurring pattern over timespan.

• Everyday, a consistent fraction (1/2) is removed, simplifying calculations as we repeatedly apply the same operation.
• It highlights the practical use of fractions in calculating real-world problems such as bank interest rates or population dynamics.
• The reduction can be expressed mathematically as \( ext{Remaining water} = rac{1}{2} \times ext{Previous day's water} \).

Such fractional analysis demonstrates the power of compounding changes in mathematical reasoning, preparing students for more advanced applications in calculus and algebra.