Problem 67 - Algebra and Trigonometry

Question

Mixing Coolant A truck radiator holds 5 gal and is filled with water. A gallon of water is removed from the radiator and replaced with a gallon of antifreeze; then, a gallon of the mixture is removed from the radiator and again replaced by a gallon of antifreeze. This process is repeated indefinitely. How much water remains in the tank after this process is repeated 3 times? 5 times?

Step-by-Step Solution

Answer

After 3 repetitions, \( \frac{64}{25} \) gallons; after 5 repetitions, \( \frac{1024}{625} \) gallons of water remain.

Step 1: Understand the Problem

Initially, the radiator is filled with 5 gallons of water. Each time one gallon is removed and replaced with antifreeze, the amount of water reduces while the mixture gains more antifreeze. The problem requires calculating the remaining water after repeating this process certain times.

Step 2: Define the Mathematical Model

Let \( W_n \) represent the amount of water after \( n \) repetitions. Initially, \( W_0 = 5 \). When a gallon of mixture is removed, the concentration of water in the radiator becomes \( \frac{W_n}{5} \). Hence, after removing one gallon of the mixture and replacing it with pure antifreeze, the amount of water left becomes:\[ W_{n+1} = W_n - \frac{W_n}{5} = \frac{4}{5}W_n \]

Step 3: Calculate After 3 Repetitions

Using the formula from Step 2, calculate the remaining amount of water after 3 repetitions:- \( W_1 = \frac{4}{5} \times 5 = 4 \)- \( W_2 = \frac{4}{5} \times 4 = \frac{16}{5} \)- \( W_3 = \frac{4}{5} \times \frac{16}{5} = \frac{64}{25} \) gallons.

Step 4: Calculate After 5 Repetitions

Using the same formula to find the remaining water after 5 repetitions:- Continuing from \( W_3 = \frac{64}{25} \):- \( W_4 = \frac{4}{5} \times \frac{64}{25} = \frac{256}{125} \)- \( W_5 = \frac{4}{5} \times \frac{256}{125} = \frac{1024}{625} \) gallons.

Step 5: Summary of Results

After 3 repetitions, the radiator contains \( \frac{64}{25} \) gallons of water. After 5 repetitions, it contains \( \frac{1024}{625} \) gallons of water.

Key Concepts

Recurrence Relations

When solving problems involving processes that happen repeatedly, like the one in our truck radiator exercise, recurrence relations become very useful. A recurrence relation is an equation that defines a sequence of numbers, where each term is defined in relation to one or more preceding terms. This is similar to how certain mathematical sequences and series operate.

In the radiator problem, we used a recurrence relation to determine how much water remains in the radiator after each repetition of removing and replacing a gallon. Specifically, the equation \( W_{n+1} = \frac{4}{5}W_n \) shows us how each subsequent term (amount of water after a new round) depends on the last one.

This relation allows us to calculate not just the immediate next value but also to predict far into the sequence with ease. Understanding recurrence relations helps simplify complex changes over iterations into manageable, calculable steps.

Exponential Decay

Exponential decay describes processes where a quantity decreases at a consistent rate over time. In our radiator exercise, every time a gallon of the mixture is replaced by antifreeze, there's an exponential decrease in water content.

The formula \( W_{n+1} = \frac{4}{5}W_n \) is an illustration of exponential decay. Since the water is reduced by a consistent fraction with each repetition, the pattern of decrease is exponential rather than linear. This means it decreases rapidly at first but never quite reaches zero.

We appreciate exponential patterns because they allow students to predict future values, like after 3 or 5 iterations, with straightforward calculations. Practicing with decay helps reinforce concepts of half-life, interest decay, and ecological decay in broader mathematical and scientific contexts.

Problem Solving Steps

Breaking down complex word problems into manageable steps is crucial for finding accurate solutions. The steps in our radiator problem are a good example.

First, always understand the problem clearly. Sketch or note what is happening descriptively. Here, it involves replacing parts of water with antifreeze.

Second, translate the problem into a mathematical model. Identify the correct variables and establish your recurrence relation for the problem.

• Define initial conditions, such as \( W_0 = 5 \) in our case, which represents the original amount of water.
• Formulate your recurrence relation, here it's \( W_{n+1} = \frac{4}{5}W_n \).

Next, conduct the calculations as required for specific iterations. This involves calculating each subsequent iteration using your model to find the solution after a specified number of operations.

By understanding and following these problem-solving steps, students can tackle similar problems in mathematics and real-world scenarios with confidence and clarity.