Problem 17 - Precalculus Mathematics for Calculus

Question

Express the given quantity in terms of the indicated variable. The concentration (in oz/gal) of salt in a mixture of 3 gals of brine containing 25 oz of salt to which some pure water has been added; \( x=\) volume of pure water added (in gal)

Step-by-Step Solution

Answer

The concentration is \( \frac{25}{3 + x} \) oz/gal.

Step 1: Understanding the Initial Mixture

The problem provides an initial brine solution consisting of 3 gallons containing 25 ounces of salt. Therefore, the initial concentration (C) of the brine in oz/gal is calculated as the amount of salt (in ounces) divided by the volume of brine (in gallons), which is \( C = \frac{25}{3} \) oz/gal.

Step 2: Adding Pure Water

Next, we are told that some pure water is added, with the variable \( x \) representing the gallons of pure water added. The total volume of the mixture after adding \( x \) gallons of water is \( 3 + x \) gallons.

Step 3: Calculating New Concentration

The concentration of salt in the new mixture is the amount of salt divided by the new total volume (which includes the added pure water). The amount of salt remains the same (25 oz), but the volume changes to \( 3 + x \) gallons. Therefore, the new concentration \( C' \) is given by \( C' = \frac{25}{3 + x} \) oz/gal.

Step 4: Expressing in Terms of Variable

Since the problem asks to express the concentration in terms of \( x \), the variable representing the volume of pure water added, we have already found it. The expression for the concentration of salt in terms of \( x \) is \( \frac{25}{3 + x} \).

Key Concepts

Brine Solution

Brine is a simple solution consisting mainly of salt (usually sodium chloride) dissolved in water. In our exercise, we started with a brine solution containing salt and water in specific quantities. Understanding the concentration of a brine solution is crucial since it tells us how much salt is present in each unit of liquid, usually measured in ounces per gallon for such problems.

When dealing with brine solutions, the concentration can change if we add more of either component. If more salt is added, the concentration increases; adding more water, like in our problem, decreases the concentration. Knowing how to calculate these changes is essential. It helps in making informed decisions, such as adjusting formulations in culinary recipes or industrial processes.

The initial concentration of the brine in our example was calculated using the formula:

• Concentration (C) = amount of salt (in ounces) divided by the amount of water (in gallons)

This foundational understanding helps one manipulate and control the concentration for various needs.

Volume and Capacity

Volume and capacity are core concepts in understanding how much liquid a container holds and how that affects concentration.

Volume tells us the amount of space that a substance (in this case, a liquid like brine) occupies. 'Capacity' often describes the maximum amount a container can hold. In our problem, we begin with 3 gallons of brine. By adding more water, we increase the volume without adding more salt, which impacts the concentration.

The total volume after addition of water can be calculated as:

• Total Volume = Original Volume + Volume of Water Added (x gallons)

This is important because even though the volume changes, the salt content does not, which is critical when calculating changes in concentration. Understanding how volume impacts concentration can be crucial when determining how additives like water or solutes like salt interact in a given space.

Ratio and Proportion

Ratios and proportions are mathematical tools used to solve problems involving concentrations and other similar concepts.

A ratio gives a comparison between two quantities, which in our example, was the amount of salt to the volume of brine. Initially, we had 25 ounces of salt for every 3 gallons of brine, expressed as the ratio 25:3.

Proportion uses these ratios to create equations that let us find unknown quantities. Here, after adding pure water, we need to find the new concentration of the solution.

• New Concentration Ratio = Amount of Salt (constant at 25 ounces) : New Total Volume (3 + x gallons)

The concept of proportion is useful as it shows how different quantities change in relation to one another. It’s a keystone concept in problems involving the mixing of liquids, allowing us to predict how adding different amounts of a substance, like water, can impact the entire solution.