Problem 6 - Intermediate Algebra

Question

a. Write an expression that represents the total value of \(x\) ounces of ginseng tea that costs \(\$ 32\) per pound. b. Write an expression that represents the amount of hydrochloric acid in \(x\) gallons of a \(3 \%\) hydrochloric acid solution.

Step-by-Step Solution

Answer

Total value: \( 2x \). Amount of acid: \( 0.03x \).

Step 1: Convert Ounces to Pounds

There are 16 ounces in one pound. To find how many pounds are in \( x \) ounces, we divide \( x \) by 16. The expression for the pounds of ginseng tea is \( \frac{x}{16} \).

Step 2: Calculate Total Cost

Since the cost of ginseng tea is \( \$32 \) per pound, we multiply the number of pounds by the cost per pound. The expression for the total value is \( 32 \times \frac{x}{16} \). Simplifying this, the expression becomes \( 2x \).

Step 3: Determine Volume of Hydrochloric Acid

A \(3\%\) hydrochloric acid solution means that 3% of the solution is hydrochloric acid. This can be expressed as 0.03. To find the amount of hydrochloric acid in \( x \) gallons, we multiply \( x \) by 0.03. Thus, the expression is \( 0.03x \).

Key Concepts

Unit Conversion

Unit conversion is a key skill in algebra and many other practical scenarios. It helps us express quantities in different units of measurement, which can make them easier to work with or understand. In the original problem, we tackled conversion of ounces to pounds. One pound is equivalent to 16 ounces. This means, when you have a quantity in ounces and you want to know how many pounds that is, you’ll divide the number of ounces by 16.
For example, if you have \( x \) ounces of something, the equivalent in pounds is \( \frac{x}{16} \) pounds. Such conversions are essential in various applications, like when dealing with foods, substances, or materials sold or measured in different units. Always remember to identify the conversion factor when switching between units, whether it's weight like ounces to poundsor other types like meters to kilometers.

Cost Calculation

Calculating total cost for items based on their weight or quantity is a practical use of algebraic expressions. Once we have an expression for weight in pounds, calculating cost becomes straightforward using multiplication.

• First, calculate the quantity in the desired unit, like pounds in our example.
• Then, multiply by the cost per unit. Here, the cost per pound for ginseng tea is \( \$32 \).

Hence, multiplying the number of pounds by \( 32 \) gives us the total cost. If \( x \) ounces convert to \( \frac{x}{16} \) pounds, then the cost expression becomes \( 32 \times \frac{x}{16} \), and simplifies to \( 2x \). This is a classic example of how unit conversions and cost calculations work hand-in-hand for pricing problems in both mathematical exercises and real-world scenarios.

Solution Concentration

Understanding solution concentration involves knowing what fraction or percentage of a mixture is a particular component. This is frequently expressed in percentages and helps in making or analyzing mixtures in chemistry, cooking, or industrial processes.A \(3\%\) hydrochloric acid solution implies that 3% of the total volume is acid, and the rest is the solvent, typically water. To find how much hydrochloric acid there is in a solution, convert this percentage to a decimal, which is 0.03. Then simply multiply by the total volume of the solution.
So, for \( x \) gallons of a \(3\%\) solution, the amount of hydrochloric acid is \( 0.03 \times x \). Expressing concentration this way allows for quick calculation and adjustments of solution strength in various fields such as laboratory sciences and agriculture.