Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

In a certain medical test designed to measure carbohydrate tolerance, an adult drinks 7 ounces of a \(30 \%\) glucose solution. When the test is administered to a child, the glucose concentration must be decreased to \(20 \%\). How much \(30 \%\) glucose solution and how much water should be used to prepare 7 ounces of \(20 \%\) glucose solution?

Step-by-Step Solution

Answer

4.67 ounces of 30% solution and 2.33 ounces of water.

Step 1: Define the Problem

To find out how much of the 30% glucose solution and water we should mix to get 7 ounces of a 20% glucose solution, let's define variables. Let \( x \) be the amount of the 30% glucose solution, and \( y \) be the amount of water needed.

Step 2: Set Up Equations

We have two equations. The first is based on the total amount of solution: \( x + y = 7 \). The second is based on the concentration of glucose: \( 0.30x = 0.20 \cdot 7 \).

Step 3: Solve for x in Second Equation

Solve the concentration equation for \( x \): \( 0.30x = 1.4 \), hence \( x = \frac{1.4}{0.30} \). Calculating gives \( x = 4.67 \).

Step 4: Solve for y Using Total Equation

Use the total amount equation \( x + y = 7 \) with \( x = 4.67 \): \( 4.67 + y = 7 \). Solving for \( y \), we get \( y = 7 - 4.67 = 2.33 \).

Step 5: Verify the Solution

Check if the total glucose is correct. With \( x = 4.67 \), the glucose content is \( 0.30 \times 4.67 = 1.401 \), and the total solution is 7 ounces. Since \( 0.20 \times 7 = 1.4 \), our calculations confirm the concentrations are correct.

Key Concepts

Carbohydrate Tolerance Test

The carbohydrate tolerance test is a medical procedure used to assess how well an individual's body processes carbohydrates. It is often part of diagnosing diabetes or other glucose metabolism disorders.
During this test, a specific amount of glucose solution is consumed, and blood glucose levels are measured at regular intervals. This helps determine how efficiently glucose is cleared from the blood.
Carbohydrate tolerance tests can have variations in glucose concentrations depending on the age or health condition of the individual. For instance, children may require different solution concentrations than adults.

Glucose Solution Concentration

Glucose solution concentration refers to the percentage of glucose present in the total solution. A 30% glucose solution means that 30% of the solution's weight is glucose, with the remaining 70% made up of water and other components.
To calculate the exact concentration in a problem like the one given, understanding what these percentages signify is crucial. For example, if a problem states a need to create a 20% glucose solution, calculations are needed to mix the correct ratio of glucose to water.

• For a 30% glucose solution (used in adult tests), 30g of glucose per 100g of solution is involved.
• For a 20% glucose solution (adjusted for children), the glucose content is lower at 20g per 100g of solution.

Mixture Problems

Mixture problems, like the one presented, involve combining different substances to achieve a desired concentration or quantity. These problems require careful calculation to determine how much of each component is needed.
Typically, such problems involve defining variables for unknowns -- here, the amount of glucose solution and water. Using the known desired outcome, proportion equations can then be formulated.

• In our example, the total solution is 7 ounces, requiring an equation like: \( x + y = 7 \).
• The concentration equation is based on the target: \( 0.30x = 0.20 \times 7 \).

These equations allow us to find the precise amounts necessary for the intended solution.

Algebraic Equations

Problem-solving with algebraic equations involves representing real-world situations using mathematical expressions to find unknown values. In the given scenario, algebra helps achieve the correct mixture for a specific glucose concentration.
Two critical equations emerge in mix problems:

• The total volume equation: \( x + y = 7 \), where \( x \) is the glucose solution and \( y \) is water.
• The concentration equation, depicting desired glucose levels: \( 0.30x = 0.20 \times 7 \).

Solutions come from isolating one variable, using substitution, then solving for others, ensuring results are practical. As shown, \( x = 4.67 \) and \( y = 2.33 \) from these equations deliver the required mixture.