Problem 65 - Algebra and Trigonometry

Question

Nutrition A researcher performs an experiment to test a hypothesis that involves the nutrients niacin and retinol. She feeds one group of laboratory rats a daily diet of precisely 32 units of niacin and \(22,000\) units of retinol. She uses two types of commercial pellet foods. Food A contains 0.12 unit of niacin and 100 units of retinol per gram. Food B contains 0.20 unit of niacin and 50 units of retinol per gram. How many grams of each food does she feed this group of rats each day?

Step-by-Step Solution

Answer

The researcher feeds 200 grams of Food A and 40 grams of Food B each day.

Step 1: Define the Variables

Let variable \( x \) denote the number of grams of Food A, and variable \( y \) denote the number of grams of Food B.

Step 2: Set Up the Equations

Based on the amounts of niacin and retinol in each food type and the daily nutritional requirements, we can set up two equations. \(0.12x + 0.20y = 32\) for niacin, and \(100x + 50y = 22000\) for retinol.

Step 3: Solve the Equation System for Retinol

Start with the equation for retinol, \(100x + 50y = 22000\). Simplify by dividing each term by 50 to reduce to \(2x + y = 440\).

Step 4: Solve for One Variable

From \(2x + y = 440\), express \(y\) in terms of \(x\): \(y = 440 - 2x\).

Step 5: Substitute into the Niacin Equation

Substitute \(y = 440 - 2x\) into the niacin equation: \(0.12x + 0.20(440 - 2x) = 32\).

Step 6: Simplify and Solve for \(x\)

Distribute \(0.20\) and simplify: \(0.12x + 88 - 0.40x = 32\). Combine like terms to get \(-0.28x = -56\). Solve for \(x\): \(x = 200\).

Step 7: Solve for \(y\)

Using the equation \(y = 440 - 2x\), substitute \(x = 200\): \(y = 440 - 2(200)\). Calculate: \(y = 40\).

Step 8: Verify the Solution

Substitute \(x = 200\) and \(y = 40\) back into the original equations to ensure they satisfy both. For niacin: \(0.12(200) + 0.20(40) = 24 + 8 = 32\). For retinol: \(100(200) + 50(40) = 20000 + 2000 = 22000\). Both check out.

Key Concepts

Understanding Simultaneous Equations

Simultaneous equations are a set of two or more equations with multiple variables that are solved together. The goal is to find a common solution for all the variables involved. In the context of the nutritional experiment, simultaneous equations are used to figure out the right combination of two types of food, each contributing a certain amount of niacin and retinol. These equations help to find the precise grams of Food A and Food B that meet both nutritional requirements simultaneously.
By setting up two equations based on niacin and retinol content, we can solve these equations simultaneously to find the solution. This means that the solution will satisfy both equations at once, giving us the exact amount of food needed to ensure the nutritional balance for the rats.

Exploring Linear Equations

Linear equations are equations where the highest power of the variable(s) is one. They are often represented graphically as straight lines. In this exercise, each equation, whether for niacin or retinol, represents a linear relationship between the food type and its nutrient content.
For instance, the equation for niacin is set as a linear equation: \(0.12x + 0.20y = 32\). This equation tells us that each gram of Food A contributes 0.12 units of niacin, and each gram of Food B contributes 0.20 units. The same principle applies to the retinol equation. Solving these linear equations allows us to see the direct relationship between the different food types and their contribution to the nutritional needs of the rats.

Demystifying the Nutrition Experiment

In this nutrition experiment, the researcher aims to determine how much of each type of food is needed to meet specific dietary requirements. These kinds of experiments are critical in nutritional science as they help discover the optimal diet for particular organisms.
During the experiment, the researcher uses Food A and Food B, each with known amounts of niacin and retinol, two essential nutrients. By conducting this experiment, one can understand how various foods can be blended or combined to achieve desired nutritional goals efficiently. Understanding the concepts of combination and meeting nutritional requirements are fundamental to success in this field.

Meeting Nutritional Requirements

Nutritional requirements refer to the specific quantities of nutrients necessary to maintain health or support growth. In this exercise, the key nutrients are niacin and retinol, which are vital for the rats' well-being.

• Niacin is essential for converting nutrients into energy.
• Retinol is important for vision and immune function.

The challenge is to provide both in precise quantities via food consumption.
By understanding the nutritional requirements and how different foods contribute to meeting these needs, the nutrition experiment allows researchers to develop diets that fulfill these essential nutrient needs to maintain health or encourage growth in the subjects tested.