Problem 44 - Precalculus: Mathematics for Calculus

Question

Nicole started a new diet that requires each meal to have 460 calories, 6 g of fiber, and 11 g of fat. The table shows the fiber, fat, and calorie content of one serving of each of three breakfast foods. How many servings of each food should Nicole eat to follow her diet? $$\begin{array}{|l|c|c|c|}\hline \text { Food } & \text { Fiber (g) } & \text { Fat (g) } & \text { Calories } \\\\\hline \text { Toast } & 2 & 1 & 100 \\\\\text { Cottage cheese } & 0 & 5 & 120 \\\\\text { Fruit } & 2 & 0 & 60 \\\\\hline\end{array}$$

Step-by-Step Solution

Answer

Nicole should eat 1 serving of Toast, 2 servings of Cottage Cheese, and 2 servings of Fruit.

Step 1: Define Variables

Let's define three variables. Let \( x \) be the number of servings of Toast, \( y \) be the number of servings of Cottage cheese, and \( z \) be the number of servings of Fruit that Nicole should eat.

Step 2: Set up the System of Equations

We need to form three equations based on the nutritional content of each food item.1. The calorie equation: \( 100x + 120y + 60z = 460 \)2. The fiber equation: \( 2x + 0y + 2z = 6 \)3. The fat equation: \( 1x + 5y + 0z = 11 \)

Step 3: Solve the Fiber Equation

The fiber equation simplifies the system significantly. From \( 2x + 2z = 6 \), divide every term by 2: \[ x + z = 3 \]Thus, \( z = 3 - x \).

Step 4: Substitute into the Fat Equation

Use the expression for \( z \) from the fiber equation in the fat equation:\[ x + 5y = 11 \]

Step 5: Substitute into the Calorie Equation

Substitute \( z = 3 - x \) into the calorie equation:\[ 100x + 120y + 60(3 - x) = 460 \]Simplify to: \[ 100x + 120y + 180 - 60x = 460 \]Resulting in:\[ 40x + 120y = 280 \]

Step 6: Solve the Simplified System of Equations

Now solve the equations:1. \( x + 5y = 11 \)2. \( 40x + 120y = 280 \)For the second equation, divide by 40:\[ x + 3y = 7 \]Now solve:1. \( x + 5y = 11 \)2. \( x + 3y = 7 \)Subtract the second from the first:\( (x + 5y) - (x + 3y) = 11 - 7 \)\[ 2y = 4 \]\[ y = 2 \]Substitute \( y = 2 \) back into \( x + 3y = 7 \):\( x + 3(2) = 7 \)\[ x + 6 = 7 \]\[ x = 1 \]

Step 7: Calculate z

Substitute \( x = 1 \) into the expression for \( z \): \( z = 3 - x = 3 - 1 = 2 \).

Step 8: Verify the Solution

Verify that the values \( x = 1 \), \( y = 2 \), and \( z = 2 \) satisfy the original caloric requirement:\( 100(1) + 120(2) + 60(2) = 460 \),\( 2x + 2z = 6 \) and \( x + 5y = 11 \) holds true.

Key Concepts

Calorie Content and Its Importance in Diets

When we talk about diets, calorie content is usually the primary focus. Calories measure the amount of energy you get from consuming a particular food. They're crucial because they fuel your body's daily activities. In Nicole’s case, her diet requires her to consume exactly 460 calories per meal.
To find out how many servings of each breakfast food she should eat, we can set up a system of equations based on this calorie content. Each type of food gives a different number of calories:

• Toast: 100 calories per serving
• Cottage cheese: 120 calories per serving
• Fruit: 60 calories per serving

Assuming you consume more calories than your body needs, the excess is stored as fat, leading to weight gain. On the other hand, consuming too few calories might leave you feeling tired or cause weight to drop unsustainably. Thus, maintaining the right calorie balance is vital for proper body function. Here's how Nicole uses her selected servings of food to balance her caloric intake and meet her dietary goals.

Understanding Fiber Content in Foods

Fiber content is another vital aspect of a balanced diet. Fiber is critical for maintaining good digestive health and can also help regulate blood sugar levels. For Nicole, she needs a total of 6 grams of fiber in her meal.
The equation derived from the fiber content of each food helps Nicole determine the right combination to meet her fiber needs while staying within her dietary limits. The serving values are:

• Toast: 2 grams of fiber per serving
• Cottage cheese: 0 grams of fiber per serving (since dairy generally has no fiber)
• Fruit: 2 grams of fiber per serving

As fiber does not provide energy in the form of calories, it instead aids in feeling full and satisfied. This can help individuals maintain or achieve a healthy weight. Therefore, Nicole's choice of foods is not only about calories, but also about balancing fiber to promote health benefits and meet her dietary requirements.

Fat Content and Its Role in Nutrition

Fat content in food often concerns individuals who are monitoring their dietary intake. However, fats are essential nutrients that the body needs to function correctly. For Nicole’s meal, she needs 11 grams of fat.
The equation setup involves assessing how each type of food contributes to this fat requirement:

• Toast: 1 gram of fat per serving
• Cottage cheese: 5 grams of fat per serving
• Fruit: 0 grams of fat per serving

While fats are calorie-dense, they play critical roles in energy storage, insulation, cell structure, and more. It's not only about the quantity of fat, but also the type of fat consumed that can impact health. Nicole ensures she reaches her fat goals in her meal by adjusting serving combinations, making sure she obtains the necessary nutrients while avoiding excess intake, which could counteract her fitness goals.