Problem 83 - College Algebra and Calculus: An Applied Approach

Question

A person enrolls in a diet program that guarantees a loss of at least \(1 \frac{1}{2}\) pounds per week. The person's weight at the beginning of the program is 180 pounds. Find the maximum number of weeks before the person attains a weight of 130 pounds.

Step-by-Step Solution

Answer

It will take approximately 34 weeks for the person to attain a weight of 130 pounds.

Step 1: Identify the variables

The weight to be lost is 50 pounds (180-130). The weight loss per week is 1.5 pounds. The time to lose the 50 pounds is the unknown variable.

Step 2: Formulate the equation

We can create the linear equation from the problem. The total weight to lose (50 pounds) divided by the weight loss per week (1.5 pounds per week) equals to the time in weeks. So, the equation is \(Time = \frac{{50}}{{1.5}}\).

Step 3: Solve the equation

Solving the equation \(Time = \frac{{50}}{{1.5}}\), we get the Time value approximately equal to 33.33.

Step 4: Interpret the results

Since we can't consider a fraction of a week in this case, it's necessary to round off the value. So, it will take 34 weeks for the person to reach the target weight of 130 pounds.

Key Concepts

Variables in Algebra

In algebra, variables are symbols used to represent unknown quantities. They are the foundation of algebraic expressions and equations, allowing us to generalize problems and find solutions that work for many different situations. For example, in the diet program exercise, the variable 'Time' is used to denote the unknown number of weeks it will take for a person to reach their target weight.

Representing variables properly is crucial as they stand in place of values we need to find. When identifying variables, think about what you don't know but need to determine. In our case, the starting weight is known (180 pounds), the target weight is known (130 pounds), thus, the weight loss is also known (50 pounds), and the rate of weight loss per week is given (1.5 pounds per week). Therefore, the only unknown is the 'Time' - the variable representing the number of weeks needed to achieve the goal.

Formulating Algebraic Equations

Formulating algebraic equations involves translating a real-world problem into a mathematical expression using variables and constants. A well-formulated equation provides a blueprint for finding the solution to a problem.

In the context of our exercise, the problem at hand was the number of weeks it would take for a person to lose a specific amount of weight. The constant factor is the weight that needs to be lost, which is 50 pounds. The rate of weight loss, 1.5 pounds per week, is another constant. By dividing the total weight to be lost by the rate of weight loss per week, we get the equation \(Time = \frac{{50}}{{1.5}}\), which accurately represents the situation described in the problem.

When you're faced with a problem, to formulate an equation:

• Identify what you know (constants).
• Determine what you don't know (variables).
• Find the relationship between known and unknown entities.
• Express this relationship as an equation.

Solving Linear Equations

Solving linear equations is the process of finding the values of the variables that make the equation true. A linear equation is one where the variable is not raised to any power other than one, and it can be simply rearranged to find the value of the variable.

In our diet program example, the linear equation was \(Time = \frac{{50}}{{1.5}}\). Solving this gives us approximately 33.33. However, because we can't have a fraction of a week, we round up to the nearest whole week, which gives us 34 weeks. This solution process involves a few key steps:

• Simplify the equation if necessary by combining like terms and using inverse operations.
• Rearrange the equation to isolate the variable on one side, resulting in a format like \(variable = value\).
• Carry out any calculations needed to find the variable's value.
• Round or adjust the solution according to the context of the problem.

Understanding how to solve these equations is crucial, as it's a skill you'll use in various areas of mathematics and real-life scenarios.