Problem 67 - Algebra A Combined Function

Question

A 150 -pound person uses 5.8 calories per minute when walking at a speed of 4 mph. How long must a person walk at this speed to use at least 200 calories? Round up to the nearest minute. (Source: Home \& Garden Bulletin No. 72)

Step-by-Step Solution

Answer

Walk for 35 minutes at 4 mph to burn at least 200 calories.

Step 1: Identify the Calorie Rate Per Minute

A 150-pound person burns 5.8 calories per minute when walking at a speed of 4 mph. This rate of 5.8 calories per minute will be used to calculate the total time needed to burn a certain number of calories.

Step 2: Set Up the Equation

We need to find the time (in minutes) required to burn at least 200 calories. The equation derived from the calorie rate is: \[5.8 \, \text{calories/minute} \times t \, \text{minutes} = 200 \, \text{calories}\] where \(t\) is the time in minutes.

Step 3: Solve for Time

To solve for \(t\), we rearrange the equation: \[t = \frac{200}{5.8}\] Calculating this gives:\[t \approx 34.48276 \, \text{minutes}\]

Step 4: Round Up

Since you can't walk for a fraction of a minute and need at least 200 calories, round up the time to the nearest whole number. So, \(t = \lceil 34.48276 \rceil = 35\) minutes.

Key Concepts

Calorie Burning

Calorie burning is a simple yet important concept when it comes to understanding energy expenditure during physical activities. When a person walks, their body uses energy, which is measured in calories. The rate at which calories are burned depends on several factors like body weight, walking speed, and overall fitness level. In our example, a 150-pound person burns 5.8 calories per minute while walking at a speed of 4 mph. This fixed rate allows us to calculate how many minutes they need to walk to achieve a specific calorie-burn goal. Understanding how calorie burning works can help you manage weight and improve fitness levels effectively. Incorporating this into everyday routines can make reaching health goals more attainable.

Solving Equations

Solving equations is a fundamental part of algebra, and in this context, it helps us determine how long a person must walk to burn a certain number of calories. In algebra, solving an equation often involves isolating the variable of interest on one side. Here, we want to find out the time \(t\) in minutes to burn at least 200 calories. By setting up the equation \(5.8 \times t = 200\), we can solve for \(t\) to find our answer. Solving equations requires careful arithmetic and logical thinking, and it provides clarity on how changes in one component affect the outcome. This skill is valuable for a wide range of real-world problems beyond just calorie calculations.

Unit Conversion

Unit conversion is the process of changing a quantity from one unit to another without altering its value. In algebra word problems, correctly addressing unit conversion is crucial, although in this particular exercise, all units are already consistent. The rate of calorie burn is given in calories per minute, which aligns with the time we wish to calculate. If the units were initially different, we would need to convert them to match for accurate calculations. For instance, if calorie burn were given per hour, we would convert it to per minute by dividing by 60. Understanding unit conversion ensures that the values can be used effectively in equations and real-world applications without error.

Mathematical Modeling

Mathematical modeling involves using mathematical language and equations to represent real-life situations. In this exercise, we've used a mathematical model to calculate walking time needed to burn a certain number of calories. By forming an equation \(5.8 \times t = 200\), we build a model that links time, calorie burn rate, and total calories burned. This abstraction allows us to make predictions and calculations about real-world scenarios. Mathematical models simplify complex systems into manageable equations, enabling analysis and decision-making. Whether in health, finance, engineering, or everyday life, modeling offers a toolkit for evaluating and solving problems effectively.