Problem 73 - Algebra 1

Question

Mary Gordon is training for a triathlon. Like most triathletes she regularly trains in two of the three events every day. On Saturdays she expects to burn about 800 calories during her workout by running and swimming. Running: 7.1 calories per minute Swimming: 10.1 calories per minute Bicycling: \(\quad 6.2\) calories per minute If Mary Gordon spends 45 minutes running, about how many minutes will she have to spend swimming to burn 800 calories?

Step-by-Step Solution

Answer

Mary needs to swim for approximately 47.5 minutes to burn a total of 800 calories combined with running.

Step 1: Define the problem

Mary Gordon burns 800 calories in her workout. She spends 45 minutes running which burns 7.1 calories per minute. The task is to determine how many minutes she needs to spend swimming (which burns 10.1 calories per minute) to get to her total goal of 800 calories.

Step 2: Calculate calories burned running

First, calculate the calories burned by running. Multiply the minutes she spent running (45 minutes) by the calories burned per minute when running (7.1 calories per minute). This results in \(45 * 7.1 = 319.5\) calories.

Step 3: Calculate remaining calories

Next, calculate how many calories are left for Mary to burn in order to reach her goal of 800 calories. We subtract the calories she burned running from the total amount that needs to be burned, so \(800 - 319.5 = 480.5\) calories left to be burned through swimming.

Step 4: Calculate minutes spent swimming

Finally, we calculate how many minutes she needs to swim to burn the remaining 480.5 calories. Since swimming burns 10.1 calories per minute, we divide the remaining calories by the rate of calories burned by swimming. Therefore Mary needs to swim for \(480.5 / 10.1 = 47.5\) minutes approximately.

Key Concepts

Linear Equations

Understanding linear equations is fundamental when solving calorie burn problems. In its simplest form, a linear equation represents a straight line when plotted on a graph. It shows a constant rate of change, which in real-life scenarios could represent the rate at which calories are burned per minute. The equation is generally written as y = mx + b, where m represents the slope or the rate of calorie burn, and b is the y-intercept, typically starting calories.

In the provided exercise, the calorie burn for Mary can be represented by two separate linear equations: one for running and one for swimming. For running, the equation could look like C_r = 7.1m, where C_r is the calories burned running and m is the minutes spent running. Similarly, the equation for swimming could be C_s = 10.1n, where C_s is the calories burned swimming and n is the minutes spent swimming. By solving these linear equations with given values, we can find out how much time Mary needs to spend in each activity to achieve her calorie goal.

Unit Rate Calculations

The concept of unit rate calculations is essential in scenarios where you want to calculate the efficiency or productivity of any activity, such as the rate of calorie burn. A unit rate is a ratio that compares a quantity to its unit of measure. This allows us to understand quantities in a tangible and comparative way. For example, understanding that Mary burns 7.1 calories per minute running tells us the efficiency of running as an exercise for calorie burn.

To calculate Mary's total calorie expenditure, we use the unit rates of calories burned per minute in each activity. To solve for unknown variables in such practical scenarios, we often set up proportions based on these unit rates. In Mary's case, we used the unit rate of swimming to find the unknown quantity of time she needs to swim to meet her calorie-burning goal.

Algebra Word Problems

Solving algebra word problems involves translating real-world scenarios into mathematical equations, often involving unknown variables. To successfully solve these problems, you need to:

• Clearly define the problem and identify the unknowns.
• Translate the written problem into mathematical expressions and equations.
• Use appropriate algebraic methods to solve for the unknowns.

In Mary's exercise, we are confronted with a word problem asking to calculate the time required for two types of exercises to burn a specific number of calories. The problem was broken down into steps, starting with defining the goal—burning 800 calories—and the known rates of burning calories. Expressing these rates as equations, we then solved for the unknown time she needs to spend swimming. Breaking down word problems into smaller, more manageable parts can greatly simplify the solving process.

Calories Burned Calculation

The calculation of calories burned during exercise is a practical application of unit rate calculations and linear equations. To calculate the total calories burned, you multiply the amount of time spent on the activity by the rate of calories burned per minute.

In our example, Mary has a clear goal for her calorie burn: 800 calories. Initially, we needed to calculate the total calories burned by running. This was done by multiplying the time spent running by the known calorie-burn rate for running. After deducing the calories burned through running, the remaining calories needed to reach Mary's goal were calculated. The time required for swimming was then found by dividing the remaining calories by the calorie-burn rate for swimming. These kinds of exercises strengthen the understanding of how algebra applies to real-life situations and provide a solid foundation for solving similar problems.