Problem 241 - Calculus Volume 1

Question

Sandra is a 25 -year old woman who weighs 120 \(\mathrm{lb}\) . She burns \(300-50 t\) calhr while walking on her treadmill. Her caloric intake from drinking Gatorade is 100\(t\) calories during the tth hour. What is her net decrease in calories after walking for 3 hours?

Step-by-Step Solution

Answer

Sandra's net decrease in calories after 3 hours is 0 calories.

Step 1: Determine Calories Burned

Sandra burns calories while walking for each hour, given by the formula \(300 - 50t\) where \(t\) is the hour. We need to find out how many calories she burns over 3 hours sum them up.- For \(t = 1\): she burns \(300 - 50(1) = 250\) calories.- For \(t = 2\): she burns \(300 - 50(2) = 200\) calories.- For \(t = 3\): she burns \(300 - 50(3) = 150\) calories.Adding these up, she burns \(250 + 200 + 150 = 600\) calories over 3 hours.

Step 2: Determine Calories Gained

Sandra consumes calories from Gatorade each hour, given by the formula \(100t\) calories per hour.- For \(t = 1\): she gains \(100 \times 1 = 100\) calories.- For \(t = 2\): she gains \(100 \times 2 = 200\) calories.- For \(t = 3\): she gains \(100 \times 3 = 300\) calories.Adding these up, Sandra gains \(100 + 200 + 300 = 600\) calories over 3 hours.

Step 3: Calculate Net Caloric Change

Net caloric change is the total calories burned minus the total calories gained.- Total calories burned in 3 hours: \(600\) calories.- Total calories gained from Gatorade in 3 hours: \(600\) calories.Net decrease in calories = Total calories burned - Total calories gained = \(600 - 600 = 0\).

Step 4: Conclusion

After considering the calories burned and the calories gained, Sandra's net decrease in calories after walking for 3 hours is \(0\) calories. She balances her caloric intake with her expenditure exactly.

Key Concepts

Caloric Expenditure Calculations

Understanding caloric expenditure calculations helps in determining how many calories are burned through physical activities. In Sandra's case, her walking activity on a treadmill shows a way to calculate this. She has a specific formula available for her calorie burn, which is \(300 - 50t\) calories per hour, as the exercise specifies. The formula takes into account how the calorie burn decreases over time, making it simple to plug in the values for each hour.

- For **hour 1**, plug \(t = 1\) into the formula: \(250\) calories burned.- For **hour 2**, use \(t = 2\): \(200\) calories burned.- For **hour 3**, try \(t = 3\): \(150\) calories burned.By adding these values, we find that Sandra expends a total of \(600\) calories over the 3 hour period. This approach helps us model how different activities can impact overall daily caloric expenditure. Calculating such parameters carefully is crucial for anyone interested in understanding their energy balance or designing an effective workout or diet plan.

Net Change in Calories

The concept of net change in calories is significant in nutrition and fitness, providing insights into the balance between calories consumed and expended. From Sandra's activity, she gains calories from drinking Gatorade, measured by the formula \(100t\) calories per hour. Each hour, she takes in more calories, adding up to her total caloric intake during the exercise.

- For **hour 1**, she gains \(100\times1 = 100\) calories.- For **hour 2**, she receives \(100\times2 = 200\) calories.- For **hour 3**, there's an intake of \(100\times3 = 300\) calories.This results in a total intake of \(600\) calories over the 3 hours. To figure out the net change, you subtract the calories gained from calories burned (\(600 - 600\)). Sandra's case turns out as a net change of zero, indicating she perfectly offset her caloric intake against her expenditure, which means no net loss or gain.

Mathematical Modeling of Physical Activity

Mathematical modeling of physical activity is essential to predict and analyze real-world activities through mathematical equations. By using the formulas to calculate both caloric expenditure and intake, we not only determine results but create a broader understanding of how everyday activities impact health.

Sandra's exercise demonstrates a practical application of such mathematical modeling. By developing specific equations like \(300 - 50t\) for calories spent and \(100t\) for calories consumed, these models equip individuals to foresee their net calorie outcomes based on their chosen physical activities and nutritional habits.

Using these methods, anyone can:- Track their progress over time.- Predict the effects of varied exercise routines.- Adjust their diet and energy expenditure to meet specific health goals.Furthermore, such models are foundational in various fields such as sports science, fitness industry, and even in personalized diet plans, showing their broad application and importance in maintaining a healthy lifestyle.