Problem 240 - Calculus Volume 1

Question

John is a 25-year old man who weighs \(160 \mathrm{lb}\). He burns \(500-50 t\) calories/hr while riding his bike for \(t\) hours. If an oatmeal cookie has 55 cal and John eats \(4 t\) cookies during the tth hour, how many net calories has he lost after 3 hours riding his bike?

Step-by-Step Solution

Answer

John gained 120 calories after 3 hours of biking.

Step 1: Determine Calories Burned

John burns calories at a rate of \(500 - 50t\) calories per hour. We need to calculate the total calories burned over 3 hours. For this, we'll evaluate the expression:\[\text{Total Calories Burned} = \sum_{t=1}^{3} (500 - 50t)\]Calculate for each hour, sum the results:- For \(t = 1\): \(500 - 50(1) = 450\)- For \(t = 2\): \(500 - 50(2) = 400\)- For \(t = 3\): \(500 - 50(3) = 350\)Now sum these values:\[450 + 400 + 350 = 1200\]Thus, John burns a total of 1200 calories over 3 hours.

Step 2: Calculate Cookies Calories

John eats \(4t\) oatmeal cookies during the \(t\)th hour, where each cookie contains 55 calories. Calculate the total cookie calories consumed over 3 hours:- For \(t = 1\): \(4(1) \times 55 = 220\) calories- For \(t = 2\): \(4(2) \times 55 = 440\) calories- For \(t = 3\): \(4(3) \times 55 = 660\) caloriesNow sum these values to find the total calories consumed from cookies:\[220 + 440 + 660 = 1320\]John consumes a total of 1320 calories from cookies over 3 hours.

Step 3: Determine Net Calories

To find the net calories lost, subtract the total calories consumed from cookies from the total calories burned while biking:\[\text{Net Calories Lost} = ext{Total Calories Burned} - ext{Total Calories from Cookies}\]Substitute the values we've calculated:\[1200 - 1320 = -120\]The negative value indicates that John did not lose any net calories; instead, he gained 120 calories after 3 hours of biking and eating cookies.

Key Concepts

calories burned

When we talk about **calories burned**, we refer to the energy expenditure that occurs when engaging in physical activities like exercising. In John's case, his activity is biking. The exercise specifies that John burns a base of 500 calories per hour when he starts. However, there's a decrease in this rate by 50 calories for each subsequent hour due to the expression \(500 - 50t\). This decrease represents many factors, such as fatigue, efficiency changes, or metabolic adjustments as time progresses, making him burn progressively fewer calories each hour.

Over the three-hour biking session, John's calorie burn was calculated for each hour.
- **First hour**: Starts strong, burns 450 calories.
- **Second hour**: Slight drop in energy expenditure, resulting in 400 calories burned.
- **Third hour**: Tapers off further, burning only 350 calories.
Thus, over these 3 hours, John accumulates a total of 1200 calories burned during his biking workout.

calorie consumption

**Calorie consumption** refers to the intake of calories through eating or drinking. Here, John contributes to his calorie count by consuming oatmeal cookies during his biking sessions. The exercise explains that he eats more cookies as time increases, quantified as 4 cookies multiplied by the hour number \(4t\).

Each cookie carries 55 calories, contributing significantly to the total calorie intake.
- **First hour**: Eats 4 cookies, adding 220 calories.
- **Second hour**: Doubles the intake to 8 cookies, thus 440 calories total.
- **Third hour**: Maxes out with 12 cookies, rounding up to 660 calories.
This brings John’s total calorie intake from cookies over 3 hours to a substantial 1320 calories, surpassing what he burns.

net calorie loss

The concept of **net calorie loss** encompasses the balance between calories burned and calories consumed. It is a crucial metric for understanding weight management and energy balance. In this scenario, the net calorie change is determined by subtracting calorie consumption from calorie expenditure while biking.

Here's how to calculate John's net calorie balance over the 3-hour period:

• **Total calories burned**: 1200 calories from biking.
• **Total calorie consumption**: 1320 calories from eating cookies.

Subtracting these amounts gives us:
\[\text{Net Calories Lost} = 1200 - 1320 = -120\]
The negative outcome means John did not lose calories on net but actually gained a total of 120 calories, demonstrating how consumption can exceed expenditure, even with physical activity.