Problem 240 - Calculus Volume 1

Question

John is a 25 -year old man who weighs 160 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 th 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.

Step 1: Calculate Calories Burned

John burns calories according to the formula \(500 - 50t\) calories per hour after riding for \(t\) hours. For up to 3 hours, calculate the total calories burned by summing individually over each hour. * For the 1st hour (\(t=1\)): Calories burned = \(500 - 50\times1 = 450\) cal* For the 2nd hour (\(t=2\)): Calories burned = \(500 - 50\times2 = 400\) cal* For the 3rd hour (\(t=3\)): Calories burned = \(500 - 50\times3 = 350\) calAdd these together to find the total calories burned in 3 hours: 450 + 400 + 350 = 1200 cal.

Step 2: Calculate Calories Consumed

He eats 4\(t\) oatmeal cookies each hour, each containing 55 calories.* For the 1st hour (\(t=1\)): Cookies eaten = \(4 \times 1 = 4\), Calories = \(4 \times 55 = 220\) cal* For the 2nd hour (\(t=2\)): Cookies eaten = \(4 \times 2 = 8\), Calories = \(8 \times 55 = 440\) cal* For the 3rd hour (\(t=3\)): Cookies eaten = \(4 \times 3 = 12\), Calories = \(12 \times 55 = 660\) calAdd these together to find the total calories consumed in 3 hours:220 + 440 + 660 = 1320 cal.

Step 3: Calculate Net Calories Lost

The net calories lost is the difference between the calories burned and the calories consumed over the 3 hours. Total Calories Burned = 1200 cal Total Calories Consumed = 1320 cal Net Calories Lost = Calories Burned - Calories Consumed: 1200 - 1320 = -120 cal.

Key Concepts

Calories Burned

When thinking about calories burned, it's essential to remember that our activities dictate how many calories we burn. John, for example, burns calories when he rides his bike. The formula given in the exercise, \(500 - 50t\) calories per hour, helps us determine the exact number of calories John burns after riding for different hours. This means he's burning fewer calories each hour as time progresses—450 calories in the first hour, 400 in the second, and 350 in the third. Summing it all up, John burns a total of 1,200 calories after his 3-hour ride.

Understanding this concept helps us quantify activity in terms of energy spent, which is crucial in weight management. Always consider the intensity and duration of the activity, as both affect the calorie burn rate.

• 1st hour: 450 cal
• 2nd hour: 400 cal
• 3rd hour: 350 cal
• Total over 3 hours: 1200 cal

Calories Consumed

Calories consumed are those taken in by eating food. In our exercise, John's calorie consumption comes from eating oatmeal cookies. Each cookie is 55 calories, and the number he eats increases by 4 each hour, equating to 4, 8, and 12 cookies in each of the three consecutive hours he rides. This leads to 220 calories in the first hour, 440 in the second, and 660 in the third. Overall, this adds up to John consuming 1,320 calories in those 3 hours.

Calories consumed directly affect our weight and energy balance. Understanding food's caloric values and how much you're eating aids in managing intake effectively.

• 1st hour: 4 cookies = 220 cal
• 2nd hour: 8 cookies = 440 cal
• 3rd hour: 12 cookies = 660 cal
• Total over 3 hours: 1320 cal

Net Calorie Loss

Net calorie loss is essentially the balance between calories burned and calories consumed. It's a simple subtraction, but a crucial metric in understanding weight change. In John's case, he burned 1,200 calories but consumed 1,320 calories through eating cookies. Therefore, after his 3-hour ride, John is left with a net calorie loss of -120 calories, meaning he has actually gained calories.

If our goal is to lose weight or maintain a certain energy level, understanding net calories is vital.

• Net Calories = Total Calories Burned - Total Calories Consumed
• For John: 1200 - 1320 = -120 cal

This negative result shows us that simply engaging in activity does not guarantee weight loss unless the calories consumed are also kept in check. Managing both sides—intake and output—is the key to maintaining a healthy balance.