you spend 30.0 minutes on a stair-climbing machine, climbing at a rate of 80 steps per minute, with each step 8.00 inches high. If you weigh 150 lb and the machine reports that 690 kcal have been burned at the end of the workout, what efficiency is the machine using in obtaining this result?
To calculate the efficiency of the stair-climbing machine in obtaining the reported calorie burn, we can use the following steps:
Step 1: Convert the time spent on the machine to hours.
Given that you spent 30.0 minutes on the machine, we can convert this to hours by dividing by 60.
30.0 minutes / 60 = 0.5 hours
Step 2: Calculate the total number of steps climbed.
The rate at which you climb is given as 80 steps per minute. Multiply this by the total time spent (in minutes) to get the total number of steps climbed.
80 steps/minute * 30.0 minutes = 2400 steps
Step 3: Convert the height of each step to feet.
The height of each step is given as 8.00 inches. To convert this to feet, divide by 12.
8.00 inches / 12 = 0.67 feet
Step 4: Calculate the total vertical distance climbed.
Multiply the total number of steps climbed by the height of each step in feet to get the total vertical distance climbed.
2400 steps * 0.67 feet = 1608 feet
Step 5: Calculate the work done.
The work done is the product of the vertical distance climbed and your weight. Convert your weight from pounds to the equivalent force in Newtons by multiplying by the acceleration due to gravity (9.8 m/s^2).
Work = (Force)*(distance)
Weight = 150 lb * 9.8 m/s^2 = 1470 Newtons
Work = 1470 Newtons * 1608 feet = 2351760 foot-pounds
Step 6: Convert the work into calories.
One calorie is equivalent to approximately 0.3239 foot-pounds. Divide the work done by 0.3239 to get the number of calories burned.
Calories = Work / 0.3239 = 2351760 foot-pounds / 0.3239 = 7260435 calories
Step 7: Calculate the efficiency.
The efficiency is given by the ratio of the calories reported by the machine to the actual calories burned.
Efficiency = (Reported Calories / Actual Calories) * 100%
Efficiency = (690 kcal / 7260435 calories) * 100% = 0.0095% (approximately)
Therefore, the machine is using an efficiency of approximately 0.0095% in obtaining the reported calorie burn.
To find the efficiency of the machine in obtaining the reported result, we need to calculate the work done and the actual energy expenditure.
Step 1: Calculate the work done:
Work = force x distance
Force = weight x gravity
Weight = 150 lb
Gravity = 32.2 ft/s^2 (acceleration due to gravity)
Force = 150 lb x 32.2 ft/s^2
Distance = number of steps x height of each step
Number of steps = 80 steps/min x 30 min
Height of each step = 8.00 inches
Distance = (80 steps/min) x (30 min) x (8.00 inches)
Convert the distance from inches to feet:
Distance = (80 x 30 x 8.00 inches) / 12 inches/ft
Step 2: Calculate the actual energy expenditure:
Energy expenditure = 690 kcal
Step 3: Calculate the efficiency:
Efficiency = actual energy expenditure / work done
Now, let's plug in the values and calculate the efficiency.