Alexander, who weighs 180 lb, decides to climb Mt. Krumpett, which is 5570 m high. For his food supply, he decides to take nutrition bars. The label on the bars states that each 100-g bar contains 10 g of fat, 40g of protein, and 50 g of carbohydrates.
In this problem you do not need to assume the work of climbing the mountain is 4 times the work of just lifting the weight straight up. You should calculate the number of kJ needed to lift the 171 lb mass 5640 meters and then double this number for the round trip. Remember to watch units!
Question-Alexander wants to know exactly how many bars to pack in his backpack for the journey. To provide a margin of safety, he assumes that he will need as much energy for the return trip as for the uphill climb. How many bars should Alexander pack?
To calculate the number of bars Alexander should pack, we need to first find the energy needed for the uphill climb and then double that value for the round trip.
Step 1: Calculate the energy needed for the uphill climb.
To find the energy needed, we need to calculate the work done in lifting Alexander's weight to the top of Mt. Krumpett. The formula for work is given by:
work = force * distance
The force can be calculated using the weight of Alexander, as weight = mass * gravitational acceleration.
Given:
Weight of Alexander = 180 lb
Gravitational acceleration = 9.8 m/s^2
Converting Alexander's weight to kilograms:
Weight of Alexander = 180 lb * (1 kg / 2.2 lb) ≈ 81.82 kg
Calculating the force:
Force = Weight of Alexander * Gravitational acceleration = 81.82 kg * 9.8 m/s^2 ≈ 802.84 N
Next, we need to calculate the distance. Given:
Height of Mt. Krumpett = 5570 m
Using the formula for work, we can find the energy needed for the uphill climb:
Energy = work = force * distance = 802.84 N * 5570 m ≈ 4,470,115.68 J
Step 2: Double the energy for the round trip
To provide a margin of safety, Alexander assumes that he will need as much energy for the return trip as for the uphill climb. Thus, we need to double the energy calculated in Step 1:
Total energy for the round trip = 2 * Energy ≈ 2 * 4,470,115.68 J ≈ 8,940,231.36 J
Step 3: Calculate the energy provided by one nutrition bar
Given:
One bar contains 10 g of fat, 40 g of protein, and 50 g of carbohydrates.
The energy provided by each macronutrient can be calculated using their respective energy values:
- Fat: 1 g = 9 kJ/g
- Protein: 1 g = 4 kJ/g
- Carbohydrates: 1 g = 4 kJ/g
Calculating the energy provided by each macronutrient:
Energy from fat = 10 g * 9 kJ/g = 90 kJ
Energy from protein = 40 g * 4 kJ/g = 160 kJ
Energy from carbohydrates = 50 g * 4 kJ/g = 200 kJ
Total energy provided by one nutrition bar = Energy from fat + Energy from protein + Energy from carbohydrates
= 90 kJ + 160 kJ + 200 kJ = 450 kJ
Step 4: Calculate the number of bars needed
To find the number of bars needed, we need to divide the total energy for the round trip by the energy provided by one nutrition bar:
Number of bars needed = Total energy for the round trip / Energy provided by one nutrition bar
= 8,940,231.36 J / 450 kJ
Both the total energy and the energy provided by one nutrition bar need to be in the same unit, so let's convert the total energy to kJ:
Total energy for the round trip ≈ 8,940,231.36 J * (1 kJ / 1000 J) ≈ 8940.23 kJ
Number of bars needed ≈ 8940.23 kJ / 450 kJ ≈ 19.867
Since we cannot have a fraction of a bar, Alexander should pack around 20 nutrition bars for his journey.
Therefore, Alexander should pack around 20 bars in his backpack for the journey.