Question

How many grams of carbohydrate does a person of mass 74 kg need to metabolize to climb five flights of stairs \((15 \mathrm{m}\) height increase)? Each gram of carbohydrate provides 17.6 kJ of energy. Assume \(10.0 \%\) efficiency-that is, \(10.0 \%\) of the available chemical energy in the carbohydrate is converted to mechanical energy. What happens to the other \(90 \%\) of the energy?

Short answer

Answer: The person needs approximately 6.18 grams of carbohydrate to climb the stairs. The remaining 90% of the energy is lost as heat due to inefficiencies in the body's energy conversion process and is dissipated through processes like sweating, breathing, and radiation.

Step-by-step solution

Calculate the Potential Energy

To calculate the potential energy gained by climbing the stairs, we can use the formula \(PE = mgh\), where \(PE\) is potential energy, \(m\) is mass, \(g\) is gravitational acceleration (\(9.81 \, \mathrm{m/s^2}\)), and \(h\) is the height climbed. In this case, \(m = 74 \, \mathrm{kg}\) and \(h = 15 \, \mathrm{m}\). \(PE = (74 \, \mathrm{kg})(9.81 \, \mathrm{m/s^2})(15 \, \mathrm{m})\) \(PE = 10887.3 \, \mathrm{J}\) (joules)

Convert Potential Energy to Mechanical Energy Requirement

Given the 10% efficiency, we need to determine the amount of mechanical energy required. Divide the potential energy by the efficiency: \(E_{\mathrm{mech}} = \frac{PE}{0.10}\) \(E_{\mathrm{mech}} = \frac{10887.3 \, \mathrm{J}}{0.10}\) \(E_{\mathrm{mech}} = 108873 \, \mathrm{J}\)

Calculate Required Grams of Carbohydrate

Now we have our mechanical energy requirement, we can calculate how many grams of carbohydrate are required. We know that each gram of carbohydrate provides \(17.6 \, \mathrm{kJ}\): \(grams = \frac{E_{\mathrm{mech}}}{17.6 \times 10^3 \, \mathrm{J/g}}\) \(grams = \frac{108873 \, \mathrm{J}}{17.6 \times 10^3 \, \mathrm{J/g}}\) \(grams \approx 6.18 \, \mathrm{g}\) So, the person needs approximately 6.18 grams of carbohydrate to climb the stairs.

Discuss the Remaining Energy

Only 10% of the energy is used for the mechanical work of climbing the stairs. The remaining 90% is lost as heat, primarily due to inefficiencies in the energy conversion process in the body. This heat is dissipated through various processes such as sweating, breathing, and radiation.