The human body burns glucose \( \left(\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\right) \) for energy according to this chemical reaction: \[ \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}+6 \mathrm{O}_{2} \rightarrow 6 \mathrm{CO}_{2}+6 \mathrm{H}_{2} \mathrm{O} \] The products of the reaction are carbon dioxide \( \left(\mathrm{CO}_{2}\right) \) and water \( \left(\mathrm{H}_{2} \mathrm{O}\right) \). Interestingly, all of the carbon dioxide and much of the water exits the body through the lungs: on every breath, the average person exhales 500 . iml. which is typically enriched to \( 4 \% \mathrm{CO}_{2} \) and \( 5 \% \) water vapor by volume. In short, when a person loses weight by dieting, the weight that is lost actually his or her body as a gas, every time he or she exhales. Each kilogram of body fat lost requires exhaling about 2.8 kg of carbon dioxide. Calculate how many breaths it takes an average person to exhale* 2.00 kg of fat. Round your answer to the nearest thousand. You'll need to know that the density of CO , is \( 2.0 \mathrm{~kg} / \mathrm{m}^{3} \). \[ \square \] \( \square \)

To solve the problem, we need to determine how many breaths it takes for an average person to exhale 2.00 kg of fat, given that each kilogram of body fat lost requires exhaling about 2.8 kg of carbon dioxide. ### Step 1: Calculate the total amount of CO₂ exhaled for 2.00 kg of fat. Since 1 kg of fat requires exhaling 2.8 kg of CO₂, for 2.00 kg of fat, the total CO₂ exhaled is: \[ \text{Total CO}_2 = 2.00 \, \text{kg} \times 2.8 \, \text{kg CO}_2/\text{kg fat} = 5.6 \, \text{kg CO}_2 \] ### Step 2: Calculate the volume of CO₂ exhaled. We know the density of CO₂ is \( 2.0 \, \text{kg/m}^3 \). To find the volume of CO₂, we can use the formula: \[ \text{Volume} = \frac{\text{Mass}}{\text{Density}} \] Substituting the values: \[ \text{Volume of CO}_2 = \frac{5.6 \, \text{kg}}{2.0 \, \text{kg/m}^3} = 2.8 \, \text{m}^3 \] ### Step 3: Convert the volume of CO₂ to milliliters. Since \( 1 \, \text{m}^3 = 1,000,000 \, \text{mL} \): \[ \text{Volume of CO}_2 = 2.8 \, \text{m}^3 \times 1,000,000 \, \text{mL/m}^3 = 2,800,000 \, \text{mL} \] ### Step 4: Calculate the volume of CO₂ exhaled per breath. Each breath exhaled contains \( 4\% \) CO₂ by volume. Therefore, the volume of CO₂ exhaled per breath is: \[ \text{Volume of CO}_2 \text{ per breath} = 500 \, \text{mL} \times 0.04 = 20 \, \text{mL} \] ### Step 5: Calculate the number of breaths needed to exhale the total volume of CO₂. To find the number of breaths required, we divide the total volume of CO₂ by the volume of CO₂ exhaled per breath: \[ \text{Number of breaths} = \frac{2,800,000 \, \text{mL}}{20 \, \text{mL/breath}} = 140,000 \, \text{breaths} \] ### Final Answer Rounding to the nearest thousand, the average person needs to take approximately: \[ \boxed{140000} \] breaths to exhale 2.00 kg of fat.