A certain person had a brain that weighed 1.30 kg and contained \( 6.71 \times 10^{10} \) cells. Part 1 of 2 Assuming that each cell was completely filled with water (density \( =1.00 \frac{\mathrm{~g}}{\mathrm{~mL}} \) ), calculate the length of one side of such a cell if it were a cube. Round your answer to 3 significant digits. \[ 0.00268 \mathrm{~cm} \] Part: \( 1 / 2 \) Part 2 of 2 If the cells were spread out into a thin layer that was a single cell thick, what would be the total surface area (in square meters) for the top side of the cell layer? Round your answer to 3 significant digits. \( \square \) \( m^{2} \)

To calculate the total volume of the cells, we start from the brain's weight. Given the weight of the brain is 1.30 kg, and knowing the density of water is \(1.00 \frac{\mathrm{g}}{\mathrm{mL}}\), we convert the mass in kilograms to grams: 1.30 kg = 1300 g. Since the density of water is \(1.00 \frac{\mathrm{g}}{\mathrm{mL}}\), the total volume (in mL) is equal to the mass in grams. Therefore, the total volume (V) of the cells is: V = 1300 mL = 1300 cm³ (as 1 mL = 1 cm³). With \(6.71 \times 10^{10}\) cells, the average volume per cell is: \[ \text{Volume per cell} = \frac{1300 \, \text{cm}^3}{6.71 \times 10^{10}} \approx 1.937 \times 10^{-8} \, \text{cm}^3. \] Assuming each cell is a cube (which means \(V = a^3\), where \(a\) is the length of one side of the cube), we can find \(a\) by taking the cube root: \[ a = \sqrt[3]{1.937 \times 10^{-8}} \approx 0.00268 \, \text{cm}. \] Thus, the length of one side of the cell rounded to three significant digits is \(0.00268 \, \text{cm}\). For Part 2, if these cells are spread out into a single layer, we need to find the total surface area. First, calculate the surface area of a single cell. Each cell is a cube with a side length of \(0.00268 \, \text{cm}\). The surface area \(A\) of a cube is given by: \[ A = 6a^2. \] Calculating \(A\): \[ A = 6(0.00268)^2 \approx 6(7.1584 \times 10^{-6}) \approx 4.295 \times 10^{-5} \, \text{cm}^2. \] Now, to find the total surface area for all \(6.71 \times 10^{10}\) cells, we calculate: \[ \text{Total Surface Area} = 6.71 \times 10^{10} \times 4.295 \times 10^{-5} \approx 2883700.3 \, \text{cm}^2. \] To convert to square meters (note \(1 \text{m}^2 = 10,000 \text{cm}^2\)): \[ \text{Total Surface Area} = \frac{2883700.3 \, \text{cm}^2}{10000} \approx 288.37 \, \text{m}^2. \] Rounded to three significant digits, the total surface area for the top side of the single-cell thick layer is \(288 \, \text{m}^2\).