Question

The atmospheric pressures at the top and the bottom of a building are read by a barometer to be 96.0 and \(98.0 \mathrm{kPa}\) If the density of air is \(1.0 \mathrm{kg} / \mathrm{m}^{3},\) the height of the building is \((a) 17 \mathrm{m}\) (b) \(20 \mathrm{m}\) \((c) 170 \mathrm{m}\) \((d) 204 \mathrm{m}\) \((e) 252 \mathrm{m}\)

Short answer

a) 100 m b) 150 m c) 200 m d) 204 m Answer: d) 204 m

Step-by-step solution

Note the given values and identify the suitable formula for solving the problem.

The given values are: the atmospheric pressure at the top of the building (\(P_{top} = 96.0\,\text{kPa}\)), the atmospheric pressure at the bottom of the building (\(P_{bottom} = 98.0\,\text{kPa}\)), and the density of the air (\(\rho=1.0\,\text{kg}/\text{m}^3\)). The formula to use for this problem is the hydrostatic pressure formula, which is given by: $$P_{bottom} - P_{top} = \rho \cdot g \cdot h$$ where \(g\) is the gravitational acceleration (\(9.81\,\text{m}/\text{s}^2\)), and \(h\) is the height of the building.

Convert the pressures from kPa to Pa.

Since the pressure must be in the same unit as the formula, we need to convert the pressures from kPa to Pa, knowing that 1 kPa equals 1000 Pa. $$P_{top} = 96.0\,\text{kPa} \times 1000\,\frac{\text{Pa}}{\text{kPa}} = 96000\,\text{Pa}$$ $$P_{bottom} = 98.0\,\text{kPa} \times 1000\,\frac{\text{Pa}}{\text{kPa}} = 98000\,\text{Pa}$$

Re-write the formula using the given values.

Replace the known values in the formula: $$98000\,\text{Pa} - 96000\,\text{Pa} = 1.0\,\frac{\text{kg}}{\text{m}^3} \cdot 9.81\,\frac{\text{m}}{\text{s}^2} \cdot h$$

Calculate the height of the building.

Solve for \(h\) in the equation: $$2000\,\text{Pa} = 1.0\,\frac{\text{kg}}{\text{m}^3} \cdot 9.81\,\frac{\text{m}}{\text{s}^2} \cdot h$$ Divide both sides of the equation by \((1.0\,\text{kg}/\text{m}^3) \cdot (9.81\,\text{m}/\text{s}^2)\): $$h = \frac{2000\,\text{Pa}}{1.0\,\frac{\text{kg}}{\text{m}^3} \cdot 9.81\,\frac{\text{m}}{\text{s}^2}} \approx 204\,\text{m}$$ Therefore, the height of the building is approximately 204 meters, so the correct answer is (d) 204 m.