Dam breaks present a serious risk of widespread property damage and loss of life. You're asked to assess a 1500 -m-wide dam holding back a lake 95 m deep. The dam was built to withstand a force of \(100 \mathrm{GN},\) which is supposed to be at least \(50 \%\) over the force it actually experiences. Should the dam be reinforced? (Hint: You'll need your calculus skills.)

Pressure in a fluid (like water) increases with depth. The equation for pressure due to an incompressible, non-moving fluid is given by \(P = \rho g h\), where P is the pressure, \(\rho\) is the density of the fluid (in this case water, which is about \(1000 \, \mathrm{kg/m^3}\)), g is the acceleration due to gravity (\(9.8 \, \mathrm{m/s^2}\)), and h is the height of the fluid above the point in question. In this case, h is equivalent to the depth of the lake, which is 95 m. So by plugging these values into the equation \(P = \rho g h\), we get \(P = 1000 \, \mathrm{kg/m^3} \times 9.8 \, \mathrm{m/s^2} \times 95 \, \mathrm{m} = 931000 \, \mathrm{Pa}\). Note that 1 GN = \(10^9 \, \mathrm{N}\) and 1 Pa = 1 N/m².

Now knowing the pressure, the force exerted on the dam by the water can be calculated using the equation \(F = P \times A\), where F is the force, P is the pressure, and A is the area of the dam. The dam is 1500 m wide and 95 m high, so A = 1500 m × 95 m = \(142500 m^2\). Thus, \(F = 931000 \, \mathrm{Pa} \times 142500 \, \mathrm{m^2} = 1.327475 \times 10^{14} \, \mathrm{N}\).

It was mentioned that a force of 100 GN (which is equal to \(100 \times 10^9 \, N\) or \(10^{11} \, N\)) is 50% over the actual force the dam experiences. So, the actual force the dam experiences should be \(10^{11} \, N / 1.50 = 0.67 \times 10^{11} \, N\) or \(67 \times 10^9 \, N\). Comparing this to the calculated force on the dam ( \(1.327475 \times 10^{14} \, N\)), it can be concluded that the dam needs to be reinforced as the force exerted by the water is more than the dam can withstand.