Final answer:
The force on the dam due to hydrostatic pressure is calculated using the formula F = pghA, where p is the density of water, g is the acceleration due to gravity, h is the depth of the water, and A is the area of contact between the water and the dam. By plugging in the given values, the force on the dam is found to be 1.03 × 10⁷ N.
Step-by-step explanation:
To find the force on the dam due to hydrostatic pressure, we can use the formula: F = pghA, where p is the density of water, g is the acceleration due to gravity, h is the depth of the water, and A is the area of contact between the water and the dam.
First, we need to calculate the average pressure, p, at the average depth of 40.0 m.
Since pressure increases linearly with depth, the average pressure is equal to the pressure at the average depth. Using the formula p = pgh, we can calculate
p = (1000 kg/m³)(9.8 m/s²)(40.0 m)
= 3.92 × 10⁵ N/m².
Next, we need to calculate the area of contact, A, between the water and the dam.
Since the dam has the shape of a trapezoid, we can use the formula for the area of a trapezoid, A = (a + b)h/2, where a and b are the widths of the top and bottom of the trapezoid, and h is the height.
Plugging in the values, we get
A = (30 m + 40 m)(18 m)/2
= 900 m².
Finally, we can plug the values of p, g, h, and A into the formula F = pghA to find the force on the dam:
F = (3.92 × 10⁵ N/m²)(9.8 m/s²)(3 m)(900 m²)
= 1.03 × 10⁷ N.