Final answer:
To calculate the work required to pump water out of a spherical tank into a spout, you use the formula for the volume of a sphere, the given density of water, and the acceleration due to gravity to find the force due to the weight of the water, then multiply by the sum of the tank's radius and spout's height to find the work, rounding to the nearest whole number.
Step-by-step explanation:
The subject of this question falls under Physics, particularly involving the concept of work done against gravity and the physical properties of fluids. The grade level is appropriate for high school students who are learning about mechanics and fluid dynamics. Let's determine the work required to pump water out of a spherical tank with a radius of 9 m and a spout that extends 3 m above the tank.
The work done to lift water is given by W = F * d, where F is the force needed to lift the water, and d is the distance the water must be lifted. The force F is the weight of the water, which is the mass (m) times the acceleration due to gravity (g), F = m * g. The mass of the water is its volume (V) times the density (ρ), m = ρ * V. The volume of a sphere is V = (4/3)π*r^3, and the volume of water to be pumped out is therefore the volume of the tank. However, we must add the additional height of the spout to the distance d. Therefore, the total distance is r + h = 9 m + 3 m.
Combining these relationships and substituting the given values for density and g, we get:
- Density of water (ρ) = 1,000 kg/m³
- Acceleration due to gravity (g) = 9.8 m/s²
- Radius of tank (r) = 9 m
- Height of spout (h) = 3 m
Thus, the volume is V = (4/3)π*9^3 m³ and the mass is m = ρ * V. The force F is m * g. The work done is W = F * (r + h).
Finally, after calculating these values, you round the result to the nearest whole number to find the work in Joules required to pump the water out of the tank.