A tank in the shape of an inverted right circular cone has height 7 meters and radius 4 meters. It is filled with 5 meters of hot chocolate. Find the work required to empty the tank by pumping the hot chocolate over the top of the tank. The density of hot chocolate is δ=1050 kg/m3.
To find the work required to empty the tank, we need to consider the force required to lift the hot chocolate to the height of the tank's top. This force can be found by calculating the weight of the hot chocolate.
First, we need to determine the volume of the hot chocolate. The tank is in the shape of an inverted right circular cone, so its volume can be calculated using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
where V is the volume, π is a mathematical constant (approximately equal to 3.14159), r is the radius, and h is the height.
For the hot chocolate, we have:
V_hot_chocolate = (1/3) * π * (4^2) * 5
Next, we need to determine the mass of the hot chocolate. The mass can be calculated using the density and volume:
m_hot_chocolate = density * V_hot_chocolate
Given the density δ = 1050 kg/m^3, we have:
m_hot_chocolate = 1050 * V_hot_chocolate
Finally, we can calculate the weight of the hot chocolate using the force of gravity:
weight_hot_chocolate = m_hot_chocolate * g
where g is the acceleration due to gravity (approximately equal to 9.8 m/s^2).
Now, we can calculate the work required to lift the hot chocolate. Work is defined as the force applied over a distance:
W = force * distance
In this case, the force is the weight of the hot chocolate, and the distance is the height of the tank.
W = weight_hot_chocolate * height_tank
Given the height of the tank is 7 meters, we have:
W = weight_hot_chocolate * 7
Now, let's plug in the values we have:
V_hot_chocolate = (1/3) * π * (4^2) * 5
≈ 83.78 m^3
m_hot_chocolate = 1050 * V_hot_chocolate
≈ 87,972.45 kg
weight_hot_chocolate = m_hot_chocolate * g
≈ 863,122.51 N
W = weight_hot_chocolate * 7
≈ 6,042,857.55 J
Therefore, the work required to empty the tank by pumping the hot chocolate over the top of the tank is approximately 6,042,857.55 Joules.