Answer:
when the water has a height of 120 cm, or 1.2 meters, the rate at which the height of the water is changing is 2.4 m/s.
Explanation:
To find the rate at which the height of the water is changing, we need to use calculus.
We can define the height of the water at any given time as a function h(t). The rate at which the water is being pumped into the trough can be represented as a constant function f(t) = 6 m/s. The rate at which the water's height is changing at any given time is equal to the derivative of the height function with respect to time, or h'(t).
The volume of water in the trough is equal to the area of the base of the trough (which is in the shape of an isoceles triangle) times the height of the water. The volume of the water in the trough can also be represented as a function V(h).
We can set up the following equation to represent the relationship between the volume of water in the trough and the height of the water:
V(h) = (1/2) * b * h * h
where b is the base of the isoceles triangle (5 meters).
We can also set up the following equation to represent the relationship between the volume of water being pumped into the trough and the rate at which the water's height is changing:
V'(h) = f(t)
We can substitute the expression for V(h) into the equation for V'(t) and solve for h'(t):
h'(t) = f(t) / [(1/2) * b]
= (6 m/s) / [(1/2) * 5 m]
= 6 m/s / (1/2) * (5 m)
= 6 m/s / (2.5 m)
= 2.4 m/s
So when the water has a height of 120 cm, or 1.2 meters, the rate at which the height of the water is changing is 2.4 m/s.