{Use of Tech} Flow from a tank A cylindrical tank ​​is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.
c. Find the rate at which water flows from the tank and plot the flow rate function. 

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the rate at which water flows from the tank. This is the derivative of the volume function V with respect to time t, which gives us the flow rate function.
2. Step 2: Identify the volume function. The volume of water in the tank at time t is given by V(t) = 100(200 - t)^2.
3. Step 3: Differentiate the volume function with respect to time t. Use the chain rule to find the derivative of V(t). The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
4. Step 4: Apply the chain rule. Let u = 200 - t, then V(t) = 100u^2. The derivative of V with respect to u is 200u, and the derivative of u with respect to t is -1. Therefore, the derivative of V with respect to t is dV/dt = 200u * (-1).
5. Step 5: Substitute back the expression for u. Since u = 200 - t, substitute this back into the derivative to get the flow rate function: dV/dt = -200(200 - t). This function represents the rate at which water flows from the tank.

Key Concepts

Torricelli's Law

Torricelli's Law states that the speed of fluid flowing out of an orifice under the force of gravity is proportional to the square root of the height of the fluid above the opening. This principle is crucial for understanding how the volume of water in a tank changes over time, as it relates the height of the water to the flow rate.

Volume Function

The volume function V(t) = 100(200−t)² describes how the volume of water in the tank decreases over time. Understanding this function is essential for determining the flow rate, as it provides the relationship between time and the remaining volume of water in the tank.

Derivative and Rate of Change

The derivative of a function represents the rate of change of that function with respect to its variable. In this context, finding the derivative of the volume function V(t) will yield the flow rate of water leaving the tank, which is a key aspect of solving the problem.