Questions: A tank is full of water when a valve at the bottom of the tank is opened. The equation V=100(242-t)^2 gives the volume of water in the tank, in cubic meters, after t hours. What is the volume of water in the tank before the valve is opened? square meters How long does it take the tank to fully empty hours Find an equation for dV/dt dV/dt= What is the flow rate after 4 hours? Select an answer When is the water flowing out of the tank the fastest? t= hours

A tank is full of water when a valve at the bottom of the tank is opened. The equation V=100(242-t)^2 gives the volume of water in the tank, in cubic meters, after t hours. What is the volume of water in the tank before the valve is opened? square meters How long does it take the tank to fully empty hours Find an equation for dV/dt dV/dt= What is the flow rate after 4 hours? Select an answer When is the water flowing out of the tank the fastest? t= hours
Transcript text: A tank is full of water when a valve at the bottom of the tank is opened. The equation $V=100(242-t)^{2}$ gives the volume of water in the tank, in cubic meters, after $t$ hours. What is the volume of water in the tank before the valve is opened? $\square$ cubic meters How long does it take the tank to fully empty $\square$ hours Find an equation for $\frac{d V}{d t}$ $\frac{d V}{d t}=$ $\square$ What is the flow rate after 4 hours? $\square$ Select an answer When is the water flowing out of the tank the fastest? $t=$ $\square$ hours
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Solution

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Solution Steps

Solution Approach

  1. Volume Before Valve is Opened: To find the volume of water before the valve is opened, evaluate the volume equation at \( t = 0 \).

  2. Time to Fully Empty: Determine when the volume \( V \) becomes zero by solving the equation \( 100(242-t)^2 = 0 \) for \( t \).

  3. Flow Rate Equation: Find the derivative of the volume function \( V(t) \) with respect to time \( t \) to get the flow rate equation \( \frac{dV}{dt} \).

Step 1: Volume Before Valve is Opened

To find the volume of water in the tank before the valve is opened, we evaluate the volume function \( V \) at \( t = 0 \):

\[ V(0) = 100(242 - 0)^2 = 100 \times 242^2 = 5856400 \, \text{cubic meters} \]

Step 2: Time to Fully Empty the Tank

To determine how long it takes for the tank to fully empty, we solve the equation \( V = 0 \):

\[ 100(242 - t)^2 = 0 \]

This simplifies to:

\[ 242 - t = 0 \implies t = 242 \, \text{hours} \]

Step 3: Flow Rate Equation

The flow rate \( \frac{dV}{dt} \) is found by differentiating the volume function \( V(t) \):

\[ \frac{dV}{dt} = 200t - 48400 \]

Final Answer

  • Volume before the valve is opened: \( \boxed{5856400} \, \text{cubic meters} \)
  • Time to fully empty the tank: \( \boxed{242} \, \text{hours} \)
  • Flow rate equation: \( \boxed{\frac{dV}{dt} = 200t - 48400} \)
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