The amount of water remaining in a certain bathtub as it drains when the plug is pulled is represented by the equation, `L=-4t^(2)-8t+128`, where L represents the number of liters of water in the bathtub and t represents the amount of time, in minutes, since the plug was pulled. Which expression represents the number of minutes it takes for half of the water that was in the bathtub before the plug was pulled to drain?

Options

A. `-1+sqrt(33)`
B. `-1+sqrt(17)`
C. `(-1+sqrt(33))/(2)`
D. `(-1+2sqrt(17))/(2)`

Answer

To solve the problem, we need to determine the time it takes for half of the water in the bathtub to drain. The equation given is:

\[ L = -4t^2 - 8t + 128 \]

where \( L \) is the amount of water in liters and \( t \) is the time in minutes since the plug was pulled.

### Step 1: Find the initial amount of water in the bathtub
To find the initial amount of water, we need to evaluate \( L \) at \( t = 0 \):

\[ L(0) = -4(0)^2 - 8(0) + 128 = 128 \text{ liters} \]

### Step 2: Determine half of the initial amount of water
Half of the initial amount of water is:

\[ \frac{128}{2} = 64 \text{ liters} \]

### Step 3: Set up the equation to find the time when \( L = 64 \)
We need to find \( t \) when \( L = 64 \):

\[ 64 = -4t^2 - 8t + 128 \]

### Step 4: Rearrange the equation
Rearranging the equation gives:

\[ 0 = -4t^2 - 8t + 128 - 64 \]
\[ 0 = -4t^2 - 8t + 64 \]

### Step 5: Simplify the equation
We can divide the entire equation by -4 to simplify:

\[ 0 = t^2 + 2t - 16 \]

### Step 6: Use the quadratic formula to solve for \( t \)
The quadratic formula is given by:

\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our case, \( a = 1 \), \( b = 2 \), and \( c = -16 \). Plugging in these values:

\[ t = \frac{-2 \pm \sqrt{2^2 - 4(1)(-16)}}{2(1)} \]
\[ t = \frac{-2 \pm \sqrt{4 + 64}}{2} \]
\[ t = \frac{-2 \pm \sqrt{68}}{2} \]
\[ t = \frac{-2 \pm 2\sqrt{17}}{2} \]
\[ t = -1 \pm \sqrt{17} \]

### Step 7: Determine the valid solution for time
Since time cannot be negative, we take the positive solution:

\[ t = -1 + \sqrt{17} \]

### Final Answer
The expression representing the number of minutes it takes for half of the water to drain is:

\[ t = -1 + \sqrt{17} \]