To analyze the equation [tex]\( W = -\frac{1}{2} m + 16 \)[/tex] and understand the meaning of the coefficient [tex]\(-\frac{1}{2}\)[/tex], let's break it down step by step.
1. Equation Components:
- [tex]\( W \)[/tex] represents the amount of water in the bottle in ounces.
- [tex]\( m \)[/tex] represents the time in minutes.
- [tex]\( 16 \)[/tex] is a constant, representing the initial amount of water in the bottle when [tex]\( m = 0 \)[/tex].
2. Understanding the Coefficient:
- The term [tex]\(-\frac{1}{2}m\)[/tex] indicates how the amount of water [tex]\( W \)[/tex] changes as time [tex]\( m \)[/tex] increases.
- The negative sign indicates that the water level is decreasing over time.
- The fraction [tex]\(\frac{1}{2}\)[/tex] represents the rate at which the water is leaking.
3. Interpreting the Rate:
- The coefficient [tex]\(-\frac{1}{2}\)[/tex] means that for every 1 minute that passes, the amount of water decreases by [tex]\(\frac{1}{2}\)[/tex] ounce.
- So, the rate of leakage is [tex]\(\frac{1}{2}\)[/tex] ounce per minute.
Let's summarize:
The [tex]\(-\frac{1}{2}\)[/tex] in the equation [tex]\( W = -\frac{1}{2} m + 16 \)[/tex] represents the constant rate at which water is leaking from the bottle. Specifically, the water is leaking at a rate of 0.5 ounces per minute.
