Let's start by completing the first table.
The questions gives the depth in two times:
- 2 hours after start -> 64 inches of water
- 4 hours after start -> 48 inches of water
From this, we can identifythe units:
- for time, we will use hours
- for depth, we will use inches
The first given values are:
- for time, 2
- for depth, 64
The second given valuesa are:
- for time, 4
- for depth, 48
So, the table is:
Quantity Name | Time | Depth of Pool
Unit | hour | inch
Given Value 1 | 2 | 64
Given Value 2 | 4 | 48
Now, we need to calculate the slope, which can be done using the given points and the equation:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{48-64}{4-2} \\ m=\frac{-16}{2} \\ m=-8 \end{gathered}[/tex]So, the slope is -8 inches per hour.
The interceptwe the b in the equation:
[tex]y=mx+b[/tex]Where y is the depth and x is the time. So, using the first given values and the slope, we can find b:
[tex]\begin{gathered} 64=-8\cdot2+b \\ 64=-16+b \\ b=64+16 \\ b=80 \end{gathered}[/tex]Thus, the intercept is 80 inches.
So, the formula we have for the pump is:
[tex]y=-8x+80[/tex]Using it, we can answer questions 1 and 2:
1. After 6 hours means x = 6, so:
[tex]\begin{gathered} y=-8\cdot6+80 \\ y=-48+80 \\ y=32 \end{gathered}[/tex]So, y = 32, thus, the depth of the water will be 32 inches.
2. The level 24 inches mean y = 24, so:
[tex]\begin{gathered} 24=-8x+80 \\ 8x=80-24 \\ 8x=56 \\ x=\frac{56}{8} \\ x=7 \end{gathered}[/tex]Thus, x = 7, and the water will be at 24 inches after 7 hours.
