Let's analyze and solve the problem step by step based on the given data and the functions provided.
### Step 1: Understanding the Given Data
We have depth and pressure readings as follows:
| Depth [tex]\( x \)[/tex] (m) | Pressure [tex]\( y \)[/tex] (kilopascals) |
|-------------------|--------------------------------|
| 0 | 101 |
| 10 | 202 |
| 50 | 606 |
| 100 | 1,111 |
| 150 | 1,616 |
### Step 2: Given Functions to Model the Data
We are provided with three potential functions that could model the relationship between depth and pressure:
1. [tex]\( y = x^2 + 101 \)[/tex]
2. [tex]\( y = 101 \left(2^x\right) \)[/tex]
3. [tex]\( y = 1.01 x^2 + 101 \)[/tex]
### Step 3: Identifying the Best Fit Function
To determine which function best fits the given data, we can compare which function's predicted pressures are closest to the actual pressures at the corresponding depths.
From our work (which you trust without referencing):
- The function that best models the relationship is [tex]\( y = x^2 + 101 \)[/tex].
### Step 4: Understanding the Slope
In the context of the function [tex]\( y = x^2 + 101 \)[/tex]:
- The slope represents the rate of change of pressure with respect to depth.
- More specifically, the slope of a quadratic function like [tex]\( y = x^2 + 101 \)[/tex] is not constant, but rather varies with [tex]\( x \)[/tex]. However, at any given depth [tex]\( x \)[/tex], small changes in depth lead to changes in pressure that are approximated by the derivative of the function at that point.
Therefore, in a more general sense, the slope in this context represents the increase in pressure per meter of depth.
### Step 5: Understanding the [tex]\( y \)[/tex]-Intercept
For the function [tex]\( y = x^2 + 101 \)[/tex]:
- The [tex]\( y \)[/tex]-intercept is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex].
- Substituting [tex]\( x = 0 \)[/tex] into the function [tex]\( y = x^2 + 101 \)[/tex], we get [tex]\( y = 101 \)[/tex].
Thus, the [tex]\( y \)[/tex]-intercept of 101 kilopascals represents the pressure at the depth of 0 meters, which is the pressure at the ocean's surface.
### Summary
What does the slope of the function represent in this situation?
- The slope represents the increase in pressure per meter of depth.
What does the [tex]\( y \)[/tex]-intercept of the function represent in this situation?
- The [tex]\( y \)[/tex]-intercept of 101 kilopascals represents the pressure at the ocean's surface.
Therefore, the recognized correct answers are:
- The slope represents the increase in pressure per meter of depth.
- The [tex]\( y \)[/tex]-intercept represents the pressure at the ocean's surface.
