Use the table to answer the question.

Question

Horizontal Distance (inches) Vertical Distance (inches)
0.41 0
0.2 1.5
0.07 3
0.2 4.5
0.41 6
The table shows the horizontal distance (or the depth) and the vertical distance (or the distance across) of a small satellite dish. Using technology, create an equation to model the situation. In the equation, what is the value of k? Round to the nearest whole number.

(1 point)
Responses

k=2
k equals 2

k=3
k equals 3

k=1
k equals 1

k=0
k equals 0

Answer 1

To model the situation of the satellite dish with the given horizontal distance (depth) and vertical distance (width), we can use the formula for a parabola, which is generally represented as:

\[ y = k(x - h)^2 + k \]

where:

• \( y \) is the vertical distance,
• \( x \) is the horizontal distance,
• \( h \) is the vertex of the parabola (the horizontal shift),
• \( k \) is the value that determines the width and direction of the parabola.

Given that we have pairs of points, we can use some of those points to derive an equation. However, the problem specifically asks for the value of \( k \).

It looks like the problem is asking for a generalized coefficient that can be derived from the data. Unfortunately, without a specific equation or context, we might take a heuristic approach to analyze the trend of the data.

Given the limited data points, ascertain whether the dish resembles some standard geometric shape (like a parabola), which often follows a quadratic relationship:

1. Mapping the data points and trying to find \( k \):
   • When considering the relationship between vertical and horizontal distances:
   • If we assume the relationship resembles a quadratic, the general shape can be represented by \( y = k \cdot x^2 \) where \( k \) determines the steepness of the parabola.

By observing the rounded values from the vertical distance in rows and the corresponding horizontal distances, we can estimate \( k \).

If we analyze the vertical distance and note how it changes with horizontal changes, we can find a suitable value for \( k \).

Looking closely at the relation might yield:

• \( (0, 0) \) as the vertex.
• The maximum (in the context of distances) occurs at crossings among points, allowing estimation of curvature to yield estimates of \( k \).

After calculations of predicted values for the curve mapping, one might arrive at an estimation of \( k \).

So if a definitive rule states, which we do not get directly from the dataset but can infer from how vertical distanced relates closely to horizontal constituents, one generalized value could be:

Choose \( k = 2 \) based on heuristic curvature and rounding.

Thus, rounding to the nearest whole number \( k = 2 \), would be your answer.

So the answer would be:

• k = 2