To model the path of light from the car's headlight using the provided data, we can start by analyzing the structure of the data points given. The table provides vertical and horizontal distances, which can be interpreted as coordinates in a two-dimensional space (vertical distance as \(y\) and horizontal distance as \(x\)).
Let's first note down the coordinates from the table:
- (16, 1.3)
- (4.2, 0.1)
- (6, 0)
- (7.8, 0.1)
- (0, 1.3)
To create a model, we need to determine the relationship between the vertical distance and horizontal distance. Since light from car headlights usually follows a parabolic path, a quadratic function might be suitable.
A general equation for a parabola can be written in vertex form:
\[ y = a(x - h)^2 + k \]
Where:
- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the width and direction of the opening of the parabola.
Finding the Vertex (h, k)
To estimate the values of \(h\) and \(k\):
- Plot the points on a graph.
- Determine where the vertex of the parabola lies.
From the data, it appears that the vertical distance is highest at either end when the horizontal distance is at a maximum (around 0 and 16). The lowest vertical distance occurs when the horizontal distance is around 6 (which has a vertical distance of 0).
Estimating \(h\)
Looking at the data, the highest points are near the outer range of horizontal distances (at \(x=0\) and \(x=16\)), so we can estimate that the average horizontal distance where the vertical distance is midway (zero in this case) is around 8.
This estimation suggests that \( h \) (the x-coordinate of the vertex) is likely to be around \(h = 8\).
Conclusion for h
Without using technology or detailed calculations (like regression analysis or fitting a quadratic function), we thus conclude:
In the equation that models this situation,
\(h \approx 8\).
However, since you specified rounding to the nearest hundredth:
- We can state \(h = 8.00\).
Round to the nearest hundredth:
h = 8.00.