To model the path of light from a car's headlight represented by the given data, we can assume that the light spreads out in a parabolic or quadratic form. Given the vertical and horizontal distances, we can use regression analysis or curve fitting to find the best-fitting equation, usually in the form of a quadratic equation \( y = ax^2 + bx + c \).
Here, we can treat horizontal distance as \( x \) and vertical distance as \( y \).
Let's compute to create a possible quadratic equation.
- First, let's organize the data:
| Horizontal Distance (x) | Vertical Distance (y) | |--------------------------|-----------------------| | 16 | 1.3 | | 4.2 | 0.1 | | 6 | 0 | | 7.8 | 0.1 | | 0 | 1.3 |
- Next, we can apply a quadratic regression using these points. Using a tool like a calculator, spreadsheet, or programming language for regression analysis, we might get an equation.
However, we can also calculate it manually here as an approximation since we don't have specific tools at hand right now.
Using points to fit a quadratic curve, let's assume we create a model fitting the structure:
Given that \( h \) represents the vertical distance where the headlight is focused when \( x = 0 \) (or some specific vertex form), we find the y-intercept \( c \) of our resulting equation, which is the height \( h \).
Let’s see the values at \( x = 0 \):
- When \( x = 0 \), \( y \) (Vertical Distance) is \( 1.3 \).
This suggests \( h \), the height of the headlight at horizontal distance zero, is \( 1.3 \).
Thus, if the equation we derive is validated, the rounded value of \( h \) is:
h = 1.30
If you have access to software or specific regression analysis tools, I'd encourage fitting these points for a more precise quadratic function. But based on the direct observation, \( h \) is 1.30 rounded to the nearest hundredth.
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