he rear window in a car is approximately a rectangle, 1.28 m wide and 0.160 m high. The inside rear-view mirror is 0.590 m from the driver's eyes, and 1.59 m from the rear window. What is the minimum height of the rear-view mirror if the driver is to be able to see the entire height of the rear window in the mirror without moving her head?
To determine the minimum height of the rear-view mirror, we need to find the height of the rear window that is visible from the driver's perspective.
Given:
Width of the rear window (w) = 1.28 m
Height of the rear window (h) = 0.160 m
Distance from driver's eyes to the rear-view mirror (d_mirror) = 0.590 m
Distance from the rear window to the rear-view mirror (d_window) = 1.59 m
We can use similar triangles to compare the dimensions of the rear window and the corresponding reflection in the mirror.
Using the similar triangles:
h_mirror / h = d_mirror / d_window
Substituting the given values:
h_mirror / 0.160 m = 0.590 m / 1.59 m
Simplifying:
h_mirror = (0.160 m * 0.590 m) / 1.59 m
h_mirror = 0.059 m
Therefore, the minimum height of the rear-view mirror should be 0.059 meters (or approximately 5.9 cm) for the driver to be able to see the entire height of the rear window without moving her head.
To determine the minimum height of the rear-view mirror, we need to consider the geometry of the situation.
Let's name some of the given measurements:
- Width of the rear window = 1.28 m
- Height of the rear window = 0.160 m
- Distance from the driver's eyes to the mirror = 0.590 m
- Distance from the mirror to the rear window = 1.59 m
Now, we can draw a diagram to visualize the setup:
+------------------------+
| | |
| | |
Rear | | |
Window | | |
| | |
+------------------------+
Mirror
From the given information, we can establish two important relationships:
1. The distance from the driver's eyes to the top of the rear window (assuming a straight line of sight) is equal to the distance from the top of the rear window to the mirror. This can be represented as:
Distance from driver's eyes to top of the rear window = Distance from top of the rear window to the mirror.
2. Similarly, the distance from the driver's eyes to the bottom of the rear window is equal to the distance from the bottom of the rear window to the mirror:
Distance from driver's eyes to bottom of the rear window = Distance from bottom of the rear window to the mirror.
Now let's calculate these distances individually:
Distance from driver's eyes to top of the rear window:
This can be calculated by subtracting the height of the rear window from the distance between the mirror and the rear window:
Distance from driver's eyes to top of the rear window = 1.59 m - 0.160 m = 1.43 m.
Distance from driver's eyes to bottom of the rear window:
This can be calculated by subtracting the height of the rear window from the distance between the mirror and the rear window:
Distance from driver's eyes to bottom of the rear window = 1.59 m - 0 = 1.59 m.
Now, since the driver wants to see the entire height of the rear window in the mirror without moving her head, the position of the mirror should be such that the distance from her eyes to the top of the rear window is equal to the distance from her eyes to the bottom of the rear window.
Therefore, the minimum height of the rear-view mirror should be the distance from the top of the rear window to the bottom of the rear window:
Minimum height of the rear-view mirror = Distance from driver's eyes to top of the rear window - Distance from driver's eyes to bottom of the rear window
= 1.43 m - 1.59 m
= -0.16 m.
However, a negative height is not meaningful in this context. Therefore, the minimum height of the rear-view mirror, in this case, would be zero as the mirror needs to be at eye level to see the full height of the rear window without moving the head.