The rear window in a car is approximately a rectangle, 1.31 m wide and 0.390 m high. The inside rear-view mirror is 0.440 m from the driver's eyes, and 1.42 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 find the minimum height of the rear-view mirror, we need to consider the geometry of the situation.
Let's understand the problem step by step:
1. The rear window is a rectangle, with a width of 1.31 m and a height of 0.390 m.
2. The inside rear-view mirror is 0.440 m from the driver's eyes and 1.42 m from the rear window.
3. We want to determine the minimum height of the rear-view mirror that allows the driver to see the entire height of the rear window without moving her head.
To solve this problem, we can utilize a similar triangle concept. The driver's eyes, the top of the rear window, and the top of the rear-view mirror form similar triangles.
Let's denote the height of the rear-view mirror as h.
Using similar triangles, we can set up the following equation:
(Height of the rear window) / (Distance from the rear window to the mirror) = h / (Distance from the driver's eyes to the mirror)
Substituting the given values:
0.390 m / 1.42 m = h / 0.440 m
Now, let's solve for h:
(h / 0.440 m) = (0.390 m / 1.42 m)
h = (0.390 m * 0.440 m) / 1.42 m
h ≈ 0.120 m
Therefore, the minimum height of the rear-view mirror is approximately 0.120 m (or 12 cm).
The driver will be able to see the entire height of the rear window in the mirror without moving her head if the mirror's height is at least 0.120 m.