The rear window in a car is approximately a rectangle, 1.1 m wide and 0.30 m high. The inside rearview mirror is 0.45 m from the driver's eyes, and 1.45m from the rear window.
Part A
What is the minimum length for the rearview mirror if the driver is to be able to see the entire width and height of the rear window in the mirror without moving her head?
Part B
What is the minimum height for the rearview mirror if the driver is to be able to see the entire width and height of the rear window in the mirror without moving her head?
Part A:
To find the minimum length for the rearview mirror, we need to ensure that the driver can see the entire width of the rear window in the mirror without moving her head.
Given:
Width of the rear window = 1.1 m
Distance from the driver's eyes to the rear window = 1.45 m
To calculate the minimum length of the mirror, we need to consider similar triangles. The ratio of the width of the rear window to the distance from the driver's eyes to the rear window should be equal to the ratio of the width of the mirror to the distance from the driver's eyes to the mirror.
Let's represent the width of the mirror as "x."
Using the similar triangles concept, we have:
Width of the rear window / Distance from driver's eyes to the rear window = Width of the mirror / Distance from driver's eyes to the mirror
1.1 m / 1.45 m = x / 0.45 m
Cross-multiplying, we get:
1.1 m * 0.45 m = 1.45 m * x
0.495 m² = 1.45 m * x
x = 0.495 m² / 1.45 m
x ≈ 0.341 m
Therefore, the minimum length for the rearview mirror to see the entire width of the rear window without moving her head is approximately 0.341 meters.
Part B:
To find the minimum height for the rearview mirror, we need to ensure that the driver can see the entire height of the rear window in the mirror without moving her head.
Given:
Height of the rear window = 0.30 m
Distance from the driver's eyes to the rear window = 1.45 m
Similarly, using similar triangles, we have:
Height of the rear window / Distance from driver's eyes to the rear window = Height of the mirror / Distance from driver's eyes to the mirror
0.30 m / 1.45 m = y / 0.45 m
Cross-multiplying, we get:
0.30 m * 0.45 m = 1.45 m * y
0.135 m² = 1.45 m * y
y = 0.135 m² / 1.45 m
y ≈ 0.093 m
Therefore, the minimum height for the rearview mirror to see the entire height of the rear window without moving her head is approximately 0.093 meters.