The rear window in a car is approximately a rectangle, 2.0 m wide and 0.37 m high. The inside rearview mirror is 0.52 m from the driver's eyes, and 1.38 m from the rear window.

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?

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?

To solve these problems, we need to consider the geometry of the situation. Let's tackle each question separately:

(a) The driver needs to be able to see the entire width and height of the rear window in the mirror without moving her head. This means that the mirror should be positioned such that the lines of sight from the driver to both the top and bottom corners of the rear window are within the field of view of the mirror.

To find the minimum length for the rearview mirror, we can use similar triangles. Let's call the length of the mirror L. Using similar triangles, we can set up the following equation:

L/0.52 = 2.0/1.38

Cross-multiplying, we get:

L = (0.52 * 2.0) / 1.38

Simplifying, we find:

L ≈ 0.754 m

So, the minimum length for the rearview mirror should be approximately 0.754 meters.

(b) Similarly, the driver needs to be able to see the entire width and height of the rear window in the mirror without moving her head. This time, we need to find the minimum height for the rearview mirror.

Again, we can use similar triangles. Let's call the height of the mirror H. Using similar triangles, we can set up the following equation:

H/0.52 = 0.37/1.38

Cross-multiplying, we get:

H = (0.52 * 0.37) / 1.38

Simplifying, we find:

H ≈ 0.14 m

So, the minimum height for the rearview mirror should be approximately 0.14 meters.