A mirror at an amusement park shows an upright image of any person who stands 1.9 m in front of it. If the image is three times the person’s height, what is the radius of curvature of the mirror?

To solve this problem, we can use the mirror equation which is 1/f = 1/v + 1/u.

Here, u = object distance = -1.9 m (negative because the object is on the same side as the light being reflected) v = image distance (which we need to find first) f = focal length R = radius of curvature = 2f

Given that the image is upright and three times the height of the object, we know that the mirror is concave and the image is virtual. For virtual images, v is positive.

Since the image is three times the height of the object, the magnification m = -v/u = 3 (negative because the image is upright).

We can solve for v using the magnification equation: v = -mu = -3(-1.9 m) = 5.7 m.

Now we can find the focal length using the mirror equation: 1/f = 1/v + 1/u = 1/5.7 m + 1/(-1.9 m) = 0.175 m^-1 - 0.526 m^-1 = -0.351 m^-1.

So, f = -1/0.351 ≈ -2.85 m. The negative sign indicates that the focus is on the same side as the object, which is consistent with a concave mirror.

Finally, we find the radius of curvature R = 2f = 2(-2.85 m) = -5.7 m. The negative sign indicates that the center of curvature is on the same side as the object, which is consistent with a concave mirror.