R = 3.0 m 

Answer:

Given, radius of curvature (R) of convex mirror = 3.0m
Distance of object (u) = -5.0m (since the object is in front of the mirror)

Finding the position of the image:

Using the mirror formula, 1/f = 1/u + 1/v, where f is the focal length and v is the distance of the image from the mirror, we can find the position of the image.

Since it is a convex mirror, the focal length (f) is negative. Using the mirror formula, we get:

1/f = 1/u + 1/v
=> 1/-f = 1/-5.0 + 1/v (substituting u = -5.0m)
=> -1/f = -1/5.0 + 1/v
=> -1/f + 1/5.0 = 1/v
=> v = 5.0f/(f - 5.0) (cross-multiplying)

Since the mirror formula only gives us the distance of the image from the mirror, we need to add the distance of the mirror from the object (-5.0m) to get the actual position of the image.

Therefore, the position of the image = -5.0 + v

Substituting the value of v, we get:

-5.0 + v = -2.5m

Therefore, the position of the image is -2.5m, which means the image is formed behind the mirror.

Determining the nature of the image:

The nature of the image formed by a convex mirror is always virtual, erect, and diminished. This is because the image formed by a convex mirror is always smaller than the object and is located behind the mirror.

Therefore, the nature of the image formed by the convex mirror in this case is virtual, erect, and diminished.

Finding the magnification:

The magnification of the image can be found using the formula:

magnification (m) = height of image (hi)/height of object (ho)

Since the image is virtual and erect, the height of the image is negative. Also, since the image is smaller than the object, the magnification is less than 1.

We can use the formula for magnification to find the height of the image. However, we first need to find the height of the object.

Since the distance of the object (u) is negative, we need to take the absolute value of u to find the actual distance of the object from the mirror.

Therefore, the distance of the object from the mirror = |u| = 5.0m

The height of the object is not given, so we cannot find the height of the image directly. However, we can use the fact that the magnification is less than 1 to estimate the height of the image.

Assuming the height of the object to be 1.0m, the height of the image can be found as follows:

magnification (m) = height of image (hi)/height of object (ho)
=> m = hi/1.0
=> hi = m

Substituting the value of m, we get:

hi = -0.4m (