Final answer:
To form an image one-half the size of the pencil, the pencil should be held approximately 0.0375 cm from the convex mirror.
Step-by-step explanation:
To find the distance at which a pencil should be held from a convex mirror to form an image one-half the size of the pencil, we need to make use of the mirror formula:
1/f = 1/v - 1/u
Where f is the focal length of the mirror, v is the distance of the image from the mirror, and u is the distance of the object from the mirror.
In this case, we are given that the image is half the size of the pencil, which means the magnification (m) is 1/2. With the magnification formula:
m = -v/u
We can substitute m as 1/2 and solve for v in terms of u:
1/2 = -v/u
v = -u/2
Now, substituting v = -u/2 in the mirror formula:
1/f = 1/-u/2 - 1/u
Simplifying the equation:
1/f = -1/u - 2/u
1/f = -3/u
u = -3/f
Given that the radius (R) of the convex mirror is equal to twice its focal length, R = 2f:
u = -3/R
Plugging in the value of the radius (R = 80 cm), we can calculate u:
u = -3/80
u ≈ -0.0375 cm
Therefore, the pencil should be held approximately 0.0375 cm from the convex mirror to form an image one-half the size of the pencil.