A convex lens is needed to produce an image that is 0.70 times the size of the object and located 28 cm from the lens on the other side. What focal length should be specified?

To find the focal length of the convex lens that will produce an image 0.70 times the size of the object and located 28 cm from the lens, we can use the lens formula:

1/f = 1/v - 1/u

Where:
- f is the focal length of the lens
- v is the image distance from the lens
- u is the object distance from the lens

Given:
v = 28 cm
The size of the image is 0.70 times the size of the object, which means the magnification (m) is 0.70.

Now, let's tackle this step-by-step:

Step 1: Finding the object distance (u)
Since the question does not explicitly state the object distance (u), we can use the magnification formula:

m = -v/u

Given:
m = 0.70
v = 28 cm

Rearranging the magnification formula, we can solve for u:

0.70 = -28/u

Multiply both sides by u:
0.70u = -28

Divide both sides by 0.70:
u = -40 cm

Note: The negative sign indicates that the object is on the same side as the incident light.

Step 2: Plugging in the values to find the focal length (f)
Now that we have both the object distance (u) and the image distance (v), we can substitute these values into the lens formula:

1/f = 1/v - 1/u

Substituting the given values:
1/f = 1/28 - 1/(-40)

Simplifying the equation:
1/f = 1/28 + 1/40

To add these fractions, we need to find a common denominator:
1/f = (40 + 28) / (28 * 40)
1/f = 68 / 1120

Now, taking the reciprocal of both sides to isolate f:
f = 1120 / 68

Simplifying further:
f ≈ 16.47 cm

Therefore, the focal length that should be specified for the convex lens is approximately 16.47 cm.