A certain digital camera having a lens with focal length 7.5 cm focuses on an object 1.85 m tall that is 4.25 m from the lens. (a) How far must the lens be from the photocells? (b) How tall is the image on the photocells? Is it erect or inverted? (c) A SLR digital camera often has pixels measuring 8.0 um by 8.0 um. How many such pixels does the height of this image cover?
I have part a and b, but I can't figure out c! help please!
To find the answer for part (c), we need to calculate the number of pixels that the height of the image covers.
1. First, we need to find the magnification of the image formed by the lens. We can use the thin lens equation:
1/f = 1/v - 1/u
where f is the focal length of the lens, v is the image distance from the lens, and u is the object distance from the lens.
Given:
- The focal length f = 7.5 cm
- The object distance u = 4.25 m (convert to cm by multiplying by 100)
- The image distance v is what we need to find
Rearranging the equation, we have:
1/v = 1/f + 1/u
Plugging in the values, we get:
1/v = 1/7.5 + 1/425
Now, solving for v:
v = 6.994 cm
2. Next, we need to find the height of the image formed by the lens. We can use the magnification equation:
magnification (m) = -v/u
Plugging in the values, we have:
m = -6.994/425
Simplifying, we get:
m ≈ -0.0164 (rounded to four decimal places)
Since the magnification is negative, it means the image is inverted.
3. Finally, we can calculate the height of the image on the photocells by multiplying the magnification by the height of the object. Given that the object height is 1.85 m, we have:
Image height = magnification * object height
Image height = -0.0164 * 1.85
Calculating this, we get:
Image height ≈ -0.030 m
However, the negative sign only indicates the inversion of the image and doesn't apply to the absolute value of the image height. So, the height of the image is approximately 0.030 m.
4. Now, to calculate the number of pixels that the height of the image covers, we divide the image height by the height of a single pixel:
Number of pixels = image height / pixel height
Number of pixels = 0.030 m / 8.0 μm (convert to meters by dividing by 1 million)
Calculating this, we get:
Number of pixels ≈ 3750 pixels
Therefore, the height of the image covers approximately 3750 pixels.