Kim says that she was less than 10 ft away from the president when she took a picture of him wit

Kim says that she was less than 10 ft away from the...

Kim says that she was less than $10 \mathrm{ft}$ away from the president when she took a picture of him with her $50-\mathrm{mm}$ focal length camera lens. The picture shows the upper half of the president's body (or $3.0 \mathrm{ft}$ of his total height). On the negative of the film, this part of his body is $18 \mathrm{~mm}$ high. How close was $\mathrm{Kim}$ to the president when she took the picture?

Key Concepts

Similar Triangles

The concept of similar triangles is used in optical imaging to establish proportional relationships between the actual object and its image. By considering the triangles formed by the light rays from the object and those from the image, one can derive the scaling factor between object size and image size, which is essential for distance and magnification calculations.

Thin Lens Equation

The thin lens equation (1/f = 1/do + 1/di) is used to relate the focal length of a lens to the distances of the object and the resultant image from the lens. This fundamental relationship in optics helps in analyzing how a camera or similar optical system captures and projects an image of a subject.

Magnification

Magnification in optics and photography is the ratio of the image size to the actual object size. It allows one to understand how the dimensions of a captured image relate to the subject, and it directly ties into the distances involved as defined by the lens's properties and the geometry of the imaging system.

Focal Length

Focal length is a key property of a lens that signifies the distance over which parallel rays of light are brought to a focus. It influences the field of view and the scale at which objects are rendered in the image, making it critical in determining how much of the scene is captured and the apparent size of objects.

Transcript

00:01 So the first thing we're going to do is draw a little diagram of our system.

00:04 We have a converging lens system.

00:07 So we're going to align it on this axis.

00:09 Here's the lens.

00:14 And i'm going to put the object over here.

00:18 Object.

00:20 And the image will be somewhere over there.

00:24 So we're going to put the two focal links on either side of the lens, the same distance f away on either side.

00:32 And i'm going to draw some light rays, going into my lens here and then going through the focal length and another light ray going through the focal length here and then going into my lens and here is what going to be my object my image my projected image and there's this invisible midline here and now i'm going to label some of the distances on this diagram so i have the height of my original object each and i have the height of my created image, h prime.

01:12 It's a real image here.

01:14 I have the distance from my object to my lens, which is going to be p, and the distance from my lens to my image, which is going to be q.

01:24 And the last thing i'm going to label here is this little segment in between the end of f and the end of q.

01:31 So the difference between the total distance to the image, q, and the focal length here, l.

01:38 And that's actually going to be important.

01:41 Now let's write down all the information that we have.

01:44 So we know that f is 50 millimeters.

01:49 I'm going to convert everything to millimeters for ease of use.

01:55 H, our height of our object, is 3 .0 feet, which is the same thing as 914 .4 millimeters.

02:08 Make sure you always have the same units for you.

02:12 Various measurements and we know that h prime the height of the image is 18 millimeters now i'm going to solve for l first for this little segment because i have this property here where this angle and this angle are actually the same angle they have to be their opposite angles they two straight lines intersect form them and then we have a right angle here and a right angle here.

02:49 So that means that these two are similar triangles and oh sorry the right angle goes over here.

02:57 And in similar triangles we know that the side lengths are proportional.

03:02 So when one side length grows by a certain ratio, the other one also does...

Please give Ace some feedback