Real-Life Math

You are a cytotechnologist employed by a laboratory. Your job is to check specimens taken from patients for any abnormalities. Abnormal cells are one of the signs that a patient has cancer.

The work is demanding and often tiring. Besides needing a love of science, you need to be the type of person who pays attention to detail.

You also need to have good equipment. You work with powerful electronic microscopes that allow you to look closely at specimens. The two things you need to consider when using a microscope are the magnification and the total field of vision.

Magnification is determined by the eyepiece, which is the lens you peer into to see the specimen. Most cytotechnologists use an eyepiece with a magnification of 10.

In addition to magnification, eyepieces have another feature called field of view index (FIN). This is the total area you could possibly see when looking through any microscope using this eyepiece.

In reality, the total area that you will see when looking through a specific microscope is determined by the relationship between the eyepiece and the objective lens. The objective lens is the lens that is located just above the slide. It's the lenspiece that you tilt when you want to get a better look at the specimen.

The formula for determining the width of the field of vision is as follows:

FIN / magnification of the objective lens = diameter of the field of vision in millimeters

For example, if the eyepiece FIN is 20, and the objective lens magnification is 10, then the diameter of the circle that you see when you peer through the eyepiece is:

20 / 10 = 2 mm

Your boss calls a staff meeting and announces that he has made the decision to switch to eyepieces that have a FIN of 22. He believes that this will help reduce the workload. A higher FIN means you can examine the entire slide more quickly. It will change the distance you need to move the slide to examine a new section.

You need to figure out how far to move the slide every time you want to examine a new section. It's important for accuracy, so you don't go over certain areas twice and miss other sections.

To solve the problem, you'll need to use a bit of geometry. Imagine the largest square that could fit into the circle that you see when staring into the eyepiece. Now, imagine that the square is made up of 2 equal triangles.

The hypotenuse of each triangle is equal to the diameter of the circle. In the example given above, where the eyepiece FIN is 20, and the objective lens magnification is 10, then the hypotenuse of the triangle would be 2 mm.

Normally, you can't figure out the length of both sides of a triangle just from knowing the hypotenuse. However, since this is a square, the two sides of each triangle must be equal in length. This allows you to use the Pythagorean theorem (A² + B² = C²) to solve the problem. The length of any side will be the distance that the slide must be moved when you want to examine a new section.

• Want to see the solution?