First off all, I think that it's imperative that everyone understand what an "alt tag" (alt being short for "alternative") is in the first place. The accessibility portion of the Penn State site describes it well: "a clear text alternative of the image for screen reader users."
You may or may not know that individuals who are using a screen reader are living with a visual impairment, or otherwise handicapped. This issue prevents them from seeing the information on a web page. The alt tag's foremost purpose is to help handicapped understand what the image is about. Secondarily, alt tags can help search engine crawlers index an image correctly.
The previous answer provided by Kristian is certainly more efficient, however I caution you to not use any type of programmatic naming convention.
This is bigger than SEO people. I work closely with a company who built a website for Dallas Arboretum, and not long after they launched, they were up against some sort of discrimination lawsuit because one or more visually impaired users did not receive accurate information through the screen reader. They won the lawsuit.
When you write alt tags, be sure to do so with the user in mind. Do not focus purely on SEO nor search engines. Yes, it is an SEO ranking factor - but the factor has more to do with whether or not you wrote the alt tag correctly vs. whether you are stuffing keywords in the alt tag.
According to moz.com, an alt tag should be sufficiently descriptive, but should not contain any spammy attempts at keyword stuffing.
Here is the alt tag test: If you can close your eyes, and have someone read the alt tag to you, and imagine a reasonably accurate version of the image - then you have passed the test.
Further reading is available here: https://moz.com/learn/seo/alt-text.
No website omission is worthy of a lawsuit. Take the time to do it right!