You need to learn about pixels vs DPI and how it all works. Frankly, it confuses too many people. It's 101 stuff. It's stuff you should know before entering the field or learn within your first few days.
1. DPI doesn't exist in the digital world. It's only pixels. Also a lesser discussed part of digital artwork is quality. I can easily provide stupid high res artwork but it might look like absolute garbage.
2. DPI in the digital world is basically a scaling factor stored in the meta data. It's also used as information provided if you want to turn something that is digital to something physical (printing). To prove this, open up Photoshop and make a new document that size. Now change the DPI from 100 to 300. Notice how the file size doesn't change. The only thing that changes is the canvas size. Now take either of those files and bring it into Illustrator and reduce the size 1/3 or increase it 3x. Notice how the DPI corresponds? Like I said, it's a scaling factor.
3. In this example, your ratios are different, but ignoring that, your 5000x3000 image when scaled to 100" high, would be 30dpi. This will likely appear pretty pixelated and crappy unless you res-up the image. That low DPI figure is essentially telling you it's a low resolution image.
4. DPI when printing is a completely different subject. It boils down to the quality of the print. High DPI = more detailed print as it's putting more droplets of ink per inch. You could also say it's higher ink density, but that can be semi controversial.
So big picture you need to look at two different things. The first is the digital quality. Is the image a good enough quality image for the size I want to print? The next question is how high of a quality print is necessary for the application.
Either way, you should really know the basics if you want to be in the digital art / printing field. This is 101 stuff.