Gabrielle would like to purchase a new which would be used to connect the company’s lan and the internet.

It would be modem. The modem connects to the internet through their ISP's line via a (usually) coaxial cable. The modem then transfers information to a router, which is used to transfer the information coming from the internet to the rest of the devices on the network, wired, wireless, or both.

Nowadays, though, modems and routers tend to come in a two-in-one, whether it's first-party provided by your ISP, or third-party where you've purchased one. This two-in-one is often just referred to as the router, so this could well be the answer to your question as well.

The reason I stated the above answer, is because a regular router on its own is unable to connect to the internet, it must be receiving internet from somewhere. The two-in-one however would have the coaxial cable to connect to the internet, as well as your usual Ethernet ports and wireless capability.

That is called a  (wimax)           because it  uses wireless transmissions for data communications as much as 35 miles apart.