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I would recommend using a module for sending mail, even for cases as simple as this. My most recent experience is with MIME::Lite, which is quite good.
You should assert that the $name field does not contain any newlines or special characters, otherwise spammers will use that field to embed another mail. A script I wrote early in my career had that problem, and it wasn't long before it was exploited.
I also recommend sending mail via SMTP instead of through a local executable, as it will allow you more flexibility and portability. The sendmail implementation of the sendmail command could have different options than the postfix implementation, or even other sendmail versions.