Yes, there is. Here's the PowerShell and AMO (Analysis Management Objects) way. 😉
[System.Reflection.Assembly]::LoadWithPartialName("Microsoft.AnalysisServices") | Out-Null;
[Microsoft.AnalysisServices.Server] $srv = New-Object Microsoft.AnalysisServices.Server;
[string]$ServerName = "Server\Instance";
$srv.Connect($ServerName);
$BackupDestination = $srv.ServerProperties.Item("BackupDir").Value;
$BackupDestination;