Which process is engaged when a project team member makes a change to project budget with the project manager's approval?

'Manage Cost Plan' is not a recognized PMBOK process name - cost planning is handled by Plan Cost Management, which establishes policies and procedures for managing costs.

Estimate Costs is a planning process focused on approximating monetary resources needed for project activities - it does not handle changes to an existing budget.

Determine Budget is a planning process that aggregates estimated costs to establish the authorized cost baseline - it does not manage changes after the baseline is set.

Control Costs is the monitoring and controlling process responsible for tracking project spending, identifying variances from the cost baseline, and managing approved changes to that baseline. When a team member modifies the budget with the project manager's approval, this represents an authorized cost baseline change - precisely what Control Costs manages. It uses tools like Earned Value Management and variance analysis to maintain cost performance and process change requests through integrated change control.