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A cost-aware Bayesian sequential decision-making strategy for domain search and object classification using a limited-range sensor is presented. On one hand, it is risky to allocate all available sensing resources at a single location while ignoring other regions. On the other hand, the sensor may miss-detect or miss-classify a critical object with insufficient observations. Therefore, we develop a decision-making strategy that balances the tolerable risks and the desired decision precision under limited resources.