Worldwide, there are approximately 10 million new cases of dementia reported each year. Due to cognitive deficits, persons living with dementia (PLWDs) have lower independence and decreased quality of life at home. Physically-embodied socially assistive robots (SARs) have the potential to engage PLWDs and assist them in their home-based activities. However, limited attention has been given to the development and assessment of SAR to interact with PLWDs within the context of home-based activities. In this case study, we designed a list of activities, including entertainment, playing games, and ten activities of daily living in a simulated home setting. We invited one older adult (female) without cognitive impairment and two (male) with dementia to perform these activities with the guidance of a humanoid SAR. The results of our observational study demonstrate that the multimodal interaction in SARs, such as verbal communication, eye contact, and gestures, can greatly facilitate the adoption and use of SARs by PLWDs. We found that PLWDs attributed human-like qualities to the robot when performing home-based activities. We also identified main challenges in SAR to assist PLWDs with these tasks.