The spherical object within a cell which controls its activity is the nucleus.

Here's why:

* Control Center: The nucleus is often called the "brain" of the cell because it contains the cell's genetic material (DNA). This DNA holds the instructions for building and maintaining the cell, as well as for carrying out all its functions.

* Spherical Shape: While not always perfectly spherical, the nucleus is generally a round or oval-shaped structure.

* Important Functions: The nucleus is responsible for:

* DNA Replication: Copying the cell's DNA before cell division.

* Transcription: Creating RNA molecules from DNA, which carry instructions to the ribosomes for protein synthesis.

* Cell Division Control: Regulating the timing and process of cell division.

Let me know if you have any other questions!