$($a$)$ If $25\%$ of the $89\mathrm{????}\mathrm{????}$ remains in the body after $90$ days, write a function of the form $\mathrm{????}(\mathrm{????})={\mathrm{????}}_0\mathrm{????}-\mathrm{????}\mathrm{????}$ to model the amount $\mathrm{????}(\mathrm{????})$ of radioactivity in the body $\mathrm{????}$ days after the initial dose. Round the value of $\mathrm{????}$ to five decimal places.