8. A dose of antibiotics is given to a patient.
The amount of the antibiotic, \(x\) milligrams, in the patient's bloodstream \(t\) hours after the dose was given, is found to satisfy the equation
$$\log _ { 10 } x = 2.74 - 0.079 t$$
- Show that this equation can be written in the form
$$x = p q ^ { - t }$$
where \(p\) and \(q\) are constants to be found. Give the value of \(p\) to the nearest whole number and the value of \(q\) to 2 significant figures.
- With reference to the equation in part (a), interpret the value of the constant \(p\).
When a different dose of the antibiotic is given to another patient, the values of \(x\) and \(t\) satisfy the equation
$$x = 400 \times 1.4 ^ { - t }$$
- Use calculus to find, to 2 significant figures, the value of \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when \(t = 5\)