4. The growth of a weed on the surface of a pond is being studied. The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), is modelled by the equation $$A = \frac { 80 p \mathrm { e } ^ { 0.15 t } } { p \mathrm { e } ^ { 0.15 t } + 4 }$$ where \(p\) is a positive constant and \(t\) is the number of days after the start of the study.
Given that

• \(30 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed at the start of the study
• \(50 \mathrm {~m} ^ { 2 }\) of the surface of the pond was covered by the weed \(T\) days after the start of the study
  1. show that \(p = 2.4\)
  2. find the value of \(T\), giving your answer to one decimal place.
     (Solutions relying entirely on graphical or numerical methods are not acceptable.)

The weed grows until it covers the surface of the pond.

Find, according to the model, the maximum possible surface area of the pond.