Spreading stain or growing patch area

A question is this type if and only if it models a growing area (oil patch, ink stain) as a function of time with given constants determined from initial conditions, then requires finding the rate of change of area at a specific time.

2 questions · Moderate -0.8

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Edexcel C1 Q8
11 marks Moderate -0.8
8. Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A \mathrm {~cm} ^ { 2 }\), after \(t\) seconds is given by $$A = ( p + q t ) ^ { 2 } ,$$ where \(p\) and \(q\) are positive constants.
Given that when \(t = 0 , A = 4\) and that when \(t = 5 , A = 9\),
  1. find the value of \(p\) and show that \(q = \frac { 1 } { 5 }\),
  2. find \(\frac { \mathrm { d } A } { \mathrm {~d} t }\) in terms of \(t\),
  3. find the rate at which the area of the stain is increasing when \(t = 15\).
OCR C1 Q6
10 marks Moderate -0.8
Some ink is poured onto a piece of cloth forming a stain that then spreads. The area of the stain, \(A\) cm\(^2\), after \(t\) seconds is given by $$A = (p + qt)^2,$$ where \(p\) and \(q\) are positive constants. Given that when \(t = 0\), \(A = 4\) and that when \(t = 5\), \(A = 9\),
  1. find the value of \(p\) and show that \(q = \frac{1}{5}\), [5]
  2. find \(\frac{\mathrm{d}A}{\mathrm{d}t}\) in terms of \(t\), [3]
  3. find the rate at which the area of the stain is increasing when \(t = 15\). [2]