A beam of light travels through a container of murky liquid. The rate of change of the light's intensity, III lux, with respect to the depth, xxx metres, is modeled by the equation:
R=−0.75I2 R = -0.75 I^2 R=−0.75I2where RRR is the rate of change dIdx\frac{dI}{dx}dxdI. The intensity of the light as it enters the liquid at the surface (where x=0x = 0x=0) is 444 lux.
By first forming a suitable differential equation, show that
I=43x+1 I = \frac{4}{3x + 1} I=3x+14Determine the rate of change of the light's intensity with respect to depth when x=1x = 1x=1.
Practise AQA A Level Maths 1.11 H: Integration with exam-style questions for A Level Maths. 436 questions covering 1.11.1 Fundamental Theorem of Calculus, 1.11.2 Integrating standard functions, 1.11.3 Definite integrals and areas, 1.11.4 Integration as the limit of a sum (A-level only), 1.11.5 Integration by substitution and by parts (A-level only), 1.11.6 Integration using partial fractions (A-level only), 1.11.7 Differential equations with separable variables (A-level only), and 1.11.8 Interpreting solutions of differential equations (A-level only), matched to the AQA A Level Maths (7357) specification and written in Paper 1, Paper 2 and Paper 3 style. Every question includes a full worked solution and mark scheme, so you can see where marks are awarded rather than just whether you got the answer right.
