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.
438 exam-style questions on OCR A Level Maths 1.8 Integration, covering 1.8.1 Fundamental theorem of calculus (A-level only), 1.8.2 Integrating x^n, 1.8.3 Integrating standard functions (A-level only), 1.8.4 Evaluating definite integrals, 1.8.5 Area between a curve and the x-axis, 1.8.6 Area between two curves, 1.8.7 Integration as the limit of a sum (A-level only), 1.8.8 Integration by substitution (A-level only), 1.8.9 Integration by parts (A-level only), 1.8.10 Use of partial fractions in integration (A-level only), 1.8.11 Differential equations with separable variables (A-level only), 1.8.12 Interpreting the solution of a differential equation (A-level only), and 1.8 Integration. Each one has a worked solution and a mark scheme showing where the marks go.