A population of bacteria in a laboratory culture is being monitored. The number of bacteria, BBB, in the culture, t t\,t hours after the initial observation, is modelled by the equation
B=400ekt7+ekt B = \frac{400e^{kt}}{7 + e^{kt}} B=7+ekt400ektwhere k k\,k is a constant.
Find the number of bacteria in the culture at the start of the study.
Given that there are 160 bacteria in the culture after 5 hours,
show that k=15ln(143)\displaystyle k = \frac{1}{5}\ln\left(\frac{14}{3}\right)k=51ln(314).
Given also that, when t=Tt = Tt=T, the number of bacteria is increasing at a rate of 25 per hour,
find the possible values of TTT, giving your answers to one decimal place.
425 exam-style questions on OCR A Level Maths 1.7 Differentiation, covering 1.7.1 Derivative as gradient of the tangent, 1.7.2 Gradient of the tangent at a point, 1.7.3 Sketching the gradient function, 1.7.4 Second derivatives, 1.7.5 Second derivative as rate of change of gradient, 1.7.6 Convex, concave and points of inflection (A-level only), 1.7.7 Differentiation from first principles for powers of x, 1.7.8 Differentiation from first principles for sin x and cos x (A-level only), 1.7.9 Differentiating x^n, 1.7.10 Differentiating e^(kx) and a^(kx) (A-level only), 1.7.11 Differentiating trigonometric functions (A-level only), 1.7.12 Derivative of ln x (A-level only), 1.7.13 Tangents and normals, 1.7.14 Stationary points, 1.7.15 Increasing and decreasing functions, 1.7.16 Points of inflection (A-level only), 1.7.17 Product and quotient rules (A-level only), 1.7.18 Chain rule (A-level only), 1.7.19 Parametric and implicit differentiation (A-level only), 1.7.20 Constructing differential equations (A-level only), and 1.7 Differentiation. Each one has a worked solution and a mark scheme showing where the marks go.