The mass MMM grams of a crystal growing in a chemical bath at time ttt hours is modelled by the differential equation
dMdt=10tMe2t,M≥0,t≥0 \frac{\text{d}M}{\text{d}t} = \frac{10t \sqrt{M}}{\text{e}^{2t}}, \quad M \ge 0, \quad t \ge 0 dtdM=e2t10tM,M≥0,t≥0Given that the initial mass of the crystal is 444 g, solve this differential equation to find an expression for M12M^{\frac{1}{2}}M21 in the form M12=f(t)M^{\frac{1}{2}} = f(t)M21=f(t).
Hence determine the equation of the horizontal asymptote to the curve with equation M12=f(t)M^{\frac{1}{2}} = f(t)M21=f(t).
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.