Given that
1625−4x2≡A25−2x+B25+2x \frac{1}{625 - 4x^2} \equiv \frac{A}{25 - 2x} + \frac{B}{25 + 2x} 625−4x21≡25−2xA+25+2xBfind the values of constants AAA and BBB.
A storage tank for a chemical reactor is in the shape of a cuboid with a base of length 2.5 metres and width 2 metres. The tank is 1.2 metres deep. Initially, the tank is empty. A chemical solution is pumped into the tank from the top at a constant rate of 0.25 cubic metres per minute.
Due to a fault in the drainage valve, the solution simultaneously leaks out from the bottom at a rate of 0.04d20.04d^20.04d2 cubic metres per minute, where ddd is the depth of the solution in metres.
At time ttt minutes after the pumping begins, the volume of solution in the tank is VVV cubic metres.
(b)(i) Show that
dVdt=625−4V22500 \frac{dV}{dt} = \frac{625 - 4V^2}{2500} dtdV=2500625−4V2Hence, find an expression for ttt in terms of VVV.
Determine how long it takes for the tank to be completely filled with the chemical solution. Give your answer to the nearest minute.
825 exam-style questions on OCR (MEI) A Level Maths 1.9 Calculus, covering 1.9.1 Gradient of a curve at a point, 1.9.2 Gradient as limit of chord gradient, 1.9.3 Derivative as gradient of tangent, 1.9.4 Sketch the gradient function, 1.9.5 Differentiate y = kx^n, 1.9.6 Second derivative as rate of change of gradient, 1.9.7 Stationary points: maxima and minima, 1.9.8 Increasing and decreasing functions, 1.9.9 Tangent and normal at a point, 1.9.10 Differentiate e^kx, a^kx and ln x (A-level only), 1.9.11 Differentiate trigonometric functions (A-level only), 1.9.12 Product rule (A-level only), 1.9.13 Quotient rule (A-level only), 1.9.14 Chain rule (A-level only), 1.9.15 Rates of change with the chain rule (A-level only), 1.9.16 Implicit differentiation (A-level only), 1.9.17 Concavity and the second derivative (A-level only), 1.9.18 Points of inflection (A-level only), 1.9.19 Integration as reverse of differentiation, 1.9.20 Integrate kx^n, 1.9.21 Constant of integration, 1.9.22 Indefinite and definite integrals, 1.9.23 Area between a graph and the x-axis, 1.9.24 Integrate e^kx, 1/x, sin kx, cos kx (A-level only), 1.9.25 Integration as the limit of a sum (A-level only), 1.9.26 Area between two curves (A-level only), 1.9.27 Integration by substitution (reverse chain rule) (A-level only), 1.9.28 Integration by substitution (other cases) (A-level only), 1.9.29 Integration by parts (A-level only), 1.9.30 Integration using partial fractions (A-level only), 1.9.31 Formulate first order differential equations (A-level only), 1.9.32 Solve first order differential equations (A-level only), and 1.9.33 Interpret solutions of differential equations (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.