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.
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.