An industrial conical mold is being manufactured with a fixed vertical height of 12 cm12\text{ cm}12 cm. The mold has a base radius of r cmr\text{ cm}r cm and a slant height of l cml\text{ cm}l cm.
Determine an expression for l in terms of rl\text{ in terms of }rl in terms of r.
During a specific expansion process, the radius of the base is increasing at a constant rate of 1.5 cm per second1.5\text{ cm per second}1.5 cm per second.
Calculate the rate at which the total surface area of the mold is changing at the instant when the radius is 5 cm5\text{ cm}5 cm. Give your answer in cm2 per second\text{cm}^2\text{ per second}cm2 per second correct to one decimal place.
[The total surface area, SSS, of a cone is given by the formula S=πr2+πrlS = \pi r^2 + \pi rlS=πr2+πrl]
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.