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1.9 Calculus

1.9 Calculus

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Question 367

A precision-engineered storage vault for high-density alloys is in the shape of a cuboid with a rectangular base of width xxx cm and length 3x3x3x cm. The height of the vault is hhh cm.

The volume of the vault is fixed at 1350 cm31350\text{ cm}^31350 cm3.

a.

Show that the surface area of the vault, S cm2S\text{ cm}^2S cm2, is given by

S=6x2+3600x S = 6x^2 + \frac{3600}{x} S=6x2+x3600​
[4]
b.

Find dSdx\frac{dS}{dx}dxdS​.

[2]
c.

Hence find the value of xxx for which SSS is stationary, giving your answer to 3 significant figures.

[3]
d.

Find d2Sdx2\frac{d^2S}{dx^2}dx2d2S​ and hence show that the value of xxx found in part (c) gives the minimum value of SSS.

[3]
e.

Hence find the minimum surface area of the vault, giving your answer to 1 decimal place.

[2]
Markscheme

1.9 Calculus Questions

  1. A Level
  2. /Maths
  3. /1.9 Calculus

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.

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