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1.9 Calculus

1.9 Calculus

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Question 495

A landscape architect is designing a decorative concrete partition for a park. The height of the partition, HHH metres, at a horizontal distance ddd metres from a central pillar, is modeled by the equation

H=24d+9,d≥−2.25 H = 2\sqrt{4d + 9}, \quad d \ge -2.25 H=24d+9​,d≥−2.25

A straight support beam, represented by line lll, is attached to the partition at the point P(4,10)P(4, 10)P(4,10). The beam is perpendicular to the curve of the partition at PPP.

a.

Use calculus to show that an equation for the line lll is

5d+4H−60=0 5d + 4H - 60 = 0 5d+4H−60=0
[4]
b.

The region RRR is the cross-sectional area of the partition's side, bounded by the curve, the ground (the line H=0H = 0H=0), and the support beam lll.

Use algebraic integration to find the exact area of RRR.

[6]
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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