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1.9 Calculus

1.9 Calculus

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Question 482

A landscape architect is designing a feature for a park. The cross-sectional elevation of the terrain is modelled by a curve CCC with equation

y=x12(15−x),x≥0 y = x^{\frac{1}{2}}(15 - x), \quad x \ge 0 y=x21​(15−x),x≥0

where yyy is the height in decametres and xxx is the horizontal distance in decametres. The curve CCC has a stationary point at PPP.

a.

Using calculus, determine the xxx-coordinate of PPP.

[3]
b.

The region R1R_1R1​, representing a hill section, is bounded by the curve CCC and the xxx-axis.

The region R2R_2R2​, representing a valley section, is bounded by the curve CCC, the xxx-axis, and the vertical line x=kx = kx=k, where k>15k > 15k>15 is a constant.

Given that the area of the hill R1R_1R1​ is equal to the area of the valley R2R_2R2​, find, using calculus, the exact value of kkk.

[5]
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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