A-Level Mathematics October/November 2024 Q7(a): Given that θ = 1.2, find the area of the metal plate. Give your answer correct to 3 sig…

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  The total area of the metal plate is the sum of the area of a major sector, the area of two triangles, and the area of two semicircles. All angles must be in radians.

  1. Area of the Major Sector The angle of the minor sector removed is $2\theta = 2 \times 1.2 = 2.4$ radians. The angle of the major sector is the rest of the circle. Angle $\alpha = 2\pi - 2.4$ radians. Radius $r = 20$. Area of major sector = $\frac{1}{2} r^2 \alpha = \frac{1}{2} (20)^2 (2\pi - 2.4)$ $= 200(2\pi - 2.4) = 776.637...$

  2. Area of the Two Triangles The area of two identical triangles with sides $20$ and an included angle of $\theta = 1.2$ radians is calculated using the formula $2 \times \frac{1}{2}ab\sin C$. Area of two triangles = $2 \times \frac{1}{2} (20)(20) \sin(1.2) = 400 \sin(1.2)$ $= 372.813...$

  3. Area of the Two Semicircles The radius of each small semicircle, $r_{small}$, is found by considering a right-angled triangle formed by bisecting the angle $\theta$. $r_{small} = 20 \sin(\frac{\theta}{2}) = 20 \sin(\frac{1.2}{2}) = 20 \sin(0.6)$. The area of two semicircles is equal to the area of one full circle with this radius. Area of two semicircles = $\pi r_{small}^2 = \pi (20 \sin(0.6))^2 = 400\pi \sin^2(0.6)$ $= 400.643...$

  4. Total Area Total Area = (Area of major sector) + (Area of two triangles) + (Area of two semicircles) Total Area = $776.637... + 372.813... + 400.643...$ Total Area = $1550.093...$

  Giving the answer correct to 3 significant figures: Area = 1550

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  How the marks are awarded

  • M1 — For correctly applying the sector area formula $\frac{1}{2}r^2\alpha$ to the major sector, using the correct radius $r=20$ and angle $\alpha = 2\pi - 2.4$. The calculation $\frac{1}{2} (20)^2 (2\pi - 2.4)$ earns this mark.
  • M1 — For finding the correct expression for the radius of the small semicircles, $r_{small} = 20 \sin(0.6)$, by using trigonometry with half the angle $\theta$.
  • DM1 — For calculating the area of the two semicircles (a full circle) using the radius found previously. The calculation $\pi (20 \sin(0.6))^2$ earns this dependent method mark.
  • M1 — For finding the area of the two triangles using the formula $2 \times \frac{1}{2}ab\sin C$, with the correct values: $2 \times \frac{1}{2} (20)(20) \sin(1.2)$.
  • A1 — For obtaining the final correct answer of 1550, after correctly summing all component areas and rounding to 3 significant figures. An unrounded answer of 1550.09... would also be sufficient for this mark.

  Common mistakes

  • Using a calculator in Degree mode instead of Radian mode for trigonometric functions, leading to incorrect values for $\sin(1.2)$ and $\sin(0.6)$.
  • Incorrectly calculating the area of the major sector, for example by using the angle $2.4$ or $1.2$ instead of the reflex angle $2\pi - 2.4$.
  • Errors in calculating the area of the two semicircles, such as only calculating the area of one semicircle (e.g., $\frac{1}{2}\pi r^2$) or forgetting to square the radius.
  • Confusing the angles, for example using $\theta=1.2$ to find the radius of the small semicircles ($20 \sin(1.2)$) instead of $\theta/2 = 0.6$.

  Examiner tip: When faced with a composite shape, methodically break it down into simpler, standard geometric figures for which you know the area formulas.

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