Higher June 2022 Paper 2 Q20
20 The diagram shows four identical circles drawn inside a square.
Diagram NOT accurately drawn
Each circle touches two other circles and two sides of the square.
The region inside the square that is outside the circles, shown shaded in the diagram, has a total area of 40 cm2
Work out the perimeter of the square.
Give your answer correct to 3 significant figures.
(4)
| Scheme | Marks |
|---|---|
eg \(2d \times 2d - 4 \times \pi \times \left(\dfrac{1}{2}d\right)^2\;(= 40)\) oe or \(4r \times 4r - 4 \times \pi \times r^2\;(= 40)\) oe or \(x^2 - 4\pi\left(\dfrac{1}{4}x\right)^2\;(= 40)\) oe or \(w^2 - \pi\left(\dfrac{1}{2}w\right)^2\;(= 10)\) oe | M1 |
\(d = \sqrt{\dfrac{40}{4 - \pi}}\) (= 6.826...) or \(2d = \sqrt{\dfrac{160}{4 - \pi}}\) (= 13.652…) oe \(r = \sqrt{\dfrac{40}{16 - 4\pi}}\) (3.413...) or \(4r = \sqrt{\dfrac{640}{16 - 4\pi}}\) (=13.652...) oe \(x = \sqrt{\dfrac{40}{1 - 0.25\pi}}\) (13.652…) or \(w = \sqrt{\dfrac{10}{1 - 0.25\pi}}\) (= 6.826…)oe | M1 |
| (perimeter =) 8 × “6.826…” (8 × diameter(or side of small square when divided)) or 16 × “3.413...” (16 × radius) oe or 4 × “13.652…”(4 × side of square) | M1ft |
| 54.6 | A1 |
| (4) | |
| (4 marks) |
Notes
M1: oe a correct expression or a correct equation for the shaded area (must be in one unknown only)
where
\(d\) = diameter
\(r\) = radius
\(x\) = side of large square
\(w\) = side of square when shape divided into 4
M1: oe a correct expression for \(d\) or \(2d\) or \(r\) or \(4r\) or \(x\) or \(w\)
M1ft: dep on first M1 For substituting values into a calculation for the perimeter
use of their \(r\), \(d\), \(x\), \(w\)
A1: 54.4 - 54.7