A household is laying a new patio, shaped as a rectangle measuring \(6\) m by \(4\) m with a quarter-circle flower bed of radius \(2\) m cut from one corner...
Question 1 Report
A household is laying a new patio, shaped as a rectangle measuring \(6\) m by \(4\) m with a quarter-circle flower bed of radius \(2\) m cut from one corner, shown below.
- Find the area of the quarter-circle flower bed, in terms of \(\pi\). (2)
- Hence find the area of paving needed, giving your answer correct to 3 significant figures. (2)
- Paving costs £38 per m\(^2\). Work out the total cost of the paving, correct to the nearest penny. (2)
- The household has budgeted £750 for paving. Determine, with a reason, whether this is enough, and if not, find by how much they are short. (1)
Answer Details
The paved area is the full rectangle with the quarter-circle flower bed removed, so it is found by subtracting the two areas.
(a) A quarter circle of radius 2 m has area \( \dfrac{1}{4} \times \pi \times 2^2 = \dfrac{1}{4} \times 4\pi = \pi \ \text{m}^2 \), left as a multiple of \( \pi \) as requested [2 marks].
(b) The full rectangle has area \( 6 \times 4 = 24 \ \text{m}^2 \). Removing the flower bed gives the paved area \( 24 - \pi = 20.858\ldots \), which rounds to \( 20.9 \ \text{m}^2 \) to 3 significant figures [2 marks].
(c) Using the unrounded paved area, the cost is \( 38 \times (24 - \pi) = 912 - 38\pi = 792.619\ldots \), which rounds to \( £792.62 \) to the nearest penny [2 marks].
(d) Since \( £792.62 \) is more than the \( £750 \) budget, the budget is not enough. The shortfall is \( £792.62 - £750 = £42.62 \) [1 mark].
Working with the exact multiple of \( \pi \) throughout, rather than a rounded decimal from part (b), is what keeps the cost in part (c) accurate to the nearest penny; rounding too early can shift the final pence figure.
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