5 (a) (i) Find the area of this flowerbed - AQA - A-Level Maths Mechanics - Question 5 - 2021 - Paper 3

First, we need to calculate the perimeter of the flowerbed, which consists of the arc length and the two straight edges:

1. Calculate the arc length using the formula: $P = r \theta$ $P = 5 \times 0.7 = 3.5 m$

2. The total perimeter is:

$Total \ perimeter = Arc \ length + 2 \times r \ = 3.5 + 2 \times 5 \ = 3.5 + 10 = 13.5 m$

3. The cost of the edging strip is then calculated by multiplying the perimeter by the cost per metre (£1.80): $Cost = Total \ perimeter \times Cost \ per \ metre = 13.5 \times 1.80 = 24.30 \pound$

Hence, the cost of the edging strip required for this flowerbed is £24.30.

The area of the flowerbed is given as 20 m². We also know the perimeter is:

$P = 20 + 2r$

For a radius of r, the arc length is:

$Arc \ length = r \theta = r \times \text{(angle in radians)} = 3.5 m$

Thus the complete equation for C is:

$C = 1.80 \times (20 + 2r) \implies C = \frac{18}{5} (20 + r)$

This completes the derivation.

To find the minimum cost, we need to differentiate C with respect to r:

1. We have: $C = \frac{18}{5} (20 + r)$
2. The derivative is: $\frac{dC}{dr} = \frac{72}{5} \cdot \frac{1}{r^2}$
3. Setting this equal to zero to find the critical points: $0 = \frac{72}{r^2} \Rightarrow r^2 = 72 \Rightarrow r = \sqrt{72} \approx 8.49$
4. Evaluating this shows:
   • First derivative changes sign indicating a minimum OC.

Finally, substituting r into the cost equation:

Thus, the minimum cost occurs at r = 4.5.