Use Sam’s measurements to calculate the area of the sector. Give your answer to four significant figures.

Find the upper bound and lower bound of the area of the sector.

Find, with justification, the largest possible percentage error if the answer to part (a) is recorded as the area of the sector.

$\mathrm{\pi }\times 2^2\times \frac{30}{360}$           (M1)

$=1.047 {\text{cm}}^2$           A1

Note: Do not award the final mark if the answer is not correct to 4 sf.

[2 marks]

attempt to substitute any two values from $1.5, 2.5, 25$ or $35$ into area of sector formula           (M1)

$\left(\text{upper bound}=\mathrm{\pi }\times 2.5^2\times \frac{35}{360}=\right) 1.91 {\text{cm}}^2 \left(1.90895\dots \right)$           A1

$\left(\text{lower bound}=\mathrm{\pi }\times 1.5^2\times \frac{25}{360}=\right) 0.491 {\text{cm}}^2 \left(0.490873\dots \right)$           A1

Note: Given the nature of the question, accept correctly rounded OR correctly truncated $3$ significant figure answers.

[3 marks]

$\left(\left|\frac{1.047-1.90895\dots }{1.90895\dots }\right|\times 100=\right) 45.2 \left(\%\right) \left(45.1532\dots \right)$           A1

$\left(\left|\frac{1.047-0.490873\dots }{0.490873\dots }\right|\times 100=\right) 113 \left(\%\right) \left(113.293\dots \right)$           A1

so the largest percentage error is $113 \%$           A1

Note: Accept $45.1 (\%)$ ($45.1428$), from use of full accuracy answers. Given the nature of the question, accept correctly rounded OR correctly truncated $3$ significant figure answers. Award A0A1A0 if $113\%$ is the only value found.

[3 marks]

In part (a), the area was almost always found correctly although some candidates gave the answer 1.0472 which is correct to 4 decimal places, not 4 significant figures as required. In part (b), many candidates failed to realize that the upper bounds for r and θ were 2.5 and 35° and lower bounds were 1.5 and 25°. Consequently, the bounds for the area were incorrect. In many cases, the incorrect values in part (b) were followed through into part (c) although in the percentage error calculations, many candidates had 1.047 in the denominator instead of the appropriate bound.