Higher January 2020 Paper 2 Q26

26 Here is a sector, \(AOB\), of a circle with centre \(O\) and angle \(AOB = x^\circ\)

Diagram NOT accurately drawn

The sector can form the curved surface of a cone by joining \(OA\) to \(OB\).

Diagram NOT accurately drawn

The height of the cone is 25 cm.
The volume of the cone is 1600 cm³

Work out the value of \(x\).
Give your answer correct to the nearest whole number.

(6)

Mark scheme

Scheme	Marks
\(1600 = \dfrac{1}{3} \times \pi \times r^2 \times 25\) oe	M1
eg \(r = \sqrt{\dfrac{1600}{\frac{1}{3}\pi \times 25}}\) or \(r = \sqrt{\dfrac{192}{\pi}}\;\left(= \sqrt{61.1(154..)} = 7.8176...\right)\)	M1
\(l = \sqrt{\text{“}7.817...\text{”}^2 + 25^2}\;\left(= \sqrt{686.1154...} = 26.193...\right)\)	M1
2 × π × “7.817…” (= 49.1196…) or π × “7.817...” × “26.193...” (= 643.315...)	M1
\(\text{“}49.1196...\text{”} = 2 \times \pi \times \text{“}26.193...\text{”} \times \dfrac{x}{360}\) or \(\text{“}643.315...\text{”} = \pi \times \text{“}26.193...\text{”}^2 \times \dfrac{x}{360}\)	M1
107°	A1
	(6)
	(6 marks)

M1: for substituting into volume formula for cone correctly and equating to 1600

M1: dep for correct rearrangement of volume formula for \(r\)

M1: Dep on M2 correct method to find slant height of cone (radius of sector)

M1: for using \(C = 2\pi r\) oe using figures from correct method
or
for using \(A = \pi rl\) using figures from correct method

M1: for using arc length = \(2\pi r \times \dfrac{x}{360}\)

for using area of sector = \(\pi r^2 \times \dfrac{x}{360}\)

A1: for 107° – 108°