A2 June 2023 Q7

7. The random variable \(R\) has a continuous uniform distribution over the interval \([2, 10]\)

(a) Write down the probability density function \(\mathrm{f}(r)\) of \(R\) (1)

A sphere of radius \(R\) cm is formed.

The surface area of the sphere, \(S\) cm², is given by \(S = 4\pi R^2\)

(b) Show that \(\mathrm{E}(S) = \dfrac{496\pi}{3}\) (4)

The volume of the sphere, \(V\) cm³, is given by \(V = \dfrac{4}{3}\pi R^3\)

Mark scheme (a)

Scheme	Marks	AO
\(\mathrm{f}(r) = \dfrac{1}{8}\) for \(2 \leqslant r \leqslant 10\) [0 otherwise]	B1	1.1b
	(1)

B1: correct probability density function with limits
Allow any letter but must be consistent

Mark scheme (b)

Scheme	Marks	AO
\(\mathrm{E}(4\pi R^2) = 4\pi\mathrm{E}(R^2)\) or \(\displaystyle\int_2^{10} 4\pi r^2\left(\tfrac{1}{8}\right)\mathrm{d}r\)	M1	1.1b
\(\mathrm{E}(R) = 6\) or \(= \left[\tfrac{1}{2}\pi\left(\dfrac{r^3}{3}\right)\right]_2^{10}\)	M1	1.1b
\(\mathrm{E}(4\pi R^2) = 4\pi\left(\mathrm{Var}(R) + [\mathrm{E}(R)]^2\right) = 4\pi\left(\tfrac{16}{3} + \text{“}6\text{”}^2\right)\) or \(= \tfrac{1}{2}\pi\left(\dfrac{10^3}{3} - \dfrac{2^3}{3}\right)\)	M1	2.1
\(= \dfrac{496\pi}{3}\)*	A1*	2.2a
	(4)

M1: use of \(\mathrm{E}(aR) = a\mathrm{E}(R)\) / correct integral with their \(\mathrm{f}(r)\) (limits not needed)

M1: \(\mathrm{E}(R) = 6\) seen or implied / use of integration to find \(\mathrm{E}(SA)\)

M1: use of \(\mathrm{E}(R^2) = \mathrm{Var}(R) + [\mathrm{E}(R)]^2\) / correct integration of their \(\mathrm{f}(r)\), limits shown

A1*: exact value from correct working

Mark scheme (c)

Scheme	Marks	AO
\(\mathrm{E}\left(\tfrac{4}{3}\pi R^3\right) = \displaystyle\int_2^{10} \tfrac{4}{3}\pi r^3\left(\tfrac{1}{8}\right)\mathrm{d}r\)	B1ft	1.1b
\(= \left[\tfrac{1}{6}\pi\left(\dfrac{r^4}{4}\right)\right]_2^{10}\)	M1	2.1
\(= \tfrac{1}{6}\pi\left(\dfrac{10^4}{4} - \dfrac{2^4}{4}\right)\)	dM1	1.1b
\(= 416\pi\)	A1	2.2a
	(4)
	(9 marks)

follow through their pdf from (a) for first 3 marks of (c)

B1ft: correct integral for \(\mathrm{E}(V)\) including limits,

M1: integration of their pdf from (a) to find \(\mathrm{E}(V)\)

dM1: (dep on previous M1) use of correct limits, may be implied

A1: \(416\pi\) (allow awrt 1310 to 3 sf)