A2 June 2022 Q7
7. A rectangle is to have an area of 40 cm2
The length of the rectangle, \(L\) cm, follows a continuous uniform distribution over the interval \([4, 10]\)
Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals. (7)
| Scheme | Marks | AO |
|---|---|---|
| \(P = 2\left(L + \tfrac{40}{L}\right)\) | M1 | 3.1a |
| \(\mathrm{f}(l) = \tfrac{1}{6}\) | B1 | 1.1b |
| \(\mathrm{E}(P) = \mathrm{E}\left(2\left(L + \tfrac{40}{L}\right)\right) = \displaystyle\int_4^{10} \tfrac{1}{6}\left(\text{“}2l + \tfrac{80}{l}\text{”}\right)\mathrm{d}l\) | M1 | 2.1 |
| \(= \left[\tfrac{1}{6}l^2 + \tfrac{40}{3}\ln|l|\right]_4^{10}\) | M1A1 | 1.1b 1.1b |
| \(= \left(\tfrac{1}{6}(100) + \tfrac{40}{3}\ln 10\right) - \left(\tfrac{1}{6}(16) + \tfrac{40}{3}\ln 4\right)\) | M1 | 1.1b |
| \(= 26.217\ldots\) awrt 26.2 | A1 | 1.1b |
| (7 marks) |
Notes
M1: Finding an expression for the perimeter in terms of \(L\)
B1: Correct distribution for \(L\) (may be implied by \(\mathrm{E}(L) = 7\))
M1: Setting up integral for expectation of perimeter
(allow 2 separate integrals e.g. \(2(7) + \displaystyle\int_4^{10} \tfrac{1}{6}\left(\tfrac{80}{l}\right)\mathrm{d}l\))
M1: Attempt to integrate an expression for expectation of perimeter (allow two separate expressions)
A1: Correct integration
depM1: (dep on previous M1) Use of correct limits 10 and 4
A1: awrt 26.2 Note: exact value is \(14 + \dfrac{40\ln(2.5)}{3}\)