Question
[Maximum mark: 14]
Georgia is on vacation in Costa Rica. She is in a hot air balloon over a lush jungle in Muelle.
When she leans forward to see the treetops, she accidentally drops her purse. The purse falls down a distance of 4 metres during the first second, 12 metres during the next second, 20 metres during the third second and continues in this way. The distances that the purse falls during each second forms an arithmetic sequence.
(a) (i) Write down the common difference, $d$, of this arithmetic sequence.
(ii) Write down the distance the purse falls during the fourth second. [2]
(b) Calculate the distance the purse falls during the 13th second. [2]
(c) Calculate the total distance the purse falls in the first 13 seconds. [2]
Georgia drops the purse from a height of 1250 metres above the ground.
(d) Calculate the time, to the nearest second, the purse will take to reach the ground. [3]
Georgia visits a national park in Muelle. It is opened at the start of 2019 and in the first year there were 20000 visitors. The number of people who visit the national park is expected to increase by $8 \%$ each year.
(e) Calculate the number of people expected to visit the national park in 2020. [2]
(f) Calculate the total number of people expected to visit the national park by the end of 2028 . [3]
Answer/Explanation
(a) (i) We have $u_1=4$ and $u_2=12$.
Hence the common difference is
$
\begin{aligned}
d & =u_2-u_1 \\
& =12-4 \\
& =8
\end{aligned}
$
(ii) Using the $n$th term formula $u_n=u_1+(n-1) d$ with $n=4$, we get
$
\begin{aligned}
u_4 & =u_1+(4-1) d \\
& =4+(4-1)(8) \\
& =28 \mathrm{~m}
\end{aligned}
$
(b) Using the $n$th term formula $u_n=u_1+(n-1) d$ with $n=13$, we obtain
$
\begin{aligned}
u_{13} & =u_1+(13-1) d \\
& =4+(13-1)(8) \\
& =100 \mathrm{~m}
\end{aligned}
$
(c) Using the sum of $n$ terms formula $S_n=\frac{n}{2}\left(2 u_1+(n-1) d\right)$ with $n=13$, we get
$
\begin{aligned}
S_{13} & =\frac{13}{2}\left(2 u_1+(13-1) d\right) \\
& =\frac{13}{2}(2(4)+(13-1)(8)) \\
& =676 \mathrm{~m}
\end{aligned}
$
Question
[Maximum mark: 14]
Tesco PLC, a British multinational grocery retailer, offers online food shopping and delivery services 7 days a week. The number of deliveries to an incorrect home address in a day from Tesco Superstore A, denoted by $X$, follows a Poisson distribution with mean of 7 .
(a) Find the probability that Superstore A delivers to
(i) exactly 7 incorrect addresses in a particular day;
(ii) less than 7 incorrect addresses in a particular day;
(iii) more than 45 incorrect addresses in a particular week. [6]
Another Tesco Superstore, B, also makes incorrect home deliveries in a day. Let $Y$ represent the number of incorrect home deliveries in a day of Superstore B and assume that $Y$ follows a Poisson distribution of mean of 5 and that these two Superstores operate independently.
(b) Find the probability that the combined number of incorrect deliveries from the two Superstores in a day is 10 or less. [3]
A new random variable is defined by $Z=2 X+4 Y$.
(c) For the random variable $Z$, find
(i) the mean;
(ii) the variance. [3]
(d) Determine whether $Z$ could satisfy a Poisson distribution. Justify your answer. [2]
Answer/Explanation
(a) (i) Since $X \sim \operatorname{Po}(7)$, we have
$
\begin{aligned}
\operatorname{Pr}(\mathrm{X}=7) & =\operatorname{poissonPdf}(7,7) \\
& \approx 0.149 \quad \text { [by using G.D.C.] }
\end{aligned}
$
(ii) Since $X \sim \operatorname{Po}(7)$, we have
$
\begin{aligned}
\operatorname{Pr}(\mathrm{X}<7) & =\operatorname{Pr}(\mathrm{X} \leq 6) \\
& =\operatorname{poissonCdf}(7,0,6) \\
& \approx 0.45 \quad[\text { by using G.D.C.] }
\end{aligned}
$
(iii) Let $W$ be the number of incorrect deliveries in a week. Since we expect there to be 7 incorrect deliveries in a day, we expect there to be $\lambda=7 \times 7=49$ incorrect deliveries in a week.
Hence $W \sim \operatorname{Po}(49)$, and then
$
\begin{aligned}
\operatorname{Pr}(\mathrm{W}>45) & =\operatorname{Pr}(\mathrm{W} \geq 46) \\
& =\operatorname{poissonCdf}(49,46,1000) \\
& \approx 0.685 \quad[\text { by using G.D.C.] }
\end{aligned}
$