Question
[Maximum mark: 15]
Premier Pet-foods Co. has developed a new dry food for dogs. They start to mass produce the food and the amount of inventory in the warehouse is recorded on a monthly basis. The table below shows the data for the amount of inventory ( $I$, in tonnes) for the first ten months $(t)$
(a) Sketch a scatter plot to represent the given data on the axes below. [2]
(b) (i) State a suitable type of function that could model this set of data points.
(ii) Hence determine the model function for this dataset. [2]
(c) Comment on the choice of model using the coefficient of determination. [2]
(d) Sketch the model function on the scatter plot and comment on how it fits to the data. [3]
(e) Using your model, determine after how many months the inventory is at a maximum. [2]
(f) Using your model, forecast when the the company will run out of inventory. [2]
(g) Using your model, determine the amount of inventory after 16 months and comment on if the model is still valid in forecasting inventory levels at this time. [2]
Answer/Explanation
(a)
(b) (i) $\text{The scatter plot shows that the data can be modelled by a quadratic function.}$
(ii) Using GDC, we obtain the model function for this dataset as follows.
$
I(t)=-0.634 t^2+7.51 t-0.896
$
Question
[Maximum mark: 19]
On Wednesday Eddy goes to a velodrome to train. He cycles the first lap of the track in 25 seconds. Each lap Eddy cycles takes him 1.6 seconds longer than the previous lap.
(a) Find the time, in seconds, Eddy takes to cycle his tenth lap. [3]
Eddy cycles his last lap in $55.4$ seconds.
(b) Find how many laps he has cycled on Wednesday. [3]
(c) Find the total time, in minutes, cycled by Eddy on Wednesday. [4]
On Friday Eddy brings his friend Mario to train. They both cycled the first lap of the track in 25 seconds. Each lap Mario cycles takes him $1.05$ times as long as his previous lap.
(d) Find the time, in seconds, Mario takes to cycle his fifth lap. [3]
(e) Find the total time, in minutes, Mario takes to cycle his first ten laps.
Each lap Eddy cycles again takes him $1.6$ seconds longer that his previous lap. After a certain number of laps Eddy takes less time per lap than Mario.
(f) Find the number of the lap when this happens. [3]
Answer/Explanation
(a) We have an arithmetic sequence with the first term $u_1=25$ and the common difference $d=1.6$.
Hence, using the $n$th term formula $u_n=u_1+(n-1) d$ with $n=10$, we get
$
\begin{aligned}
u_{10} & =u_1+(10-1) d \\
& =25+(10-1)(1.6) \\
& =39.4 \text { seconds }
\end{aligned}
$
(b) Solving the equation $u_n=55.4$ for $n$, we obtain
$
\begin{aligned}
u_1+(n-1) d & =55.4 \\
25+(n-1)(1.6) & =55.4 \\
n & =20 \text { laps }\quad\text{[by using G.D.C.]}
\end{aligned}
$
(c) Using the sum of $n$ terms formula $S_n=\frac{n}{2}\left(2 u_1+(n-1) d\right)$ with $n=20$, we get
$
\begin{aligned}
S_{20} & =\frac{20}{2}\left(2 u_1+(20-1) d\right) \\
& =\frac{20}{2}(2(25)+(20-1)(1.6)) \\
& =804 \\
& =13.4 \text { minutes }
\end{aligned}
$
(d) We have a geometric sequence with the first term $v_1=25$ and the common ratio $r=1.05$.
Hence, using the $n$th term formula $v_n=v_1 r^{n-1}$ with $n=5$, we obtain
$
\begin{aligned}
v_5 & =v_1 r^{5-1} \\
& =25(1.05)^{5-1} \\
& \approx 30.4 \text { seconds }
\end{aligned}
$