No questions found
This investigation looks at the way a virus spreads in plants in a field.
In a field there are a large number of plants in a straight line.
The diagram shows the plants near the middle of the field.
..................
On Day 1, one of the plants is infected with a virus (V).
.........V........
On Day 2, that plant is dead (D) and the virus infects the plants next to it.
......VDV......
This continues from day to day so this is the pattern on Day 3.
.....VDDDV.....
The diagram shows that the virus infects two more plants on day 3.
So the total number of plants that are infected or dead is five.
(a) Draw the pattern for Day 4.
.................................
(b) Complete this table.
[Table_1]
\[ \begin{array}{|c|c|} \hline \text{Day (n)} & \text{Total number of plants that are infected or dead (t)} \\ \hline 1 & 1 \\ 2 & 3 \\ 3 & 5 \\ 4 & \\ 5 & \\ \hline \end{array} \]
(c) What is the name of the numbers in the \( t \) column in \textit{part (b)}?
..........................................
(d) Find the total number of plants that are infected or dead on Day 9.
..........................................
(e) Find a formula for \( t \) in terms of \( n \).
..........................................
(f) On which day are there a total of 97 plants that are infected or dead?
..........................................
514
589
510
In another field there are a large number of plants in equally spaced rows and columns. The diagram shows the plants near the middle of the field. One of the plants is infected with the virus (V).
On Day 2, that plant is dead (D) and the virus infects the plants next to it. These plants form a cross.
On Day 3, the virus spreads along the arms of the cross.
This continues from day to day.
(a) Draw the pattern for Day 4.
(b) Complete this table.
[Table_1]
(c) Find a formula for $t$ in terms of $n$.
597
560
553
In another field, one of the plants is infected with a different virus (Z). This virus affects all the plants next to it.
On Day 2 that plant is dead (D) and the plants next to it are infected.
On Day 3 there are 5 dead plants and 8 infected plants.
(a) Draw the pattern for Day 4.
(b) Complete this table to show the number of infected plants each day.
[Table_1]
You may use this grid to help you.
(c) Work out a formula for the number of infected plants \( p \) in terms of the day \( n \) for \( n \geq 2 \).
..................................................
(d) Complete this table to show the total number of infected or dead plants each day.
[Table_2]
(e) The formula for \( t \) in terms of \( n \) is \( t = 2n^2 + bn + c \).
Find the value of \( b \) and the value of \( c \).
\( b = \) ..................................................
\( c = \) ..................................................
(f) Show that your formula works when \( n = 6 \).
558
557
567
